Integral inequalities using signed measures corresponding to majorization

Abstract

In this paper we derive majorization type integral inequalities using measure spaces with signed measures. We obtain necessary and sufficient conditions for the studied integral inequalities to be satisfied. To apply our results, we first generalize Hardy–Littlewood–Pólya and Fuchs inequalities. Then we deal with the nonnegativity of some integrals with nonnegative convex functions. As a consequence, the known characterization of Steffensen–Popoviciu measures on compact intervals is extended to arbitrary intervals. Finally, we give necessary and sufficient conditions for the satisfaction of the integral Jensen inequality and the integral Lah–Ribarič inequality for signed measures. All the considered problems are also studied for special classes of convex functions. To prove the main assertions some approximation results for nonnegative convex functions are also developed.

1 Introduction

The essence of the famous Hardy–Littlewood–Pólya inequality is the next: it contains a characterization of convex functions using the basic concept of majorization. Majorization is a binary relation (preorder) for finite sequences of real numbers, and the theory of majorization is a significant topic in mathematics (see Marshall and Olkin [11]).

Definition 1

Let $\textbf{s}:=\left( s_{1},\ldots ,s_{n}\right) \in \mathbb {R}^{n}$ and $\textbf{t}:=\left( t_{1},\ldots ,t_{n}\right) \in \mathbb {R}^{n}$.

• (a) We say that $\textbf{s}$ is weakly majorized by $\textbf{t}$, written $\textbf{s}\prec _{w}\textbf{t}$, if

  $$\begin{aligned} \sum \limits _{i=1}^{k}s_{\left[ i\right] }\le \sum \limits _{i=1}^{k}t_{\left[ i\right] },\quad k=1,\ldots ,n, \end{aligned}$$

  (1)

  where $s_{\left[ 1\right] }\ge s_{\left[ 2\right] }\ge \cdots \ge s_{\left[ n\right] }$ and $t_{\left[ 1\right] }\ge t_{\left[ 2\right] }\ge \cdots \ge t_{\left[ n\right] }$ are the entries of $\textbf{s}$ and $\textbf{t}$, respectively, in decreasing order.

• (b) We say that $\textbf{s}$ is majorized by $\textbf{t}$, written $\textbf{s}\prec \textbf{t}$, if (1) holds, and in addition

  $$\begin{aligned} \sum \limits _{i=1}^{n}s_{\left[ i\right] }=\sum \limits _{i=1}^{n}t_{\left[ i\right] }. \end{aligned}$$

Theorem 1

Let $C\subset \mathbb {R}$ be an interval, and let $\textbf{s}:=\left( s_{1},\ldots ,s_{n}\right) \in C^{n}$ and $\textbf{t}:=\left( t_{1},\ldots ,t_{n}\right) \in C^{n}$.

• (a) Hardy–Littlewood-Pólya inequality (see Hardy, Littlewood and Pólya [6] and Niculescu and Persson [14]) If $\textbf{s}\prec \textbf{t}$, then

  $$\begin{aligned} \sum \limits _{i=1}^{n}f\left( s_{i}\right) \le \sum \limits _{i=1}^{n}f\left( t_{i}\right) \end{aligned}$$

  (2)

  for every convex function $f:C\rightarrow \mathbb {R}$. Conversely, if inequality (2) holds for every convex function $f:C\rightarrow \mathbb {R}$, then $\textbf{s}\prec \textbf{t}$.

• (b) (see Niculescu and Persson [14]) If $f:C\rightarrow \mathbb {R}$ is an increasing and convex function and $\textbf{s}\prec _{w}\textbf{t}$, then (2) also holds.

Let $\left( X,\mathcal {A}\right)$ be a measurable space. The unit mass at $x\in X$ (the Dirac measure at x) is denoted by $\varepsilon _{x}$. The set of all subsets of X is denoted by $P\left( X\right)$.

By $\mathbb {N}_{+}$ we denote the set of positive integers.

The previous result can be reformulated by using discrete measures. Let $\textbf{s}:=\left( s_{1},\ldots ,s_{n}\right) \in C^{n}$ and $\textbf{t}:=\left( t_{1},\ldots ,t_{n}\right) \in C^{n}$, and consider the discrete measures

$$\begin{aligned} \mu :=\sum \limits _{i=1}^{n}\varepsilon _{s_{i}}\quad \text {and}\quad \nu :=\sum \limits _{i=1}^{n}\varepsilon _{t_{i}} \end{aligned}$$

on $P\left( C\right)$. Let $\mu \prec \nu$ $\left( \mu \prec _{w}\nu \right)$ mean the same as $\textbf{s}\prec \textbf{t}$ $\left( \textbf{s}\prec _{w}\textbf{t}\right)$. In these notations Theorem 1 can be written in the form

$$\begin{aligned} \mu \prec \nu \text { if and only if }\int \limits _{C}fd\mu \le \int \limits _{C}fd\nu \end{aligned}$$

for every convex function $f:C\rightarrow \mathbb {R}$, and

$$\begin{aligned} \mu \prec _{w}\nu \text { implies }\int \limits _{C}fd\mu \le \int \limits _{C}fd\nu \end{aligned}$$

for every increasing and convex function $f:C\rightarrow \mathbb {R}$.

Among the weighted versions of Hardy–Littlewood–Pólya inequality, we highlight the following inequality by Fuchs [5].

Theorem 2

Let $C\subset \mathbb {R}$ be an interval, and let $f:C\rightarrow \mathbb {R}$ be a convex function. If $\left( s_{1},\ldots ,s_{n}\right) \in C^{n}$, $\left( t_{1},\ldots ,t_{n}\right) \in C^{n}$ and $q_{1},\ldots ,q_{n}$ are real numbers such that

• (a) $s_{1}\ge \cdots \ge s_{n}$ and $t_{1}\ge \cdots \ge t_{n}$,

• (b) $\sum \nolimits _{i=1}^{k}q_{i}s_{i}\le \sum \nolimits _{i=1}^{k}q_{i}t_{i}$ $\left( k=1,\ldots ,n-1\right)$,

• (c) $\sum \nolimits _{i=1}^{n}q_{i}s_{i}=\sum \nolimits _{i=1}^{n}q_{i}t_{i}$,

then

$$\begin{aligned} \sum \limits _{i=1}^{n}q_{i}f\left( s_{i}\right) \le \sum \limits _{i=1} ^{n}q_{i}f\left( t_{i}\right) . \end{aligned}$$

(3)

Of course, as with the Hardy–Littlewood–Pólya inequality, (3) can be rewritten in the equivalent form

$$\begin{aligned} \int \limits _{C}fd\mu \le \int \limits _{C}fd\nu , \end{aligned}$$

where $\mu$ and $\nu$ are discrete signed measures on $P\left( C\right)$.

The above two inequalities and their applicability alone justify the investigation of the following problem: Let $\left( X,\mathcal {A},\mu \right)$ and $\left( Y,\mathcal {B},\nu \right)$ be measure spaces, where $\mu$ and $\nu$ are finite signed measures. Let $C\subset \mathbb {R}$ be an interval with nonempty interior, and let $\varphi :X\rightarrow C$, $\psi :Y\rightarrow C$ be $\mu$-integrable and $\nu$-integrable functions, respectively. Under which conditions does inequality

$$\begin{aligned} \int \limits _{X}f\circ \varphi d\mu \le \int \limits _{Y}f\circ \psi d\nu , \end{aligned}$$

hold for any convex function $f:C\rightarrow \mathbb {R}$ for which $f\circ \varphi$ is $\mu$-integrable and $f\circ \psi$ is $\nu$-integrable? Especially, the same problem can be asked if we consider only increasing, decreasing, or nonnegative convex functions.

As we have seen, the problem is closely related to the theory of majorization and it has been studied by many authors and solved in many specific cases. If $\mu$ and $\nu$ are measures and $C=\mathbb {R}^{n}$, Moein, Pereira and Plosker [13] give a complete analysis of the problem (see also Chong [2] and Dahl [3]). Another case, studied in detail, where the functions $\varphi$ and $\psi$ are defined on a compact interval $X=Y=\left[ a,b\right]$ (see for example, Barnett, Cerone and Dragomir [1], Maligranda, Pečarić and Persson [10] and Ruch, Schranner and Seligman [17]). The theory of majorization can be extended using Steffensen–Popoviciu measures, and similar results are also found in this topic (see Niculescu and Persson [14]). In Horváth [7] we give a comprehensive and uniform treatment of the problem to give conditions for the inequality

$$\begin{aligned} \int \limits _{\left[ a,b\right] }f\circ \varphi d\mu \le \int \limits _{\left[ a,b\right] }f\circ \psi d\nu , \end{aligned}$$

to be valid, where $\mu$ and $\nu$ are finite signed measures on a $\sigma$-algebra containing the Borel sets of $\left[ a,b\right]$.

The aim of this paper is to provide a precise solution to the problem posed above (necessary and sufficient conditions), and then to illustrate the strength of the results by applying them. Our main results significantly extend previous results in this direction and provide a new approach. In applications, we first generalize Hardy–Littlewood–Pólya and Fuchs inequalities, giving necessary and sufficient conditions for inequalities of the form

$$\begin{aligned} \sum \limits _{i\in X}f\left( s_{i}\right) p_{i}\le \sum \limits _{j\in Y}f\left( t_{j}\right) q_{j} \end{aligned}$$

to be satisfied for any convex function $f:C\rightarrow \mathbb {R}$, where X is either $\left\{ 1,\ldots ,m\right\}$ for some $m\ge 1$ or $\mathbb {N} _{+}$, Y is either $\left\{ 1,\ldots ,n\right\}$ for some $n\ge 1$ or $\mathbb {N}_{+}$, $\left( s_{i}\right) _{i\in X}$ and $\left( t_{j}\right) _{j\in Y}$ are sequences from C and $\left( p_{i}\right) _{i\in X}$ and $\left( q_{j}\right) _{j\in Y}$ are real sequences. Easy to check necessary and sufficient conditions are obtained when X and Y are both finite sets. Next, we deal with the nonnegativity of the integral

$$\begin{aligned} \int \limits _{C}fd\mu \end{aligned}$$

for every nonnegative convex function f on the real interval C. As a consequence, the known characterization of Steffensen–Popoviciu measures on compact intervals is extended to arbitrary intervals in $\mathbb {R}$. Our last application consists of two parts. We give necessary and sufficient conditions for the satisfaction of inequalities

$$\begin{aligned} f\left( \frac{1}{\mu \left( X\right) }\int \limits _{X}\psi d\mu \right) \le \frac{1}{\mu \left( X\right) }\int \limits _{X}f\circ \psi d\mu \end{aligned}$$

(4)

and

$$\begin{aligned} \frac{1}{\mu \left( X\right) }\int \limits _{X}f\circ \psi d\mu \le \frac{b-t_{\psi ,\mu }}{b-a}f\left( a\right) +\frac{t_{\psi ,\mu }-a}{b-a}f\left( b\right) , \end{aligned}$$

(5)

where

$$\begin{aligned} t_{\psi ,\mu }:=\frac{1}{\mu \left( X\right) }\int \limits _{X}\psi d\mu \end{aligned}$$

for finite signed measures. For finite measures inequality (4) is the classical integral Jensen inequality, while inequality (5) is the well-known integral Lah–Ribarič inequality (see Lah and Ribarič [9]). Looking at the integral Jensen inequality, similar results are found for real Stieltjes measures: Krnić, Pečarić and Rodić [8] and Pečarić, Perić and Lipanović [15] use Green functions, while Mihai and Niculescu [12] uses Steffensen–Popoviciu measures. A result for the integral Lah–Ribarič inequality in this direction is found in Florea and Niculescu [4], where $X=\left[ a,b\right]$ and $\psi$ is the identity function on $\left[ a,b\right]$.

We also consider all the previous questions for special classes of convex functions.

To prove the main assertions some approximation results for nonnegative convex functions are also developed.

2 Preliminary results

2.1 Inequalities associated to majorization

Positive and negative parts of a real number x are denoted by $x^{+}$ and $x^{-}$, respectively.

We now introduce some usual short-hand notations: for a real function $f:D_{f}\rightarrow \mathbb {R}$

$$\begin{aligned} \left\{ f\ge w\right\} :=\left\{ x\in D_{f}\mid f\left( x\right) \ge w\right\} ,\quad w\in \mathbb {R}, \end{aligned}$$

and the sets $\left\{ f<w\right\}$, $\left\{ f=w\right\}$, etc. are defined analogously.

The characteristic function of $A\subset X$ is denoted by $\chi _{A}$.

Let $\left( X,\mathcal {A}\right)$ be a measurable space ($\mathcal {A}$ always means a $\sigma$-algebra of subsets of X). If $\mu$ is either a measure or a signed measure on $\mathcal {A}$, then the real vector space of $\mu$-integrable real functions on X is denoted by $L\left( \mu \right)$. The integrable functions are considered to be measurable.

Let $C\subset \mathbb {R}$ be an interval with nonempty interior. The following notations are introduced for some special functions defined on C:

$$\begin{aligned} id_{C}\left( t\right) :=t,\quad p_{C,w}\left( t\right) :=\left( t-w\right) ^{+},\quad n_{C,w}\left( t\right) :=\left( t-w\right) ^{-}\quad t,w\in C. \end{aligned}$$

We begin with a simple but essential statement.

Lemma 1

Let $\left( X,\mathcal {A},\mu \right)$ and $\left( Y,\mathcal {B},\nu \right)$ be measure spaces, where $\mu$ and $\nu$ are finite signed measures with $\mu \left( X\right) =\nu \left( Y\right)$. Assume $\varphi \in L\left( \mu \right)$ and $\psi \in L\left( \nu \right)$ such that $\int \nolimits _{X}\varphi d\mu =\int \nolimits _{Y}\psi d\nu$. Then for every $w\in \mathbb {R}$ the following two assertions are equivalent.

• (a)
  $$\begin{aligned} \int \limits _{X}p_{\mathbb {R},w}\circ \varphi d\mu \le \int \limits _{Y} p_{\mathbb {R},w}\circ \psi d\nu . \end{aligned}$$

  (6)
• (b)
  $$\begin{aligned} \int \limits _{X}n_{\mathbb {R},w}\circ \varphi d\mu \le \int \limits _{Y} n_{\mathbb {R},w}\circ \psi d\nu . \end{aligned}$$

Proof

We only prove that (b) follows from (a), the converse statement can be handled similarly.

It is easy to check that

$$\begin{aligned} \int \limits _{X}n_{\mathbb {R},w}\circ \varphi d\mu= & {} \int \limits _{\left\{ \varphi <w\right\} }\left( w-\varphi \right) d\mu =\int \limits _{X}\left( w-\varphi \right) d\mu -\int \limits _{\left\{ \varphi \ge w\right\} }\left( w-\varphi \right) d\mu \\= & {} \int \limits _{X}\left( w-\varphi \right) d\mu +\int \limits _{X}p_{\mathbb {R},w}\circ \varphi d\mu . \end{aligned}$$

Thus the conditions $\mu \left( X\right) =\nu \left( X\right)$, $\int \nolimits _{X}\varphi d\mu =\int \nolimits _{X}\psi d\nu$ and (6) imply that

$$\begin{aligned} \int \limits _{X}n_{\mathbb {R},w}\circ \varphi d\mu= & {} \int \limits _{Y}\left( w-\psi \right) d\nu +\int \limits _{X}p_{\mathbb {R},w}\circ \varphi d\mu \\\le & {} \int \limits _{Y}\left( w-\psi \right) d\nu +\int \limits _{Y}p_{\mathbb {R},w}\circ \psi d\nu \\= & {} \int \limits _{Y}\left( w-\psi \right) d\nu -\int \limits _{\left\{ \psi \ge w\right\} }\left( w-\psi \right) d\nu =\int \limits _{Y}n_{\mathbb {R},w} \circ \psi d\nu . \end{aligned}$$

The proof is complete.$\square$

The next result contains integral inequalities for some special functions.

Lemma 2

Let $\left( X,\mathcal {A}\right)$ be a measurable space, and let $\mu$ and $\nu$ be finite measures on $\mathcal {A}$. Assume $\varphi \in L\left( \mu \right)$ and $\psi \in L\left( \nu \right)$ such that

$$\begin{aligned} \int \limits _{\left\{ \varphi \ge w\right\} }\varphi d\mu \le \int \limits _{\left\{ \varphi \ge w\right\} }\psi d\nu ,\quad w\in \mathbb {R}. \end{aligned}$$

(7)

• (a) If the function f is either $id_{\mathbb {R}}$ or $p_{\mathbb {R},w}$ for some $w\in \mathbb {R}$, then

  $$\begin{aligned} \int \limits _{X}f\circ \varphi d\mu \le \int \limits _{X}f\circ \psi d\nu . \end{aligned}$$

  (8)

• (b) Assume

  $$\begin{aligned} \mu \left( X\right) =\nu \left( X\right) \end{aligned}$$

  (9)

  and

  $$\begin{aligned} \int \limits _{X}\varphi d\mu =\int \limits _{X}\psi d\nu \end{aligned}$$

  (10)

  are also satisfied. If $f=n_{\mathbb {R},w}$ for some $w\in \mathbb {R}$, then inequality (8) holds too.

Proof

• (a) Let the functions $\varphi _{n}$, $\psi _{n}:X\rightarrow \mathbb {R}$ be defined by

  $$\begin{aligned} \varphi _{n}:=\varphi \cdot \chi _{\left\{ \varphi \ge -n\right\} },\quad \psi _{n}:=\psi \cdot \chi _{\left\{ \varphi \ge -n\right\} },\quad n\in \mathbb {N}^{+}. \end{aligned}$$

  Then $\left( \varphi _{n}\right) _{n\in \mathbb {N}^{+}}$ is a decreasing sequence of $\mu$-integrable functions on X such that $\varphi _{n}\rightarrow \varphi$ pointwise on X. By the monotone convergence theorem,

  $$\begin{aligned} \int \limits _{\left\{ \varphi \ge -n\right\} }\varphi d\mu =\int \limits _{X} \varphi _{n}d\mu \rightarrow \int \limits _{X}\varphi d\mu . \end{aligned}$$

  (11)

  Since

  $$\begin{aligned} \bigcup \limits _{n\in \mathbb {N}^{+}}\left\{ \varphi \ge -n\right\} =X, \end{aligned}$$

  $\psi _{n}\rightarrow \psi$ pointwise on X. It follows from the definition of $\psi _{n}$ that

  $$\begin{aligned} \left| \psi _{n}\right| \le \left| \psi \right| ,\quad n\in \mathbb {N}^{+}. \end{aligned}$$

  Based on our previous two findings, the dominated convergence theorem implies that

  $$\begin{aligned} \int \limits _{\left\{ \varphi \ge -n\right\} }\psi d\nu =\int \limits _{X} \psi _{n}d\nu \rightarrow \int \limits _{X}\psi d\nu . \end{aligned}$$

  (12)

  By applying (11), (12) and (7), we obtain that

  $$\begin{aligned} \int \limits _{X}\varphi d\mu \le \int \limits _{X}\psi d\nu , \end{aligned}$$

  and therefore the result holds for $f=id_{\mathbb {R}}$. Now assume that $f=p_{\mathbb {R},w}$ for some $w\in \mathbb {R}$. In this case

  $$\begin{aligned} \int \limits _{X}f\circ \varphi d\mu =\int \limits _{\left\{ \varphi \ge w\right\} }\left( \varphi -w\right) d\mu \quad \text {and}\quad \int \limits _{X}f\circ \psi d\nu =\int \limits _{\left\{ \psi \ge w\right\} }\left( \psi -w\right) d\nu . \end{aligned}$$

  (13)

  If $\left\{ \varphi \ge w\right\} =\emptyset$, inequality (8) trivially follows from (13), and thus it can be supposed that $\left\{ \varphi \ge w\right\} \ne \emptyset$. Then by the first part of (13) and (7),

  $$\begin{aligned}{} & {} \int \limits _{X}f\circ \varphi d\mu \le \int \limits _{\left\{ \varphi \ge w\right\} }\left( \psi -w\right) d\nu \\{} & {} \quad =\int \limits _{\left\{ \varphi \ge w\right\} \cap \left\{ \psi \ge w\right\} }\left( \psi -w\right) d\nu +\int \limits _{\left\{ \varphi \ge w\right\} \cap \left\{ \psi <w\right\} }\left( \psi -w\right) d\nu , \end{aligned}$$

  and therefore it follows from the definition of the set $\left\{ \psi \ge w\right\}$ and from the second part of (13) that

  $$\begin{aligned} \int \limits _{X}f\circ \varphi d\mu \le \int \limits _{\left\{ \varphi \ge w\right\} \cap \left\{ \psi \ge w\right\} }\left( \psi -w\right) d\nu \le \int \limits _{\left\{ \psi \ge w\right\} }\left( \psi -w\right) d\nu =\int \limits _{X}f\circ \psi d\nu . \end{aligned}$$

• (b) Lemma 1 can be applied. The proof is complete. $\square$

The following result is a natural complement to the previous statement.

Lemma 3

Let $\left( X,\mathcal {A}\right)$ be a measurable space, and let $\mu$ and $\nu$ be finite measures on $\mathcal {A}$. Assume $\varphi \in L\left( \mu \right)$ and $\psi \in L\left( \nu \right)$ such that

$$\begin{aligned} \int \limits _{\left\{ \psi \le w\right\} }\varphi d\mu \le \int \limits _{\left\{ \psi \le w\right\} }\psi d\nu ,\quad w\in \mathbb {R}. \end{aligned}$$

(14)

• (a) If the function f is either $-id_{\mathbb {R}}$ or $n_{\mathbb {R},w}$ for some $w\in \mathbb {R}$, then

  $$\begin{aligned} \int \limits _{X}f\circ \varphi d\mu \ge \int \limits _{X}f\circ \psi d\nu . \end{aligned}$$

  (15)

• (b) Assume (9) and (10) are also satisfied. If $f=p_{C,w}$ for some $w\in \mathbb {R}$, then inequality (15) holds too.

Proof

• (a) Under the conditions $-\psi \in L\left( \nu \right)$, $-\varphi \in L\left( \mu \right)$, and

  $$\begin{aligned} \int \limits _{\left\{ -\psi \ge -w\right\} }-\psi d\nu \le \int \limits _{\left\{ -\psi \ge -w\right\} }-\varphi d\mu ,\quad w\in \mathbb {R}, \end{aligned}$$

  and therefore Lemma 2 (a) implies that

  $$\begin{aligned} \int \limits _{X}f\circ \left( -\psi \right) d\nu \le \int \limits _{X} f\circ \left( -\varphi \right) d\mu , \end{aligned}$$

  where f is either $id_{\mathbb {R}}$ or $p_{\mathbb {R},w}$ for some $w\in \mathbb {R}$. This gives the result by using that $\left( -w\right) ^{+}=w^{-}$.

• (b) By (10),

  $$\begin{aligned} \int \limits _{X}-\psi d\nu =\int \limits _{X}-\varphi d\mu . \end{aligned}$$

  Since $\left( -w\right) ^{-}=w^{+}$, Lemma 2 (b) can be applied. The proof is complete. $\square$

2.2 Approximation results

The interior of an interval $C\subset \mathbb {R}$ is denoted by $C^{\circ }$.

We need some approximation results from Horváth [7].

Definition 2

Let $C\subset \mathbb {R}$ be an interval with nonempty interior. A function $f:C\rightarrow \mathbb {R}$ is called piecewise linear if it is continuous and there exists finite points $t_{1}<t_{2}<\cdots <t_{k}$ in the interior of C such that the restriction of f to each interval $C\bigcap ] -\infty ,t_{1}]$, $\left[ t_{1},t_{2}\right]$, $\ldots$, $C\bigcap [ t_{k},\infty [$ is an affine function.

Theorem 3

Let $C\subset \mathbb {R}$ be an interval with nonempty interior, and let $f:C\rightarrow \mathbb {R}$ be a continuous convex function.

• (a) The function f is the pointwise limit of an increasing sequence of piecewise linear convex functions on C.

• (b) If f is increasing, then f is the pointwise limit of an increasing sequence of piecewise linear, increasing and convex functions on C.

• (c) If f is decreasing, then f is the pointwise limit of an increasing sequence of piecewise linear, decreasing and convex functions on C.

• (d) In all three cases the convergence is uniform on every compact subinterval of C.

Remark 1

Let $C\subset \mathbb {R}$ be an interval with nonempty interior, and let $f:C\rightarrow \mathbb {R}$ be a piecewise linear convex function. If C is compact, then it is well known (see Niculescu and Persson [14]) that f has a simple structure. The same is true for the functions described in Definition 2, the proof can be copied too. For the sake of completeness, and because we need the representations in the proofs, we give them.

• (a) The function f has the form

  $$\begin{aligned} f\left( t\right) =\alpha t+\beta +\sum \limits _{i=1}^{k}\gamma _{i}\left( \left( t-t_{i}\right) ^{+}+\left( t-t_{i}\right) ^{-}\right) ,\quad t\in C \end{aligned}$$

  for suitable points $t_{1}<t_{2}<\cdots <t_{k}$ in the interior of C, $\alpha$, $\beta \in \mathbb {R}$ and $\gamma _{i}>0$ $\left( i=1,\ldots ,k\right)$.

• (b) If f is increasing, then f is of the form

  $$\begin{aligned} f\left( t\right) =\alpha t+\beta +\sum \limits _{i=1}^{k}\gamma _{i}\left( t-t_{i}\right) ^{+},\quad t\in C \end{aligned}$$

  for suitable points $t_{1}<t_{2}<\cdots <t_{k}$ in the interior of C, $\alpha \ge 0$, $\beta \in \mathbb {R}$ and $\gamma _{i}>0$ $\left( i=1,\ldots ,k\right)$.

• (c) If f is decreasing, then f is of the form

  $$\begin{aligned} f\left( t\right) =\alpha t+\beta +\sum \limits _{i=1}^{n}\gamma _{i}\left( t-t_{i}\right) ^{-},\quad t\in C \end{aligned}$$

  for suitable points $t_{1}<t_{2}<\cdots <t_{k}$ in the interior of C, $\alpha \le 0$, $\beta \in \mathbb {R}$ and $\gamma _{i}>0$ $\left( i=1,\ldots ,k\right)$.

For nonnegative convex functions, we can formulate a more precise result than Theorem 3.

Theorem 4

Let $C\subset \mathbb {R}$ be an interval with nonempty interior, and let $f:C\rightarrow \mathbb {R}$ be a nonnegative, continuous and convex function.

• (a) Theorem 3 (a) is satisfied by a sequence whose elements have the form

  $$\begin{aligned} f\left( t\right) =\beta +\sum \limits _{i=1}^{k}\gamma _{i}\left( t-t_{i}\right) ^{+}+\sum \limits _{j=1}^{l}\delta _{j}\left( t-s_{j}\right) ^{-},\quad t\in C \end{aligned}$$

  for suitable points $t_{1}<t_{2}<\cdots <t_{k}$, $s_{1}<s_{2}<\cdots <s_{l}$ in the interior of C, $\beta \ge 0$ and $\gamma _{i}$, $\delta _{j}>0$ $\left( i=1,\ldots ,k,\text { }j=1,\ldots ,l\right)$,

• (b) If f is increasing, then Theorem 3 (b) is satisfied by a sequence whose elements have the form

  $$\begin{aligned} f\left( t\right) =\beta +\sum \limits _{i=1}^{k}\gamma _{i}\left( t-t_{i}\right) ^{+},\quad t\in C \end{aligned}$$

  for suitable points $t_{1}<t_{2}<\cdots <t_{k}$ in the interior of C, $\beta \ge 0$ and $\gamma _{i}>0$ $\left( i=1,\ldots ,k\right)$.

• (c) If f is decreasing, then Theorem 3 (c) is satisfied by a sequence whose elements have the form

  $$\begin{aligned} f\left( t\right) =\beta +\sum \limits _{i=1}^{k}\gamma _{i}\left( t-t_{i}\right) ^{-},\quad t\in C \end{aligned}$$

  for suitable points $t_{1}<t_{2}<\cdots <t_{k}$ in the interior of C, $\beta \ge 0$ and $\gamma _{i}>0$ $\left( i=1,\ldots ,k\right)$.

Proof

(b) Since f is increasing and nonnegative, the right-hand limit p of f exists at the left-hand endpoint $a\in [ -\infty ,\infty [$ of C and $p\ge 0$. It can be supposed that there exists $t_{1}\in C^{\circ }$ such that $f\left( t_{1}\right) >p$. Let $y=f_{-}^{\prime }\left( t_{1}\right) \left( t-t_{1}\right) +f\left( t_{1}\right)$ be the equation of the left-hand tangent line to the graph of f at $t_{1}$. Obviously, $f_{-}^{\prime }\left( t_{1}\right) >0$, and hence there is exactly one solution $t_{0}$ of the equation

$$\begin{aligned} f_{-}^{\prime }\left( t_{1}\right) \left( t-t_{1}\right) +f\left( t_{1}\right) =p. \end{aligned}$$

It is easy to think that either $a<t_{0}<t_{1}$ or $t_{0}=a$. In the first case

$$\begin{aligned} f_{-}^{\prime }\left( t_{1}\right) \left( t-t_{1}\right) +f\left( t_{1}\right) =p+f_{-}^{\prime }\left( t_{1}\right) \left( t-t_{0}\right) ^{+},\quad t\ge t_{0}, \end{aligned}$$

while in the second case

$$\begin{aligned} f\left( t\right) =f_{-}^{\prime }\left( t_{1}\right) \left( t-t_{1} \right) +f\left( t_{1}\right) ,\quad t\in C,\quad t\le t_{1}. \end{aligned}$$

A suitable sequence can now be obtained by simply modifying the sequence defined in Theorem 8 (a) in Horváth [7] to take the above into account.

(c) Apply (b) to the nonnegative, increasing, continuous and convex function $\widehat{f}:-C\rightarrow \mathbb {R}$, $\widehat{f}\left( t\right) :=f\left( -t\right)$.

(a) If f is not monotonic on C, then there exists a point $t_{0}\in C^{\circ }$ such that f is decreasing on $C\bigcap ] -\infty ,t_{0}]$ and increasing on $C\bigcap [ t_{0},\infty [$. Define the functions $f_{1}$, $f_{2}:C\rightarrow \mathbb {R}$ by

$$\begin{aligned} f_{1}\left( t\right) :=\left\{ \begin{array}{ll} 0,&{}\quad \text {if }t\in C\bigcap ] -\infty ,t_{0} ] \\ f\left( t\right) -f\left( t_{0}\right) ,&{}\quad \text {if }t\in C\bigcap ] t_{0},\infty [ \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} f_{2}\left( t\right) :=\left\{ \begin{array}{ll} f\left( t\right) -f\left( t_{0}\right) , &{}\quad \text {if }t\in C\bigcap ] -\infty ,t_{0} ] \\ 0,&{}\quad \text {if }t\in C\bigcap ] t_{0},\infty [ \end{array} \right. . \end{aligned}$$

Then $f_{1}$ and $f_{2}$ are nonnegative, continuous and convex function, $f_{1}$ is increasing, $f_{2}$ is decreasing and

$$\begin{aligned} f=f\left( t_{0}\right) +f_{1}+f_{2}, \end{aligned}$$

and therefore the result follows from (b) and (c).

The proof is complete. $\square$

2.3 Some recent results

The $\sigma$-algebra of Borel sets on an interval $C\subset \mathbb {R}$ is denoted by $\mathcal {B}_{C}$. The Lebesgue measure on $\mathcal {B}_{C}$ is denoted by $\lambda _{C}$.

To discuss our results and to make reading the paper more comfortable, we recall the following known result.

Theorem 5

(see Theorem 11 in Horváth [7]) Let $C\subset \mathbb {R}$ be an interval with nonempty interior, and let $f:C\rightarrow \mathbb {R}$ be a convex function. Let $\left[ a,b\right] \subset \mathbb {R}$ with $a<b$, and let $\left( \left[ a,b\right] ,\mathcal {A}\right)$ be a measurable space such that $\mathcal {B}_{\left[ a,b\right] }\subset \mathcal {A}$.

(a):

Suppose that one of the following two conditions is met:

$(\textrm{i}_\textrm{a})$ :

Let $\mu$ be a finite measure on $\mathcal {A}$. Assume $\varphi :\left[ a,b\right] \rightarrow C$ is a decreasing function, and $\psi :\left[ a,b\right] \rightarrow C$ is a $\mu$-integrable function for which $f\circ \psi$ is also $\mu$-integrable.

$(\textrm{ii}_\textrm{a})$ :

Let $\mu$ be a finite signed measure on $\mathcal {A}$. Assume $\varphi$, $\psi :\left[ a,b\right] \rightarrow C$ are decreasing functions.

$(\textrm{a}_{1})$ :

If f is increasing and

$$\begin{aligned} \int \limits _{\left[ a,x\right] }\varphi d\mu \le \int \limits _{\left[ a,x\right] }\psi d\mu ,\quad x\in \left[ a,b\right] , \end{aligned}$$

(16)

is satisfied, then

$$\begin{aligned} \int \limits _{\left[ a,b\right] }f\circ \varphi d\mu \le \int \limits _{\left[ a,b\right] }f\circ \psi d\mu . \end{aligned}$$

(17)

$(\textrm{a}_{2})$ :

If (16) and

$$\begin{aligned} \int \limits _{\left[ a,b\right] }\varphi d\mu =\int \limits _{\left[ a,b\right] }\psi d\mu \end{aligned}$$

(18)

are satisfied, then inequality (17) holds too.

(b):

Suppose that one of the following two conditions is met:

$(\textrm{i}_{\textrm{b}})$ :

Let $\mu$ be a finite measure on $\mathcal {A}$. Assume $\varphi :\left[ a,b\right] \rightarrow C$ is a $\mu$-integrable function for which $f\circ \varphi$ is also $\mu$-integrable, and $\psi :\left[ a,b\right] \rightarrow C$ is an increasing function.

$(\textrm{ii}_{\textrm{b}})$ :

Let $\mu$ be a finite signed measure on $\mathcal {A}$. Assume $\varphi$, $\psi :\left[ a,b\right] \rightarrow C$ are increasing functions.

$(\textrm{b}_{1})$ :

If f is decreasing and (16) is satisfied, then

$$\begin{aligned} \int \limits _{\left[ a,b\right] }f\circ \varphi d\mu \ge \int \limits _{\left[ a,b\right] }f\circ \psi d\mu . \end{aligned}$$

(19)

$(\textrm{b}_{2})$ :

If (16) and (18) are satisfied, then inequality (19) holds too.

Remark 2

The result contains the weighted version of Hardy–Littlewood–Pólya inequality (see Niculescu and Persson [14]) and Fuchs inequality, and even extends them to countably infinite sequences (see Corollary 13 in Horváth [7]). The integral version of the Hardy–Littlewood–Pólya inequality (see Pečarić, Proschan and Tong [16]) is also a special case of Theorem 5.

3 Main results

We start by introducing some special function sets: Let $C\subset \mathbb {R}$ be an interval with nonempty interior, and let $\left( X,\mathcal {A},\mu \right)$ and $\left( Y,\mathcal {B},\nu \right)$ be measure spaces, where $\mu$ and $\nu$ are finite signed measures. Furthermore, let $\varphi :X\rightarrow C$, $\psi :Y\rightarrow C$ be functions such that $\varphi \in L\left( \mu \right)$ and $\psi \in L\left( \nu \right)$.

• (a) Let denote $F_{C}^{i}$ the set of all increasing and convex functions on C. We define $F_{C}^{i}\left( \varphi ,\psi \right)$ as the set of all functions $f\in F_{C}^{i}$ such that $f\circ \varphi \in L\left( \mu \right)$ and $f\circ \psi \in L\left( \nu \right)$.

• (b) Let denote $F_{C}^{d}$ the set of all decreasing and convex functions on C. We define $F_{C}^{d}\left( \varphi ,\psi \right)$ as the set of all functions $f\in F_{C}^{d}$ such that $f\circ \varphi \in L\left( \mu \right)$ and $f\circ \psi \in L\left( \nu \right)$.

• (c) Let denote $F_{C}$ the set of all convex functions on C. We define $F_{C}\left( \varphi ,\psi \right)$ as the set of all functions $f\in F_{C}$ such that $f\circ \varphi \in L\left( \mu \right)$ and $f\circ \psi \in L\left( \nu \right)$.

If $\left( X,\mathcal {A},\mu \right) =\left( Y,\mathcal {B},\nu \right)$ and $\varphi =\psi$, the shorter notations $F_{C}^{i}\left( \varphi \right)$, $F_{C}^{d}\left( \varphi \right)$ and $F_{C}\left( \varphi \right)$ are used instead of $F_{C}^{i}\left( \varphi ,\psi \right)$, $F_{C}^{d}\left( \varphi ,\psi \right)$ and $F_{C}\left( \varphi ,\psi \right)$.

With these preliminaries, we can state and prove our main result.

Theorem 6

Let $\left( X,\mathcal {A},\mu \right)$ and $\left( Y,\mathcal {B},\nu \right)$ be measure spaces, where $\mu$ and $\nu$ are finite signed measures. Let $C\subset \mathbb {R}$ be an interval with nonempty interior, and let $\varphi :X\rightarrow C$, $\psi :Y\rightarrow C$ be functions such that $\varphi \in L\left( \mu \right)$ and $\psi \in L\left( \nu \right)$. Then

(a$_{1})$:

For every $f\in F_{C}^{i}\left( \varphi ,\psi \right)$ inequality

$$\begin{aligned} \int \limits _{X}f\circ \varphi d\mu \le \int \limits _{Y}f\circ \psi d\nu \end{aligned}$$

(20)

holds if and only if $\mu \left( X\right) =\nu \left( Y\right)$ and it is satisfied in the following special cases: the function f is either $id_{C}$ or $p_{C,w}$ $\left( w\in C^{\circ }\right)$.

(a$_{2})$:

For every nonnegative $f\in F_{C}^{i}\left( \varphi ,\psi \right)$ inequality (20) holds if and only if $\mu \left( X\right) \le \nu \left( Y\right)$ and it is satisfied in the following special cases: f $=p_{C,w}$ $\left( w\in C^{\circ }\right)$.

(b$_{1})$:

For every $f\in F_{C}^{d}\left( \varphi ,\psi \right)$ inequality (20) holds if and only if $\mu \left( X\right) =\nu \left( Y\right)$ and it is satisfied in the following special cases: the function f is either $-id_{C}$ or $n_{C,w}$ $\left( w\in C^{\circ }\right)$.

(b$_{2})$:

For every nonnegative $f\in F_{C}^{d}\left( \varphi ,\psi \right)$ inequality (20) holds if and only if $\mu \left( X\right) \le \nu \left( Y\right)$ and it is satisfied in the following special cases: f $=n_{C,w}$ $\left( w\in C^{\circ }\right)$.

(c$_{1})$:

For every $f\in F_{C}\left( \varphi ,\psi \right)$ inequality (20) holds if and only if $\mu \left( X\right) =\nu \left( Y\right)$ and it is satisfied in the following special cases: the function f is either $id_{C}$ or $-id_{C}$ or $p_{C,w}$ $\left( w\in C^{\circ }\right)$.

(c$_{2})$:

For every nonnegative $f\in F_{C}\left( \varphi ,\psi \right)$ inequality (20) holds if and only if $\mu \left( X\right) \le \nu \left( Y\right)$ and it is satisfied in the following special cases: the function f is either $p_{C,w}$ or $n_{C,w}$ $\left( w\in C^{\circ }\right)$.

Proof

(a$_{1}$) The constant functions $f_{1}$, $f_{2}:C\rightarrow \mathbb {R}$, $f_{1}\left( t\right) :=1$ and $f_{2}\left( t\right) :=-1$ belong to $F_{C}^{i}\left( \varphi ,\psi \right)$, and hence (20) implies $\mu \left( X\right) =\nu \left( Y\right)$. The functions $id_{C}$ and $p_{C,w}$ $\left( w\in C^{\circ }\right)$ are increasing and convex, and since $\varphi \in L\left( \mu \right)$, $\psi \in L\left( \nu \right)$ and $\mu$, $\nu$ are finite, they belong to $F_{C}^{i}\left( \varphi ,\psi \right)$. This shows that the condition is necessary.

To prove sufficiency we distinguish two cases.

(i) We first take the case that f is continuous.

By Theorem 3 (b), f is the pointwise limit of an increasing sequence of piecewise linear, increasing and convex functions on C. If $\left( f_{n}\right)$ is such a sequence, then $\left( f_{n}\circ \varphi \right)$ is also increasing and converges pointwise to $f\circ \varphi$ on X. Similarly, $\left( f_{n}\circ \psi \right)$ is increasing too, and converges pointwise to $f\circ \psi$ on Y.

By Remark 1 (b), if g is a piecewise linear, increasing and convex function on C, then g is of the form

$$\begin{aligned} g\left( t\right) =\alpha t+\beta +\sum \limits _{i=1}^{k}\gamma _{i}\left( t-t_{i}\right) ^{+},\quad t\in C \end{aligned}$$

(21)

for suitable points $t_{1}<t_{2}<\cdots <t_{k}$ in the interior of C, $\alpha \ge 0$, $\beta \in \mathbb {R}$ and $\gamma _{i}>0$ $\left( i=1,\ldots ,k\right)$. Since $\varphi \in L\left( \mu \right)$ and $\mu$ is finite, $g\circ \varphi \in L\left( \mu \right)$. Similarly, $g\circ \psi \in L\left( \nu \right)$, and hence $g\in F_{C}^{i}\left( \varphi ,\psi \right)$.

By applying B. Levi’s theorem, we obtain that

$$\begin{aligned} \int \limits _{X}f_{n}\circ \varphi d\mu \rightarrow \int \limits _{X}f\circ \varphi d\mu \quad \text {and}\quad \int \limits _{Y}f_{n}\circ \psi d\nu \rightarrow \int \limits _{Y}f\circ \psi d\nu . \end{aligned}$$

In summary, it is enough to prove (20) for piecewise linear increasing and convex functions on C. Since such a function is of the form (21), it follows from the conditions.

(ii) Now assume that f is not continuous at the right-hand endpoint of the interval C.

Then it is not hard to think that there exists a decreasing sequence $\left( f_{n}\right) _{n=1}^{\infty }$ from $F_{C}^{i}\left( \varphi ,\psi \right)$ such that $f_{n}$ is continuous $\left( n\in \mathbb {N}_{+}\right)$ and $\left( f_{n}\right)$ converges pointwise to f on C. In this case the sequences $\left( f_{n}\circ \varphi \right)$ and $\left( f_{n}\circ \psi \right)$ are also decreasing, and therefore the result follows from the first part of the proof and B. Levi’s theorem.

(a$_{2}$) It can be proved similarly to (a$_{1}$) by using Theorem 4 (b).

(b$_{1}$) It can be proved similarly to (a$_{1}$) by using Theorem 3 (c), and taking Remark 1 (c) into account.

(b$_{2}$) It can be proved similarly to (a$_{1}$) by using Theorem 4 (c).

(c$_{1}$) The condition that inequality (20) holds for both $f=id_{C}$ and $f=-id_{C}$ is, of course, equivalent to

$$\begin{aligned} \int \limits _{X}\varphi d\mu =\int \limits _{Y}\psi d\nu . \end{aligned}$$

It can be proved similarly to (a$_{1}$) by using Theorem 3 (a), and taking Remark 1 (a) and Lemma 1 into account.

(c$_{2}$) It can be proved similarly to (a$_{1}$) by using Theorem 4 (a).

The proof is complete.$\square$

Remark 3

The previous result extends Theorem 10 in Horváth [7], where only the following cases have been considered: $X=Y=\left[ a,b\right] \subset \mathbb {R}$ and $\mathcal {B}_{\left[ a,b\right] }\subset \mathcal {A}=\mathcal {B}$. Moreover, analogous of statements about nonnegative functions ((a$_{2}$), (b$_{2}$) and (c$_{2}$)) are not included.

Using the previous statement, we give sufficient conditions for satisfying inequality (20) that are easier to check.

Theorem 7

Let $\left( X,\mathcal {A}\right)$ be a measurable space, and let $\mu$ and $\nu$ be finite measures on $\mathcal {A}$ such that $\mu \left( X\right) =\nu \left( X\right)$. Let $C\subset \mathbb {R}$ be an interval with nonempty interior, and let $\varphi$, $\psi :X\rightarrow C$ be functions such that $\varphi \in L\left( \mu \right)$ and $\psi \in L\left( \nu \right)$.

(a$_{1})$:

If (7) is satisfied, then inequality

$$\begin{aligned} \int \limits _{X}f\circ \varphi d\mu \le \int \limits _{X}f\circ \psi d\nu \end{aligned}$$

(22)

holds for every $f\in F_{C}^{i}\left( \varphi ,\psi \right)$.

(a$_{2})$:

If (7) and (10) are satisfied, then inequality (22) holds for every $f\in F_{C}\left( \varphi ,\psi \right)$.

(b$_{1})$:

If (14) is satisfied, then inequality

$$\begin{aligned} \int \limits _{X}f\circ \varphi d\mu \ge \int \limits _{X}f\circ \psi d\nu \end{aligned}$$

(23)

holds for every $f\in F_{C}^{d}\left( \varphi ,\psi \right)$.

(b$_{2})$:

If (14) and (10) are satisfied, then inequality (23) holds for every $f\in F_{C}\left( \varphi ,\psi \right)$.

Proof

(a$_{1}$):

It follows from Theorem 6 (a$_{1}$) by applying Lemma 2 (a).

(a$_{2}$):

It follows from Theorem 6 (c$_{1}$), by applying Lemma 2.

(b$_{1}$):

It follows from Theorem 6 (b$_{1}$), by applying Lemma 3 (a).

(b$_{2}$):

It follows from Theorem 6 (c$_{1}$), by applying Lemma 3. The proof is complete.$\square$

Remark 4

• (a) This result contains Theorem 5 (a) under the condition (i$_{a}$) and Theorem 5 (b) under the condition (i$_{b}$). It is illustrated by considering Theorem 5 (a$_{1}$): if $\varphi :\left[ a,b\right] \rightarrow C$ is a decreasing function, and $\psi :\left[ a,b\right] \rightarrow C$ is a $\mu$-integrable function such that

  $$\begin{aligned} \int \limits _{\left[ a,x\right] }\varphi d\mu \le \int \limits _{\left[ a,x\right] }\psi d\mu ,\quad x\in \left[ a,b\right] \end{aligned}$$

  holds, then by using Lemma 4 in Horváth [7],

  $$\begin{aligned} \int \limits _{\left\{ \varphi \ge w\right\} }\varphi d\mu \le \int \limits _{\left\{ \varphi \ge w\right\} }\psi d\mu ,\quad w\in \mathbb {R} \end{aligned}$$

  is also satisfied.

• (b) An interesting question is under what conditions the previous result would be valid for signed measures. Theorem 5 under the conditions either (ii$_{a}$) or (ii$_{b}$) shows that there should be some relation between the functions $\varphi$ and $\psi$.

4 Applications

We first state Theorem 6 for discrete measures.

Theorem 8

Let the set X denote either $\left\{ 1,\ldots ,m\right\}$ for some $m\ge 1$ or $\mathbb {N}_{+}$, and let the set Y denote either $\left\{ 1,\ldots ,n\right\}$ for some $n\ge 1$ or $\mathbb {N}_{+}$. Assume $\left( p_{i}\right) _{i\in X}$ and $\left( q_{j}\right) _{j\in Y}$ are real sequences such that

$$\begin{aligned} \sum \limits _{i\in X}\left| p_{i}\right|<\infty ,\quad \sum \limits _{j\in Y}\left| q_{j}\right| <\infty . \end{aligned}$$

Let $C\subset \mathbb {R}$ be an interval with nonempty interior, and let $\left( s_{i}\right) _{i\in X}$ and $\left( t_{j}\right) _{j\in Y}$ be sequences from C such that

$$\begin{aligned} \sum \limits _{i\in X}\left| p_{i}s_{i}\right|<\infty ,\quad \sum \limits _{j\in Y}\left| q_{j}t_{j}\right| <\infty . \end{aligned}$$

Denote by $F_{C}^{i}\left( \left( s_{i}\right) ,\left( t_{j}\right) \right)$ $\left( F_{C}^{d}\left( \left( s_{i}\right) ,\left( t_{j}\right) \right) ,\text { }F_{C}\left( \left( s_{i}\right) ,\left( t_{j}\right) \right) \right)$ the set of all functions $f\in F_{C}^{i}$ $\left( f\in F_{C}^{d},\text { }f\in F_{C}\right)$ for which

$$\begin{aligned} \sum \limits _{i\in X}\left| p_{i}f\left( s_{i}\right) \right|<\infty ,\quad \sum \limits _{j\in Y}\left| q_{j}f\left( t_{j}\right) \right| <\infty . \end{aligned}$$

Then

(a$_{1})$:

For every $f\in F_{C}^{i}\left( \left( s_{i}\right) ,\left( t_{j}\right) \right)$ inequality

$$\begin{aligned} \sum \limits _{i\in X}p_{i}f\left( s_{i}\right) \le \sum \limits _{j\in Y} q_{j}f\left( t_{j}\right) \end{aligned}$$

(24)

holds if and only if

$$\begin{aligned} \sum \limits _{i\in X}p_{i}=\sum \limits _{j\in Y}q_{j} \end{aligned}$$

(25)

and

$$\begin{aligned} \sum \limits _{i\in X}p_{i}s_{i}\le \sum \limits _{j\in Y}q_{j}t_{j} \end{aligned}$$

and

$$\begin{aligned} \sum \limits _{\left\{ i\in X\mid s_{i}\ge w\right\} }p_{i}\left( s_{i}-w\right) \le \sum \limits _{\left\{ j\in Y\mid t_{j}\ge w\right\} }q_{j}\left( t_{j}-w\right) ,\quad w\in C^{\circ } \end{aligned}$$

(26)

are satisfied.

(a$_{2})$:

For every nonnegative $f\in F_{C}^{i}\left( \left( s_{i}\right) ,\left( t_{j}\right) \right)$ inequality (24) holds if and only if

$$\begin{aligned} \sum \limits _{i\in X}p_{i}\le \sum \limits _{j\in Y}q_{j} \end{aligned}$$

(27)

and (26) are satisfied.

(b$_{1})$:

For every $f\in F_{C}^{d}\left( \left( s_{i}\right) ,\left( t_{j}\right) \right)$ inequality (24) holds if and only if (25) and

$$\begin{aligned} \sum \limits _{i\in X}p_{i}s_{i}\ge \sum \limits _{j\in Y}q_{j}t_{j} \end{aligned}$$

and

$$\begin{aligned} \sum \limits _{\left\{ i\in X\mid s_{i}<w\right\} }p_{i}\left( w-s_{i} \right) \le \sum \limits _{\left\{ j\in Y\mid t_{j}<w\right\} }q_{j}\left( w-t_{j}\right) ,\quad w\in C^{\circ } \end{aligned}$$

(28)

are satisfied.

(b$_{2})$:

For every nonnegative $f\in F_{C}^{d}\left( \left( s_{i}\right) ,\left( t_{j}\right) \right)$ inequality (24) holds if and only if (27) and (28) are satisfied.

(c$_{1})$:

For every $f\in F_{C}\left( \left( s_{i}\right) ,\left( t_{j}\right) \right)$ inequality (24) holds if and only if (25) and

$$\begin{aligned} \sum \limits _{i\in X}p_{i}s_{i}=\sum \limits _{j\in Y}q_{j}t_{j} \end{aligned}$$

and (26) are satisfied.

(c$_{2})$:

For every nonnegative $f\in F_{C}\left( \left( s_{i}\right) ,\left( t_{j}\right) \right)$ inequality (24) holds if and only if (27) and (26) and (28) are satisfied.

Proof

Let the set functions $\mu :P\left( X\right) \rightarrow \mathbb {R}$ and $\nu :P\left( Y\right) \rightarrow \mathbb {R}$ by

$$\begin{aligned} \mu :=\sum \limits _{i\in X}p_{i}\varepsilon _{i},\quad \nu :=\sum \limits _{j\in Y}q_{j}\varepsilon _{j}. \end{aligned}$$

Then $\left( X,P\left( X\right) ,\mu \right)$ and $\left( Y,P\left( Y\right) ,\nu \right)$ are measure spaces with finite signed measures $\mu$ and $\nu$.

Let the functions $\varphi :X\rightarrow C$ and $\psi :Y\rightarrow C$ be defined by

$$\begin{aligned} \varphi \left( i\right) :=s_{i},\quad \psi \left( j\right) :=t_{j}. \end{aligned}$$

Then $\varphi \in L\left( \mu \right)$ and $\psi \in L\left( \nu \right)$.

Now the result is an immediate consequence of Theorem 6, since $F_{C}^{i}\left( \left( s_{i}\right) ,\left( t_{j}\right) \right) =F_{C}^{i}\left( \varphi ,\psi \right)$, $F_{C}^{d}\left( \left( s_{i}\right) ,\left( t_{j}\right) \right) =F_{C}^{d}\left( \varphi ,\psi \right)$ and $F_{C}\left( \left( s_{i}\right) ,\left( t_{j}\right) \right) =F_{C}\left( \varphi ,\psi \right)$.

The proof is complete.$\square$

When X and Y are finite sets, we can obtain the following important and interesting consequence of the previous theorem.

Theorem 9

Let $X:=\left\{ 1,\ldots ,m\right\}$ for some $m\ge 1$, and let $Y:=\left\{ 1,\ldots ,n\right\}$ for some $n\ge 1$. Assume $\left( p_{i}\right) _{i=1}^{m}$ and $\left( q_{j}\right) _{j=1}^{n}$ are real sequences, and $\left( s_{i}\right) _{i=1}^{m}$ and $\left( t_{j}\right) _{j=1}^{n}$ are sequences from C. Let $u_{1}>u_{2}>\cdots >u_{k}$ be the different elements of $\left( s_{i}\right) _{i=1}^{m}$ and $\left( t_{j}\right) _{j=1}^{n}$ in decreasing order $\left( 1\le k\le m+n\right)$.

Then

(a$_{1})$:

For every $f\in F_{C}^{i}$ inequality

$$\begin{aligned} \sum \limits _{i=1}^{m}p_{i}f\left( s_{i}\right) \le \sum \limits _{j=1} ^{n}q_{j}f\left( t_{j}\right) \end{aligned}$$

(29)

holds if and only if

$$\begin{aligned} \sum \limits _{i=1}^{m}p_{i}=\sum \limits _{j=1}^{n}q_{j} \end{aligned}$$

(30)

and

$$\begin{aligned} \sum \limits _{i=1}^{m}p_{i}s_{i}\le \sum \limits _{j=1}^{n}q_{j}t_{j} \end{aligned}$$

and

$$\begin{aligned}{} & {} \sum \limits _{\left\{ i\in X\mid s_{i}\ge u_{l}\right\} }p_{i}s_{i} -\sum \limits _{\left\{ j\in Y\mid t_{j}\ge u_{l}\right\} }q_{j}t_{j}\nonumber \\{} & {} \qquad \quad \le u_{l}\left( \sum \limits _{\left\{ i\in X\mid s_{i}\ge u_{l}\right\} }p_{i}-\sum \limits _{\left\{ j\in Y\mid t_{j}\ge u_{l}\right\} } q_{j}\right) ,\quad l=1,\ldots ,k. \end{aligned}$$

(31)

(a$_{2})$:

For every nonnegative $f\in F_{C}^{i}$ inequality (29) holds if and only if

$$\begin{aligned} \sum \limits _{i=1}^{m}p_{i}\le \sum \limits _{j=1}^{n}q_{j} \end{aligned}$$

(32)

and (31) are satisfied.

(b$_{1})$:

For every $f\in F_{C}^{d}$ inequality (29) holds if and only if (30) and

$$\begin{aligned} \sum \limits _{i=1}^{m}p_{i}s_{i}\ge \sum \limits _{j=1}^{n}q_{j}t_{j} \end{aligned}$$

and

$$\begin{aligned}{} & {} \sum \limits _{\left\{ i\in X\mid s_{i}<u_{l}\right\} }p_{i}s_{i} -\sum \limits _{\left\{ j\in Y\mid t_{j}<u_{l}\right\} }q_{j}t_{j}\nonumber \\{} & {} \qquad \quad \ge u_{l}\left( \sum \limits _{\left\{ i\in X\mid s_{i}<u_{l}\right\} } p_{i}-\sum \limits _{\left\{ j\in Y\mid t_{j}<u_{l}\right\} }q_{j}\right) ,\quad l=1,\ldots ,k. \end{aligned}$$

(33)

are satisfied.

(b$_{2})$:

For every nonnegative $f\in F_{C}^{d}$ inequality (29) holds if and only if (32) and (33) are satisfied.

(c$_{1})$:

For every $f\in F_{C}$ inequality (29) holds if and only if (30) and

$$\begin{aligned} \sum \limits _{i=1}^{m}p_{i}s_{i}=\sum \limits _{j=1}^{n}q_{j}t_{j} \end{aligned}$$

and (31) are satisfied.

(c$_{2})$:

For every nonnegative $f\in F_{C}$ inequality (29) holds if and only if (32) and (31) and (33) are satisfied.

Proof

Only (a$_{1}$) is shown, the proofs of the others follow a similar line of thought.

By Theorem 8 (a$_{1}$), it is enough to show that (31) is equivalent to (26).

It is obvious that if (26) holds, then (31) is also satisfied.

Conversely, assume (31) is satisfied.

If $u_{l+1}<w\le u_{l}$ for some $1\le l<k$ then

$$\begin{aligned} \sum \limits _{\left\{ i\in X\mid s_{i}\ge u_{l}\right\} }p_{i}s_{i} -\sum \limits _{\left\{ j\in Y\mid t_{j}\ge u_{l}\right\} }q_{j}t_{j} =\sum \limits _{\left\{ i\in X\mid s_{i}\ge w\right\} }p_{i}s_{i} -\sum \limits _{\left\{ j\in Y\mid t_{j}\ge w\right\} }q_{j}t_{j} \end{aligned}$$

(34)

and

$$\begin{aligned} \sum \limits _{\left\{ i\in X\mid s_{i}\ge u_{l}\right\} }p_{i} -\sum \limits _{\left\{ j\in Y\mid t_{j}\ge u_{l}\right\} }q_{j} =\sum \limits _{\left\{ i\in X\mid s_{i}\ge w\right\} }p_{i}-\sum \limits _{\left\{ j\in Y\mid t_{j}\ge w\right\} }q_{j}. \end{aligned}$$

(35)

Moreover,

$$\begin{aligned} \sum \limits _{\left\{ i\in X\mid s_{i}\ge u_{l+1}\right\} }p_{i}s_{i} -\sum \limits _{\left\{ j\in Y\mid t_{j}\ge u_{l+1}\right\} }q_{j} t_{j}{} & {} =\left( \sum \limits _{\left\{ i\in X\mid s_{i}\ge u_{l}\right\} } p_{i}s_{i}-\sum \limits _{\left\{ j\in Y\mid t_{j}\ge u_{l}\right\} } q_{j}t_{j}\right) \\{} & {} \qquad +\left( \sum \limits _{\left\{ i\in X\mid s_{i}=u_{l+1}\right\} }p_{i} s_{i}-\sum \limits _{\left\{ j\in Y\mid t_{j}\ge u_{l+1}\right\} }q_{j} t_{j}\right) \\{} & {} =\left( \sum \limits _{\left\{ i\in X\mid s_{i}\ge u_{l}\right\} }p_{i} s_{i}-\sum \limits _{\left\{ j\in Y\mid t_{j}\ge u_{l}\right\} }q_{j} t_{j}\right) \\{} & {} \qquad +u_{l+1}\left( \sum \limits _{\left\{ i\in X\mid s_{i}=u_{l+1}\right\} } p_{i}-\sum \limits _{\left\{ j\in Y\mid t_{j}\ge u_{l+1}\right\} } q_{j}\right) \\{} & {} \le u_{l+1}\left( \sum \limits _{\left\{ i\in X\mid s_{i}\ge u_{l}\right\} }p_{i}-\sum \limits _{\left\{ j\in Y\mid t_{j}\ge u_{l}\right\} } q_{j}\right) \\{} & {} \qquad +u_{l+1}\left( \sum \limits _{\left\{ i\in X\mid s_{i}=u_{l+1}\right\} } p_{i}-\sum \limits _{\left\{ j\in Y\mid t_{j}\ge u_{l+1}\right\} } q_{j}\right) , \end{aligned}$$

and therefore

$$\begin{aligned} \left( \sum \limits _{\left\{ i\in X\mid s_{i}\ge u_{l}\right\} }p_{i} s_{i}-\sum \limits _{\left\{ j\in Y\mid t_{j}\ge u_{l}\right\} }q_{j} t_{j}\right) \le u_{l+1}\left( \sum \limits _{\left\{ i\in X\mid s_{i}\ge u_{l}\right\} }p_{i}-\sum \limits _{\left\{ j\in Y\mid t_{j}\ge u_{l}\right\} }q_{j}\right) .\qquad \end{aligned}$$

(36)

It now follows from (34–35), (31) and (36) that

$$\begin{aligned} \sum \limits _{\left\{ i\in X\mid s_{i}\ge w\right\} }p_{i}s_{i} -\sum \limits _{\left\{ j\in Y\mid t_{j}\ge w\right\} }q_{j}t_{j}\le u_{l}\left( \sum \limits _{\left\{ i\in X\mid s_{i}\ge w\right\} }p_{i} -\sum \limits _{\left\{ j\in Y\mid t_{j}\ge w\right\} }q_{j}\right) \end{aligned}$$

and

$$\begin{aligned} \sum \limits _{\left\{ i\in X\mid s_{i}\ge w\right\} }p_{i}s_{i} -\sum \limits _{\left\{ j\in Y\mid t_{j}\ge w\right\} }q_{j}t_{j}\le u_{l+1}\left( \sum \limits _{\left\{ i\in X\mid s_{i}\ge w\right\} } p_{i}-\sum \limits _{\left\{ j\in Y\mid t_{j}\ge w\right\} }q_{j}\right) , \end{aligned}$$

which imply (26) for all $u_{l+1}<w\le u_{l}$.

The proof is complete.$\square$

Remark 5

• (a) We emphasize that the satisfaction of inequality (29) can be determined by examining a finite number of easily verifiable conditions.

• (b) The theorem gives exact conditions for inequality (29), where discrete signed measures are used, and therefore it is a substantial extension of both Hardy–Littlewood–Pólya inequality and Fuchs inequality.

• (c) Assume $m=n$, $s_{1}\ge \cdots \ge s_{n}$, $t_{1}\ge \cdots \ge t_{n}$ and $p_{i}=q_{i}$ $\left( i=1,\ldots ,n\right)$. By applying part (c$_{1}$), it is not hard to think that the conditions (b) and (c) in Fusch inequality not only sufficient but also necessary for the inequality (3) to be satisfied for any convex function on C.

• (d) It is also new, to the best of the author’s knowledge, that necessary and sufficient conditions are given separately for the function classes $F_{C} ^{i}$, $F_{C}^{d}$ and $F_{C}$ and their subsets of nonnegative functions.

Our next result is related to Steffensen–Popoviciu measures.

Theorem 10

Let $\left( X,\mathcal {A}\right)$ be a measurable space, and let $\mu$ be a finite signed measure on $\mathcal {A}$. Let $C\subset \mathbb {R}$ be an interval with nonempty interior, and let $\psi :X\rightarrow C$ be a $\mu$-integrable function.

• (a) For every nonnegative function $f\in F_{C}^{i}\left( \psi \right)$ inequality

  $$\begin{aligned} \int \limits _{X}f\circ \psi d\mu \ge 0 \end{aligned}$$

  (37)

  holds if and only if

  $$\begin{aligned} \mu \left( X\right) \ge 0 \end{aligned}$$

  (38)

  and

  $$\begin{aligned} \int \limits _{X}p_{C,w}\circ \psi d\mu =\int \limits _{\left\{ \psi \ge w\right\} }\left( \psi -w\right) d\mu \ge 0,\quad w\in C^{\circ }. \end{aligned}$$

  (39)

• (b) For every nonnegative function $f\in F_{C}^{d}\left( \psi \right)$ inequality (37) holds if and only if (38) is satisfied and

  $$\begin{aligned} \int \limits _{X}n_{C,w}\circ \psi d\mu =\int \limits _{\left\{ \psi <w\right\} }\left( w-\psi \right) d\mu \ge 0,\quad w\in C^{\circ }. \end{aligned}$$

  (40)

• (c) For every nonnegative function $f\in F_{C}\left( \psi \right)$ inequality (37) holds if and only if all three (38), (39) and (40) are fulfilled.

Proof

(a):

Since $\mu$ is a finite signed measure and $\psi \in L\left( \mu \right)$, $p_{C,w}\circ \psi$ is also $\mu$-integrable for every $w\in C^{\circ }$. Moreover, $f_{1}:C\rightarrow \mathbb {R}$, $f_{1}\left( t\right) :=1$ and $p_{C,w}$ $\left( w\in C^{\circ }\right)$ are nonnegative elements of $F_{C}^{i}$, and therefore conditions (38) and (39) are necessary. Conversely, suppose conditions (38) and (39) are satisfied. Then the proof that (37) holds is very like the proof of the sufficiency part of Theorem 6 (a$_{1}$) by using Theorem 4 (b) instead of Theorem 1 (b). We leave it to the reader.

(b–c):

They can be proved similarly to (a) by using Theorem 4 (c) and (a). The proof is complete.$\square$

A simple corollary of the previous statement is the following theorem, which extends the well known result on the characterization of Steffensen–Popoviciu measures defined on the Borel sets of compact intervals.

Theorem 11

Let $C\subset \mathbb {R}$ be an interval with nonempty interior, and let $\mu$ be a finite signed measure on $\mathcal {B}_{C}$ such that $id_{C}$ is $\mu$-integrable.

• (a) For every nonnegative and $\mu$-integrable function $f\in F_{C}^{i}$ inequality

  $$\begin{aligned} \int \limits _{C}fd\mu \ge 0 \end{aligned}$$

  (41)

  holds if and only if

  $$\begin{aligned} \mu \left( C\right) \ge 0 \end{aligned}$$

  (42)

  and

  $$\begin{aligned} \int \limits _{C}p_{C,w}d\mu =\int \limits _{C\cap [ w,\infty [ }\left( t-w\right) d\mu \left( t\right) \ge 0,\quad w\in C^{\circ }. \end{aligned}$$

  (43)

• (b) For every nonnegative and $\mu$-integrable function $f\in F_{C}^{d}$ inequality (41) holds if and only if (42)) is satisfied and

  $$\begin{aligned} \int \limits _{C}n_{C,w}d\mu =\int \limits _{C\cap ] -\infty ,w] }\left( w-t\right) d\mu \left( t\right) \ge 0,\quad w\in C^{\circ }. \end{aligned}$$

  (44)

• (c) For every nonnegative and $\mu$-integrable function $f\in F_{C}$ inequality (41) holds if and only if all three (42), (43) and (44) are fulfilled.

Proof

Apply Theorem 10 to the measure space $\left( C,\mathcal {B}_{C},\mu \right)$.

The proof is complete.$\square$

Our last application is the extension of the classical integral Jensen and Lah–Ribarič inequalities to signed measures.

Theorem 12

Let $\left( X,\mathcal {A}\right)$ be a measurable space, and let $\mu$ be a finite signed measure on $\mathcal {A}$ such that $\mu \left( X\right) >0$. Let $C\subset \mathbb {R}$ be an interval with nonempty interior, and let $\psi :X\rightarrow C$ be a $\mu$-integrable function. Then

• (a) If (39) and (40) are satisfied, then

  $$\begin{aligned} t_{\psi ,\mu }:=\frac{1}{\mu \left( X\right) }\int \limits _{X}\psi d\mu \in C. \end{aligned}$$

  (45)

• (b) For every function $f\in F_{C}\left( \psi \right)$ inequality

  $$\begin{aligned} f\left( \frac{1}{\mu \left( X\right) }\int \limits _{X}\psi d\mu \right) \le \frac{1}{\mu \left( X\right) }\int \limits _{X}f\circ \psi d\mu \end{aligned}$$

  (46)

  holds if and only if (39) and (40) are satisfied.

• (c) Assume $C=\left[ a,b\right]$ and $t_{\psi ,\mu }\in \left[ a,b\right]$. For every function $f\in F_{C}\left( \psi \right)$ inequality

  $$\begin{aligned} \frac{1}{\mu \left( X\right) }\int \limits _{X}f\circ \psi d\mu \le \frac{b-t_{\psi ,\mu }}{b-a}f\left( a\right) +\frac{t_{\psi ,\mu }-a}{b-a}f\left( b\right) \end{aligned}$$

  (47)

  holds if and only if

  $$\begin{aligned} \frac{b-w}{b-a}\int \limits _{\left\{ \psi \ge w\right\} }\left( \psi -a\right) d\mu +\frac{w-a}{b-a}\int \limits _{\left\{ \psi <w\right\} }\left( b-\psi \right) d\mu \ge 0,\quad w\in ] a,b [. \end{aligned}$$

  (48)

Proof

(a) Assume C is left-bounded with left-hand endpoint a.

Choose a decreasing sequence $\left( a_{n}\right) _{n=1}^{\infty }$ from $C^{\circ }$ such that $a_{n}\rightarrow a$. Then $\left\{ \psi \ge a_{n}\right\} \uparrow \left\{ \psi >a\right\}$ which means that the sets $\left\{ \psi<a_{1}\right\} \subset \left\{ \psi <a_{2}\right\} \subset \cdots$ satisfy $\left\{ \psi >a\right\} =\bigcup \nolimits _{n=1} ^{\infty }\left\{ \psi \ge a_{n}\right\}$. It follows that

$$\begin{aligned} \mu \left( \left\{ \psi >a\right\} \right) =\lim \limits _{n\rightarrow \infty }\mu \left( \left\{ \psi \ge a_{n}\right\} \right) . \end{aligned}$$

(49)

By (39),

$$\begin{aligned} \int \limits _{\left\{ \psi \ge a_{n}\right\} }\left( \psi -a_{n}\right) d\mu \ge 0, \end{aligned}$$

and hence (49) implies that

$$\begin{aligned} \int \limits _{X}\left( \psi -a\right) d\mu =\int \limits _{\left\{ \psi >a\right\} }\left( \psi -a\right) d\mu =\lim \limits _{n\rightarrow \infty }\int \limits _{\left\{ \psi \ge a_{n}\right\} }\left( \psi -a_{n}\right) d\mu \ge 0, \end{aligned}$$

that is

$$\begin{aligned} \frac{1}{\mu \left( X\right) }\int \limits _{X}\psi d\mu \ge a. \end{aligned}$$

Now assume that $\psi >a$ and

$$\begin{aligned} \frac{1}{\mu \left( X\right) }\int \limits _{X}\psi d\mu =a. \end{aligned}$$

Then for each $w\in C^{\circ }$ we have

$$\begin{aligned} \int \limits _{\left\{ \psi \ge w\right\} }\left( \psi -w\right) d\mu{} & {} =\int \limits _{\left\{ \psi \ge w\right\} }\left( \psi -w\right) d\mu -\int \limits _{X}\left( \psi -a\right) d\mu \nonumber \\{} & {} =\left( a-w\right) \mu \left( \left\{ \psi \ge w\right\} \right) -\int \limits _{\left\{ \psi<w\right\} }\left( \psi -a\right) d\mu \nonumber \\{} & {} \le \left( w-a\right) \left( \mu ^{-}\left( \left\{ \psi <w\right\} \right) -\mu \left( \left\{ \psi \ge w\right\} \right) \right) , \end{aligned}$$

(50)

where $\mu ^{-}$ denotes the negative part of $\mu$. Since $\psi >a$,

$$\begin{aligned} \lim \limits _{w\rightarrow a+}\mu \left( \left\{ \psi \ge w\right\} \right) =\mu \left( X\right) >0\quad \text {and}\quad \lim \limits _{w\rightarrow a+}\mu ^{-}\left( \left\{ \psi <w\right\} \right) =0, \end{aligned}$$

and hence there exists $w_{0}\in C^{\circ }$ such that

$$\begin{aligned} \mu ^{-}\left( \left\{ \psi<w_{0}\right\} \right) -\mu \left( \left\{ \psi \ge w_{0}\right\} \right) <0. \end{aligned}$$

It follows from (50) that

$$\begin{aligned} \int \limits _{\left\{ \psi \ge w_{0}\right\} }\left( \psi -w_{0}\right) d\mu <0 \end{aligned}$$

which contradicts to (39).

We can prove in a similar way when C is right-bounded by using (40).

(b) Assume first that (46) holds.

To interpret the left-hand side of inequality (46), condition (45) is required. Since for every nonnegative function $f\in F_{C}\left( \psi \right)$ inequality

$$\begin{aligned} \int \limits _{X}f\circ \psi d\mu \ge 0 \end{aligned}$$

follows from (46) and $\mu \left( X\right) >0$, Theorem 10 (c) shows that conditions (39) and (40) are also necessary.

Now assume that (45), (39) and (40) are satisfied.

By choosing the constant function

$$\begin{aligned} \varphi :X\rightarrow \mathbb {R},\quad \varphi \left( x\right) :=\frac{1}{\mu \left( X\right) }\int \limits _{X}\psi d\mu , \end{aligned}$$

Theorem 6 (c$_{1}$) implies that inequality (46) is true for every $f\in F_{C}\left( \varphi ,\psi \right) =F_{C}\left( \psi \right)$ if it is satisfied in the following special cases: the function f is either $id_{C}$ or $-id_{C}$ or $p_{C,w}$ or $n_{C,w}$ $\left( w\in C^{\circ }\right)$.

If f is either $id_{C}$ or $-id_{C}$, then (46) is trivial.

Let $f=p_{C,w}$ for some $w\in C^{\circ }$. Then

$$\begin{aligned} \mu \left( X\right) \cdot f\left( \frac{1}{\mu \left( X\right) } \int \limits _{X}\psi d\mu \right) =\left\{ \begin{array}{ll} 0,&{}\quad \text { if }\int \limits _{X}\left( \psi -w\right) d\mu <0\\ \int \limits _{X}\left( \psi -w\right) d\mu ,&{}\quad \text { if }\int \limits _{X}\left( \psi -w\right) d\mu \ge 0 \end{array} \right. , \end{aligned}$$

(51)

while

$$\begin{aligned} \int \limits _{X}f\circ \psi d\mu= & {} \int \limits _{\left\{ \psi \ge w\right\} }\left( \psi -w\right) d\mu \end{aligned}$$

(52)

$$\begin{aligned}= & {} \int \limits _{X}\left( \psi -w\right) d\mu +\int \limits _{\left\{ \psi <w\right\} }\left( w-\psi \right) d\mu , \end{aligned}$$

(53)

and therefore the first line in (51) follows from (52) and the second line in (51) follows from (53).

The case $f=n_{C,w}$ $\left( w\in C^{\circ }\right)$ can be handled in a similar way or Lemma 1 can be used.

(c) Let the function $\vartheta :\left[ a,b\right] \rightarrow \left[ a,b\right]$ be defined by

$$\begin{aligned} \vartheta =\left\{ \begin{array}{ll} b,&{}\quad \text {if}\quad t\in [ a,t_{\psi ,\mu } [ \\ t_{\psi ,\mu },&{}\quad \text {if}\quad t=t_{\psi ,\mu }\\ a,&{}\quad \text {if}\quad t\in ] t_{\psi ,\mu },b ] \end{array} \right. , \end{aligned}$$

and introduce the measure $\nu :=\frac{\mu \left( X\right) }{b-a} \lambda _{\left[ a,b\right] }$ on $\mathcal {B}_{\left[ a,b\right] }$.

Then $\nu \left( \left[ a,b\right] \right) =\mu \left( X\right)$ and

$$\begin{aligned} \int \limits _{\left[ a,b\right] }\vartheta d\nu= & {} \frac{\mu \left( X\right) }{b-a}\left( b\left( \frac{1}{\mu \left( X\right) }\int \limits _{X}\psi d\mu -a\right) \right) \\{} & {} +a\left( b-\frac{1}{\mu \left( X\right) }\int \limits _{X}\psi d\mu \right) =\int \limits _{X}\psi d\mu \end{aligned}$$

and

$$\begin{aligned} \int \limits _{\left[ a,b\right] }f\circ \vartheta d\nu =\left( \frac{b-t_{\psi ,\mu }}{b-a}f\left( a\right) +\frac{t_{\psi ,\mu }-a}{b-a}f\left( b\right) \right) \mu \left( X\right) \end{aligned}$$

for all convex function $f:\left[ a,b\right] \rightarrow \mathbb {R}$.

It now follows from Theorem 6 (c$_{1}$) that inequality (47) holds for all convex function $f:\left[ a,b\right] \rightarrow \mathbb {R}$ if and only if

$$\begin{aligned} \int \limits _{X}p_{\left[ a,b\right] ,w}\circ \psi d\mu \le \int \limits _{\left[ a,b\right] }p_{\left[ a,b\right] ,w}\circ \vartheta d\nu ,\quad w\in ] a,b [ \end{aligned}$$

is satisfied, but some easy calculations show that this inequality is equivalent to (48).

The proof is complete.

Remark 6

• (a) The proof of the first part of the statement shows that if C is not bounded from above (below), then only condition (39) (40) is sufficient to satisfy (45).

• (b) The result extends Theorem 14 (a) in Horváth [7], where X is a compact interval in $\mathbb {R}$ and $\psi$ is the identity function on this interval. Theorem 3 in Mihai and Niculescu [12] is also related to our result, but it contains only the case when X is an open interval in $\mathbb {R}$ and $\psi$ is the identity function on this interval, and it gives just sufficient conditions.

• (c) In part (c) we obtain the complete characterization of the finite signed measures for which (47) holds. This is given first in Florea and Niculescu [4], when $X=\left[ a,b\right] \subset \mathbb {R}$ and $\psi$ is the identity function on this interval. Our result extends Theorem 1 in [4]. When $\mu$ is a measure, inequality (47) is the well-known Lah–Ribarič inequality (see Lah and Ribarič [9]).

• (d) Assume $X:=\left[ a,b\right] \subset \mathbb {R}$, $\psi$ is the identity function on this interval and $\mathcal {A}:=\mathcal {B}_{\left[ a,b\right] }$. It follows from the result that inequality

  $$\begin{aligned} f\left( \frac{1}{\mu \left( \left[ a,b\right] \right) }\int \limits _{\left[ a,b\right] }td\mu \left( t\right) \right)\le & {} \frac{1}{\mu \left( \left[ a,b\right] \right) }\int \limits _{\left[ a,b\right] }f\left( t\right) d\mu \left( t\right) \\\le & {} \frac{b-t_{\psi ,\mu }}{b-a}f\left( a\right) +\frac{t_{\psi ,\mu }-a}{b-a}f\left( b\right) \end{aligned}$$

  holds for every function convex function f on $\left[ a,b\right]$ if and only if (39), (40) and (48) are satisfied. This gives the important Hermite–Hadamard inequality (see e.g. Pečarić, Proschan and Tong [16]) in a very general form. This result can already be found in Horváth [7].