1 Introduction

The theory of majorization is perhaps most remarkable for its simplicity. It is a powerful, easy-to-use, and flexible mathematical tool which can be applicable to a wide number of fields. The key contributors in majorization are Dalton [14], Hardy et al. [16], Lorenz [30], Muirhead [36], and Schur [43]. Many important contributions were also made by other authors. Particularly, the comprehensive survey by Ando [8] gives alternative derivations, generalizations, and a different viewpoint. For an elementary discussion of majorization, see Marshall and Olkin’s monograph [32].

In 2018, Latif et al. [29] studied generalized results related to the majorization inequality by using Taylor’s polynomial in combination with newly introduced Green’s functions. In the same year, Siddique et al. [44] gave generalized majorization results via Lidstone’s polynomial and newly defined Green’s functions. The theory of majorization is widely used in many fields of application. In [21], Khan et al. presented significant material on majorization along with its applications in the field of information theory.

In this paper, our main goal is to obtain generalized results about majorization via new Green’s functions and an extension of the Montgomery identity. We further make connection of majorization with information theory and discuss our generalized majorization inequality in terms of divergences and entropies. The results we obtain in this paper are closely related to the contents given in [1–5]. Moreover, some related results with the present topic can also be found in [10, 11, 27, 41, 42].

The following definition of majorization is from [39, page 319].

Definition 1

Let \(\mathbf {x}= (x_{1},\ldots, x_{m} )\), \(\mathbf {y} = (y_{1},\ldots, y_{m} )\) be two real m-tuples. Then we say that x majorizes y (denoted by \(\mathbf{x}\succ \mathbf{y}\)) if, for \(\lambda =1,2,\ldots,m-1\),

$$ \sum_{i=1}^{\lambda }y_{[i]}\leq \sum _{i=1}^{\lambda }x_{[i]} $$

holds and

$$ \sum_{i=1}^{m}x_{i} = \sum _{i=1}^{m}y_{i}, $$

where \(x_{[i]}\) and \(y_{[i]}\) denote their nonincreasing order.

Note that, in the definition of majorization, the original order of \(x_{i}\)s and \(y_{i}\)s plays no role because real m-tuples can always be reordered nonincreasingly.

The following theorem is famed in literature as classical majorization theorem and is given in [33, page 11] (see also [39, page 320]).

Theorem 1

Let \(I=[\zeta _{1}, \zeta _{2}]\subset \mathbb{R}\)and \(\mathbf {x}= (x_{1},\ldots, x_{m} )\), \(\mathbf {y} = (y_{1},\ldots, y_{m} )\in I^{m}\)be two nonincreasing m-tuples. Then x majorizes y if and only if the following inequality holds:

$$ \sum_{i=1}^{m} f (y_{i} ) \leq \sum_{i=1}^{m} f (x_{i} ), $$
(1)

where \(f:[\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\)is a continuous convex function.

A generalization of the aforementioned theorem is regarded as weighted majorization theorem and is proved by Fuchs in [15] (see also [39, page 323]).

Theorem 2

Let \(I=[\zeta _{1}, \zeta _{2}]\subset \mathbb{R}\)and \(\mathbf {x}= (x_{1},\ldots, x_{m} )\), \(\mathbf {y} = (y_{1},\ldots, y_{m} )\in I^{m}\)be two nonincreasing m-tuples. Let \(\mathbf{p}=(p_{1},\ldots, p_{m})\in \mathbb{R}^{m}\)be such that

$$ \sum_{i=1}^{\lambda }p_{i} y_{i} \leq \sum_{i=1}^{\lambda }p_{i} x_{i} \quad \textit{for } \lambda =1,2,\ldots, m-1, $$
(2)

and

$$ \sum_{i=1}^{m} p_{i} y_{i} = \sum_{i=1}^{m} p_{i} x_{i}. $$
(3)

Then the following inequality holds:

$$ \sum_{i=1}^{m} p_{i} f (y_{i} ) \leq \sum_{i=1}^{m} p_{i} f (x_{i} ), $$
(4)

where \(f:[\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\)is a continuous convex function.

The following theorem represents an integral form of Theorem 2 and is in fact a simple consequence of Theorem 1 given in [37] (see also [39, page 328]).

Theorem 3

Let \(\phi , \psi : [a, b]\rightarrow [\zeta _{1},\zeta _{2}]\)be two continuous nonincreasing functions and \(p : [a, b]\rightarrow \mathbb{R}\)be continuous. If

$$ \int _{a}^{\lambda }p(w)\psi (w)\,dw\leq \int _{a}^{\lambda }p(w)\phi (w)\,dw \quad \textit{for every } \lambda \in [a, b], $$
(5)

and

$$ \int _{a}^{b}p(w)\psi (w)\,dw = \int _{a}^{b}p(w)\phi (w)\,dw $$
(6)

hold, then

$$ \int _{a}^{b}p(w)f\bigl(\psi (w)\bigr)\,dw \leq \int _{a}^{b}p(w)f\bigl(\phi (w)\bigr)\,dw, $$
(7)

where \(f:[\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\)is a continuous convex function.

For other forms of an integral version and generalization of the majorization theorem, see [33, page 583], [9, 22, 24–26, 28, 31]. In this paper, we present our results for nonincreasing functions ϕ and ψ which satisfy the conditions of Theorem 3, but those results hold too for nondecreasing ϕ and ψ satisfying the following inequality:

$$ \int _{\lambda }^{b}p(w)\psi (w)\,dw\leq \int _{\lambda }^{b}p(w)\phi (w)\,dw, \quad \text{for every }\lambda \in [a, b], $$
(8)

and condition (6). For instance, see example in [33, page 584].

Definition 2

Let \(I=[\zeta _{1}, \zeta _{2}]\subset \mathbb{R}\) and \(f:[\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\) be a function. Then nth order divided difference of f at distinct points \(x_{0},\ldots,x_{n}\in {}[ \zeta _{1},\zeta _{2}]\) is defined recursively (see [6, 39]) by

$$ f{}[ x_{i}]=f(x_{i}), \quad (i=0, 1,\ldots,n) $$

and

$$ f{}[ x_{0},\ldots,x_{n}]= \frac{f{}[ x_{1},\ldots,x_{n}]-f{}[ x_{0},\ldots,x_{n-1}]}{x_{n}-x_{0}}. $$

Note that nth order divided difference of a function f does not depend on the order of points.

We can extend this definition by considering the condition that some (or all) points coincide. Assuming that \(f^{(k-1)}\) exists, we define

$$\begin{aligned} f\underbrace{[x,\ldots, x]}_{k\text{-times}}=\frac{f^{(k-1)}(x)}{(k-1)!}. \end{aligned}$$
(9)

Popoviciu [40] initially discussed the notion of n-convexity. We follow the definition given by Karlin [20].

Definition 3

A function \(f:[\zeta _{1},\zeta _{2}]\rightarrow \mathbb{R}\) is n-convex, \(n\geq 0\) if

$$\begin{aligned} f[x_{0},\ldots, x_{n}]\geq 0 \end{aligned}$$

holds for all choices of \((n+1)\) distinct points \(x_{0},\ldots,x_{n}\in {}[ \zeta _{1},\zeta _{2}]\).

Aljinović et al. in [7] proved the following proposition which gives an extension of the Montgomery identity via Taylor’s formula.

Proposition 1

Let \(f: I \rightarrow \mathbb{R}\)be such that \(f^{(n-1)}\)is absolutely continuous, where \(n\in \mathbb{N}\)and \(I\subset \mathbb{R}\)is an open interval. Then, for \(\zeta _{1}, \zeta _{2}\in I\)with \(\zeta _{1}<\zeta _{2}\), the following identity holds:

$$\begin{aligned} f(x) =&\frac{1}{\zeta _{2}-\zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}f(s)\,ds + \sum _{k=0}^{n-2}\frac{(x-\zeta _{1})^{k+2}f^{(k+1)}(\zeta _{1})}{k!(k+2)(\zeta _{2}-\zeta _{1})} \\ & {}- \sum_{k=0}^{n-2} \frac{(x-\zeta _{2})^{k+2}f^{(k+1)}(\zeta _{2})}{k!(k+2)(\zeta _{2}-\zeta _{1})} + \frac{1}{(n-1)!} \int _{\zeta _{1}}^{\zeta _{2}}T_{n}(x, s)f^{(n)}(s)\,ds, \end{aligned}$$
(10)

where

$$ T_{n}(x, s)= \textstyle\begin{cases} -\frac{(x-s)^{n}}{n(\zeta _{2}-\zeta _{1})}+ \frac{x-\zeta _{1}}{\zeta _{2}-\zeta _{1}}(x-s)^{n-1} ,& \zeta _{1} \leq s \leq x, \\ -\frac{(x-s)^{n}}{n(\zeta _{2}-\zeta _{1})}+ \frac{x-\zeta _{2}}{\zeta _{2}-\zeta _{1}}(x-s)^{n-1} ,& x< s \leq \zeta _{2}. \end{cases} $$
(11)

As a special case, for \(n=1\), the sum \(\sum_{k=0}^{n-2}\cdots \) in (10) is empty, so (10) reduces to the following famous Montgomery identity (see [35]):

$$\begin{aligned} f(x)=\frac{1}{\zeta _{2}-\zeta _{1}} \int _{\zeta _{1}}^{\zeta _{2}}f(s)\,ds + \int _{\zeta _{1}}^{\zeta _{2}}P(x, s)f'(s)\,ds, \end{aligned}$$
(12)

where \(P(x, s)\) is the Peano kernel given by

$$ P(x, s)= \textstyle\begin{cases} \frac{s-\zeta _{1}}{\zeta _{2}-\zeta _{1}} ,& \zeta _{1} \leq s \leq x, \\ \frac{s-\zeta _{2}}{\zeta _{2}-\zeta _{1}} ,& x< s \leq \zeta _{2}. \end{cases} $$
(13)

As stated in [34], the complete reference about Abel–Gontscharoff polynomial and a theorem for ‘two-point right focal problem’ is given in [6].

Remark 1

Abel–Gontscharoff polynomial as a special choice for ‘two-point right focal’ interpolating polynomial for \(n=2\) is as follows:

$$\begin{aligned} f(z)= f(\zeta _{1}) + (z - \zeta _{1} ) f'(\zeta _{2}) + \int _{\zeta _{1}}^{\zeta _{2}} G_{\varOmega , 2} (z, w) f''(w) \,dw, \end{aligned}$$
(14)

where \(G_{\varOmega , 2} (z, w): [\zeta _{1}, \zeta _{2}]\times [\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\) is Green’s function for ‘two-point right focal problem’ given by

$$ G_{1}(z, w)=G_{\varOmega , 2} (z, w)= \textstyle\begin{cases} (\zeta _{1} - w ), & \zeta _{1} \leq w\leq z, \\ (\zeta _{1} - z ), & z\leq w\leq \zeta _{2}. \end{cases}$$
(15)

Motivated by Abel–Gontscharoff Green’s function for ‘two-point right focal problem’, Mehmood et al. (see [34]) presented some new types of Green’s functions which are continuous as well as convex, as follows:

Let \([\zeta _{1}, \zeta _{2}]\subset \mathbb{R}\). Define new types of Green’s functions \(G_{d}: [\zeta _{1}, \zeta _{2}]\times [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\), where \(d= 2, 3, 4\), as follows:

$$\begin{aligned}& G_{2}(z, w)= \textstyle\begin{cases} (z-\zeta _{2}) ,& \zeta _{1}\leq w \leq z, \\ (w-\zeta _{2}) ,& z \leq w\leq \zeta _{2}, \end{cases}\displaystyle \end{aligned}$$
(16)
$$\begin{aligned}& G_{3}(z, w)= \textstyle\begin{cases} (z-\zeta _{1}) ,& \zeta _{1}\leq w \leq z, \\ (w-\zeta _{1}) ,& z \leq w\leq \zeta _{2}, \end{cases}\displaystyle \end{aligned}$$
(17)
$$\begin{aligned}& G_{4}(z, w)= \textstyle\begin{cases} (\zeta _{2}-w) ,& \zeta _{1}\leq w \leq z, \\ (\zeta _{2}-z) ,& z \leq w\leq \zeta _{2}. \end{cases}\displaystyle \end{aligned}$$
(18)

The following lemma, given by Mehmood et al. [34], will help us to obtain the new generalizations of majorization inequality.

Lemma 1

Let \(f: [\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\)be such that \(f \in C^{2}([\zeta _{1}, \zeta _{2}])\)and \(G_{d}\), (\(d=1,2,3,4\)) be Green’s functions given in (15)–(18) respectively. Then along with identity (14) the following identities hold:

$$\begin{aligned}& f(z)= f (\zeta _{2}) + (\zeta _{2} - z) f'(\zeta _{1}) + \int _{ \zeta _{1}}^{\zeta _{2}} G_{2} (z, w) f'' (w) \,dw, \end{aligned}$$
(19)
$$\begin{aligned}& f(z)= f (\zeta _{2}) - (\zeta _{2}- \zeta _{1}) f'(\zeta _{2}) + (z- \zeta _{1})f'(\zeta _{1}) + \int _{\zeta _{1}}^{\zeta _{2}} G_{3} (z, w) f'' (w) \,dw, \end{aligned}$$
(20)
$$\begin{aligned}& f(z)= f (\zeta _{1}) + (\zeta _{2}- \zeta _{1}) f'(\zeta _{1}) - ( \zeta _{2} - z)f'(\zeta _{2}) + \int _{\zeta _{1}}^{\zeta _{2}} G_{4} (z, w) f'' (w) \,dw. \end{aligned}$$
(21)

We organize this paper in the following way:

In Sect. 2, we give generalized results of the majorization inequality and related bounds by using an extension of the Montgomery identity and new Green’s functions. In Sect. 3, we use Csiszár f-divergence and generalized majorization-type inequalities to obtain new generalized results. We further discuss our obtained generalized results in terms of the Shannon entropy and the Kullback–Leibler distance.

2 Generalized majorized identities and related bounds via Montgomery identity and new Green’s functions

Before starting this section, we first define some notations which will be used throughout this article.

Majorization difference for a continuous convex function f is denoted as follows:

$$\begin{aligned} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, f( \cdot ) \bigr):= \sum_{i=1}^{m} p_{i} f (x_{i} ) - \sum_{i=1}^{m} p_{i} f (y_{i} ), \end{aligned}$$
(22)

where x, y, and p are as defined in Theorem 2. Similarly, the integral majorization difference for a continuous convex function f is denoted as follows:

$$\begin{aligned} \widetilde{\mathbb{D}} \bigl(\phi , \psi , p, f(\cdot ) \bigr):= \int _{a}^{b}p(w) \bigl(f\bigl(\phi (w)\bigr)-f \bigl(\psi (w)\bigr) \bigr)\,dw, \end{aligned}$$
(23)

where ϕ, ψ, and p are as defined in Theorem 3.

The following theorem gives two equivalent statements between the weighted majorization inequality for a continuous convex function and the inequality involving newly defined Green’s functions.

Theorem 4

Let \(I=[\zeta _{1}, \zeta _{2}]\subset \mathbb{R}\)and \(\mathbf {x}= (x_{1},\ldots, x_{m} )\), \(\mathbf {y} = (y_{1},\ldots, y_{m} )\in I^{m}\)be two nonincreasing m-tuples. Let \(\mathbf{p}=(p_{1},\ldots, p_{m})\in \mathbb{R}^{m}\)be such that it satisfies (3) and \(G_{d} \) (\(d=1,2,3,4\)) be as defined in (15)–(18) respectively. Then the following two assertions are equivalent:

  1. (i)

    If \(f : [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\)is a continuous convex function, we have

    $$\begin{aligned} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, f( \cdot ) \bigr) \geq 0. \end{aligned}$$
    (24)
  2. (ii)

    For \(s \in [\zeta _{1}, \zeta _{2}]\), the following inequality holds:

    $$\begin{aligned} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr)\geq 0, \quad d= 1, 2, 3, 4. \end{aligned}$$
    (25)

Proof

Let assertion (i) hold. Then \(G_{d}(\cdot , s)\) (\(s \in [\zeta _{1}, \zeta _{2}]\)), being continuous and convex, for fixed \(d=1, 2, 3, 4\) satisfies inequality (24), i.e., inequality (25) holds.

On the other hand, let assertion (ii) hold and \(f: [\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\) be a convex function such that \(f\in C^{2} ([\zeta _{1}, \zeta _{2}] )\). Then we can write the function f in the forms (14), (19), (20), and (21) for Green’s functions \(G_{d}\), \(d=1,2,3,4\), respectively. Hence using (3) and performing simple calculations, for all \(s \in [\zeta _{1}, \zeta _{2}]\), we have

$$\begin{aligned} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, f( \cdot ) \bigr)= \int _{\zeta _{1}}^{\zeta _{2}}\mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr)f''(s) \,ds, \quad d= 1, 2, 3, 4. \end{aligned}$$
(26)

Since f is convex, \(f^{\prime \prime }(s)\geq 0\) for all \(s \in [\zeta _{1}, \zeta _{2}]\). Also, inequality (25) holds, so from (26) we get inequality (24).

One must note that in this proof, the demand for the existence of the second derivative of f is not necessary ([39], page 172). We can directly eliminate this condition because it is possible to approximate uniformly continuous convex functions by convex polynomials. □

The following theorem gives weighted majorization difference by using extension of the Montgomery identity and newly defined Green’s functions.

Theorem 5

Let all the assumptions of Theorem 2hold. Let \(f: I \rightarrow \mathbb{R}\)be such that \(f^{(n-1)}\)is absolutely continuous, where \(n\in \mathbb{N} \) (\(n\geq 3\)) and \(I\subset \mathbb{R}\)is an open interval. Then, for \(\zeta _{1}, \zeta _{2}\in I\)with \(\zeta _{1}<\zeta _{2}\)and for all \(s\in [\zeta _{1}, \zeta _{2}]\), we have the following identities:

$$\begin{aligned} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, f(\cdot ) \bigr) &= \sum_{k=1}^{n-1}\frac{k}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr) \\ &\quad{} \times \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}}\,ds \\ &\quad{} + \frac{1}{(n-3)!} \int _{\zeta _{1}}^{\zeta _{2}} f^{(n)}(t) \biggl( \int _{\zeta _{1}}^{\zeta _{2}} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr)\widehat{T}_{n-2}(s, t)\,ds \biggr) \,dt, \end{aligned}$$
(27)

where

$$ \widehat{T}_{n-2}(s, t)= \textstyle\begin{cases} \frac{1}{\zeta _{2}-\zeta _{1}} [\frac{(s-t)^{n-2}}{n-2}+(s- \zeta _{1})(s-t)^{n-3} ] ,& \zeta _{1} \leq t \leq s, \\ \frac{1}{\zeta _{2}-\zeta _{1}} [\frac{(s-t)^{n-2}}{n-2}+(s- \zeta _{2})(s-t)^{n-3} ] ,& s< t \leq \zeta _{2}, \end{cases} $$
(28)

and \(G_{d}\) (\(d=1,2,3,4\)) are Green’s functions defined in (15)–(18) respectively. Moreover, we have

$$\begin{aligned} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, f(\cdot ) \bigr) &= \sum_{k=1}^{n-1}\frac{k-2}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr) \\ &\quad {} \times \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}}\,ds \\ &\quad {} + \frac{1}{(n-3)!} \int _{\zeta _{1}}^{\zeta _{2}} f^{(n)}(t) \biggl( \int _{\zeta _{1}}^{\zeta _{2}} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr)T_{n-2}(s, t) \,ds \biggr) \,dt, \end{aligned}$$
(29)

where \(T_{n-2}\)is as defined in (11).

Proof

Using identities (14), (19), (20), and (21), for fixed \(d=1,2,3,4\), into weighted majorization difference (22), we get

$$\begin{aligned} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, f( \cdot ) \bigr)= \int _{\zeta _{1}}^{\zeta _{2}}\mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr)f''(s) \,ds. \end{aligned}$$
(30)

Now, using an extension of the Montgomery identity given in (10) for the function \(f(s)\) and after differentiating it twice with respect to s, we get

$$\begin{aligned} f''(s) &=\sum_{k=1}^{n-1} \frac{k}{(k-1)!} \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}} \\ &\quad {} + \frac{1}{(n-3)!} \int _{\zeta _{1}}^{\zeta _{2}}\widehat{T}_{n-2}(s, t)f^{(n)}(t)\,dt. \end{aligned}$$
(31)

Using (31) in (30), we have

$$\begin{aligned} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, f(\cdot ) \bigr) &= \sum_{k=1}^{n-1}\frac{k}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr) \\ &\quad {} \times \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}}\,ds \\ &\quad {} + \frac{1}{(n-3)!} \int _{\zeta _{1}}^{\zeta _{2}} \mathbb{D} \bigl( \mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr) \biggl( \int _{\zeta _{1}}^{\zeta _{2}} \widehat{T}_{n-2}(s, t)f^{(n)}(t)\,dt \biggr) \,ds. \end{aligned}$$
(32)

Applying Fubini’s theorem in the last term of (32), we get (27).

Also, replacing f by \(f''\) and n by \(n-2\) (\(n\geq 3\)) in (10) and then rearranging indices, we have

$$\begin{aligned} f''(s) &=\frac{f'(\zeta _{2})-f'(\zeta _{1})}{\zeta _{2}-\zeta _{1}}+\sum _{k=3}^{n-1} \frac{k-2}{(k-1)!} \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}} \\ &\quad {} + \frac{1}{(n-3)!} \int _{\zeta _{1}}^{\zeta _{2}}T_{n-2}(s, t)f^{(n)}(t) \,dt, \end{aligned}$$
(33)

which can also be written as

$$\begin{aligned} f''(s) &=\sum_{k=1}^{n-1} \frac{k-2}{(k-1)!} \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}} \\ &\quad {} + \frac{1}{(n-3)!} \int _{\zeta _{1}}^{\zeta _{2}}T_{n-2}(s, t)f^{(n)}(t) \,dt. \end{aligned}$$
(34)

Using (34) in (30) and then applying Fubini’s theorem, we obtain (29). □

An integral version of Theorem 5 is as follows.

Theorem 6

Let all the assumptions of Theorem 3hold. Let \(f: I \rightarrow \mathbb{R}\)be such that \(f^{(n-1)}\)is absolutely continuous, where \(n\in \mathbb{N} \) (\(n\geq 3\)) and \(I\subset \mathbb{R}\)is an open interval. Then, for \(\zeta _{1}, \zeta _{2}\in I\)with \(\zeta _{1}<\zeta _{2}\)and for all \(s\in [\zeta _{1}, \zeta _{2}]\), we have the following identities:

$$\begin{aligned} \widetilde{\mathbb{D}} \bigl(\phi , \psi , p, f(\cdot ) \bigr) &= \sum _{k=1}^{n-1}\frac{k}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \widetilde{\mathbb{D}} \bigl(\phi , \psi , p, G_{d}(\cdot , s) \bigr) \\ &\quad {} \times \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}}\,ds \\ &\quad {} + \frac{1}{(n-3)!} \int _{\zeta _{1}}^{\zeta _{2}} f^{(n)}(t) \biggl( \int _{\zeta _{1}}^{\zeta _{2}}\widetilde{\mathbb{D}} \bigl( \phi , \psi , p, G_{d}(\cdot , s) \bigr)\widehat{T}_{n-2}(s, t)\,ds \biggr) \,dt, \end{aligned}$$
(35)

where \(\widehat{T}_{n}\)is as defined in (28)and \(G_{d}\) (\(d=1,2,3,4\)) are the Green’s functions defined in (15)–(18) respectively. Moreover,

$$\begin{aligned} \widetilde{\mathbb{D}} \bigl(\phi , \psi , p, f(\cdot ) \bigr) &= \sum _{k=1}^{n-1}\frac{k-2}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \widetilde{\mathbb{D}} \bigl(\phi , \psi , p, G_{d}(\cdot , s) \bigr) \\ &\quad {} \times \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}}\,ds \\ &\quad {} + \frac{1}{(n-3)!} \int _{\zeta _{1}}^{\zeta _{2}} f^{(n)}(t) \biggl( \int _{\zeta _{1}}^{\zeta _{2}}\widetilde{\mathbb{D}} \bigl( \phi , \psi , p, G_{d}(\cdot , s) \bigr)T_{n-2}(s, t)\,ds \biggr) \,dt, \end{aligned}$$
(36)

where \(T_{n-2}\)is as defined in (11).

Proof

Using identities (14), (19), (20), and (21), for fixed \(d=1,2,3,4\), into the integral weighted majorization difference (23) and following similar steps as in the proof of Theorem 5, we get required results. □

A refinement of the weighted majorization-type inequality is presented in the following theorem.

Theorem 7

Let all the assumptions of Theorem 5hold. Let \(f: I\rightarrow \mathbb{R}\)be an n-convex function. If, for \(d=1,2,3,4\),

$$\begin{aligned} \int _{\zeta _{1}}^{\zeta _{2}}\mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr)\widehat{T}_{n-2}(s, t)\,ds \geq 0 \quad \textit{for all }t \in [\zeta _{1}, \zeta _{2}], \end{aligned}$$
(37)

then

$$\begin{aligned} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, f(\cdot ) \bigr) &\geq \sum_{k=1}^{n-1}\frac{k}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr) \\ &\quad {} \times \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}}\,ds. \end{aligned}$$
(38)

Moreover, if

$$\begin{aligned} \int _{\zeta _{1}}^{\zeta _{2}}\mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr)T_{n-2}(s, t) \,ds \geq 0 \quad \textit{for all } t \in [\zeta _{1}, \zeta _{2}], \end{aligned}$$
(39)

then

$$\begin{aligned} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, f(\cdot ) \bigr) &\geq \sum_{k=1}^{n-1}\frac{k-2}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr) \\ &\quad{} \times \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}}\,ds. \end{aligned}$$
(40)

If we reverse the sign of inequalities in (37) and (39), then inequalities (38) and (40) are also reversed.

Proof

As f is an n-convex function, it follows that \(f^{(n)}\geq 0\) (see [39], page 19 and page 293). Using this fact and substituting (37) and (39) in (27) and (29), respectively, we get the desired results. □

An integral version of Theorem 7 is as follows.

Theorem 8

Let all the assumptions of Theorem 6hold. Let \(f: I\rightarrow \mathbb{R}\)be an n-convex function. If, for \(d=1,2,3,4\),

$$\begin{aligned} \int _{\zeta _{1}}^{\zeta _{2}}\widetilde{\mathbb{D}} \bigl(\phi , \psi , p, G_{d}(\cdot , s) \bigr)\widehat{T}_{n-2}(s, t)\,ds \geq 0 \quad \textit{for all }t \in [\zeta _{1}, \zeta _{2}], \end{aligned}$$
(41)

then

$$\begin{aligned} \widetilde{\mathbb{D}} \bigl(\phi , \psi , p, f(\cdot ) \bigr) &\geq \sum_{k=1}^{n-1}\frac{k}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \widetilde{\mathbb{D}} \bigl(\phi , \psi , p, G_{d}(\cdot , s) \bigr) \\ &\quad {} \times \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}}\,ds. \end{aligned}$$
(42)

Moreover, if

$$\begin{aligned} \int _{\zeta _{1}}^{\zeta _{2}}\widetilde{\mathbb{D}} \bigl(\phi , \psi , p, G_{d}(\cdot , s) \bigr)T_{n-2}(s, t)\,ds \geq 0 \quad \textit{for all } t \in [\zeta _{1}, \zeta _{2}], \end{aligned}$$
(43)

then

$$\begin{aligned} \widetilde{\mathbb{D}} \bigl(\phi , \psi , p, f(\cdot ) \bigr) &\geq \sum_{k=1}^{n-1}\frac{k-2}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \widetilde{\mathbb{D}} \bigl(\phi , \psi , p, G_{d}(\cdot , s) \bigr) \\ &\quad {} \times \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}}\,ds. \end{aligned}$$
(44)

If we reverse the sign of inequalities in (41) and (43), then inequalities (42) and (44) are also reversed.

Proof

Using (41) and (43) in (35) and (36) respectively and following similar steps as in the proof of Theorem 7, we get required results. □

Theorem 9

Let all the assumptions of Theorem 5be true. If f is n-convex, where n is even, then inequalities (38) and (40) hold.

Proof

Since \(G_{d}\) is continuous as well as convex for \(d= 1, 2, 3, 4\), therefore from Theorem 2 we can write

$$ \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr)\geq 0. $$
(45)

Note that, when \(n-2\) is even, \(\widehat{T}_{n-2}(s, t)\) and \(T_{n-2}(s, t)\) are nonnegative, so (37) and (39) hold. Now, using Theorem 7, we get the required results. □

An integral version of Theorem 9 is as follows.

Theorem 10

Let all the assumptions of Theorem 6be true. If f is n-convex, where n is even, then inequalities (42) and (44) hold.

Proof

Similar to the proof of Theorem 9. □

The following corollary gives a generalized majorization theorem, i.e., Fuchs’s theorem for n-convex functions.

Corollary 1

Let all the assumptions of Theorem 9be true. If the functions \(\mathcal{F}_{1}, \mathcal{F}_{2}:[\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\), given by

$$\begin{aligned} \mathcal{F}_{1}(\cdot )= \sum _{k=1}^{n-1}\frac{k}{(k-1)!} \int _{ \zeta _{1}}^{\zeta _{2}}G_{d}(\cdot , s) \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}}\,ds \end{aligned}$$
(46)

and

$$\begin{aligned} \mathcal{F}_{2}(\cdot )= \sum _{k=1}^{n-1}\frac{k-2}{(k-1)!} \int _{ \zeta _{1}}^{\zeta _{2}}G_{d}(\cdot , s) \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}}\,ds, \end{aligned}$$
(47)

are convex, then the right-hand sides of (42) and (44) are nonnegative, i.e., (4) is satisfied.

Proof

Note that inequalities (38) and (40) can be written as follows:

$$\begin{aligned} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, f(\cdot) \bigr)\geq \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, \mathcal{F}_{i}(\cdot ) \bigr), \quad i=1, 2. \end{aligned}$$
(48)

Now, the use of convex functions \(\mathcal{F}_{i}\), \(i=1, 2\), in (4) lead us to the nonnegativity of the right-hand side of (48), which gives the required result. □

Remark 2

As given for previous theorems, we can obtain an integral version of Corollary 1, which is a generalization of the integral majorization theorem.

Remarks 1

  1. (i)

    We can obtain upper bounds like Grüss- and Ostrowski-type inequalities for our obtained generalized identities. We can also present Lagrange and Cauchy-type mean value theorems by using linear functionals deduced from our generalized results (see for example [29, 38, 44]).

  2. (ii)

    We can use an elegant method introduced by Jakšetić and Pečarć [18, 19] (see also [23, 34]) to give n-exponential convexity, exponential convexity, and log-convexity, with the help of linear functionals deduced from our generalized results, on a given family with the same property for both discrete and integral cases. For more details, see [38].

3 Csiszár f-divergence for majorization

This section belongs to the study of generalized majorization-type inequality (38) in the form of divergences and entropies. We use Csiszár f-divergence and generalized majorization-type inequalities to obtain new generalized results. Moreover, results related to the Shannon entropy and the Kullback–Leibler (K–L) distance are also discussed.

The following notion of f-divergence was introduced by Csiszár in [12]. For more details, see [13].

Definition 4

Let \(f:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\) be a convex function. If \(\mathbf {r}= (r_{1},\ldots, r_{m} )\) and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\) are two positive probability distributions, then the f-divergence functional is

$$ I_{f}(\mathbf {r}, \mathbf {w}):=\sum _{i=1}^{m}w_{i}f \biggl( \frac{r_{i}}{w_{i}} \biggr). $$

Note that in the f-divergence functional, nonnegative probability distributions can also be used by defining

$$ f(0):=\lim_{t\rightarrow 0^{+}}f(t); \quad \quad 0f \biggl( \frac{0}{0} \biggr):=0; \quad \quad 0f \biggl(\frac{a}{t} \biggr):=0, \quad a>0. $$

In [17], Horváth et al. considered the following functionality based on the previous definition.

Definition 5

Let \(J\subset \mathbb{R}\) be an interval and \(f: J \rightarrow \mathbb{R}\) be an n-convex function. Let \(\mathbf {r}= (r_{1},\ldots, r_{m} )\in \mathbb{R}^{m}\) and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\in \mathbb{R}_{+}^{m}\) such that \(\frac{r_{i}}{w_{i}}\in J\), \(i=1, 2,\ldots, m\). Then

$$ \widetilde{I}_{f}(\mathbf {r}, \mathbf {w}):=\sum _{i=1}^{m}w_{i}f \biggl( \frac{r_{i}}{w_{i}} \biggr). $$

Let \(\mathbf {r}= (r_{1},\ldots, r_{m} )\) and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\) be two m-tuples. Onwards now, we use the following notations in this article, i.e.,

$$ \frac{\mathbf {r}}{\mathbf {w}}:= \biggl(\frac{r_{1}}{w_{1}}, \frac{r_{2}}{w_{2}},\ldots, \frac{r_{m}}{w_{m}} \biggr) \quad \text{and} \quad \widetilde{I}_{G_{d}}( \mathbf {r}, \mathbf {w}, s):=\sum_{i=1}^{m} w_{i}G_{d} \biggl(\frac{r_{i}}{w_{i}} ,s \biggr). $$

The following theorem connects the generalized majorization-type inequality given in Theorem 9 and Csiszár f-divergence.

Theorem 11

Let \(f: I \rightarrow \mathbb{R}\)be such that \(f^{(n-1)}\)is absolutely continuous, where \(n\in \mathbb{N}\) (\(n> 3\)) and \(I\subset \mathbb{R}\)is an open interval. Let \(G_{d}\) (\(d=1,2,3,4\)) be as defined in (15)–(18) respectively. Also, let \(\mathbf {q}= (q_{1},\ldots, q_{m} )\), \(\mathbf {r}= (r_{1},\ldots, r_{m} )\in \mathbb{R}^{m}\), and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\in \mathbb{R}_{+}^{m}\). Let

$$ \sum_{i=1}^{\lambda }r_{i} \leq \sum_{i=1}^{\lambda }q_{i} $$
(49)

for \(\lambda =1, 2,\ldots, m-1\)and

$$ \sum_{i=1}^{m} r_{i} = \sum_{i=1}^{m} q_{i}, $$
(50)

with \(\frac{q_{i}}{w_{i}}, \frac{r_{i}}{w_{i}}\in I\) (\(i=1, 2,\ldots, m\)). If \(\frac{\mathbf {q}}{\mathbf {w}}\)and \(\frac{\mathbf {r}}{\mathbf {w}}\)are decreasing and f is an n-convex function for \(n=\mathrm{even}\) (\(n>3\)), then

$$\begin{aligned} \widetilde{I}_{f}(\mathbf {q}, \mathbf {w}) &\geq \widetilde{I}_{f}(\mathbf {r}, \mathbf {w})+ \sum _{k=1}^{n-1} \frac{k}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \bigl(\widetilde{I}_{G_{d}}( \mathbf {q}, \mathbf {w}, s)-\widetilde{I}_{G_{d}}(\mathbf {r}, \mathbf {w}, s) \bigr) \\ &\quad{} \times \frac{f^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-f^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}- \zeta _{1}}\,ds. \end{aligned}$$
(51)

Proof

Take \(x_{i}=\frac{q_{i}}{w_{i}}\), \(y_{i}=\frac{r_{i}}{w_{i}}\), and \(p_{i}=w_{i}>0 \) (\(i=1, 2,\ldots, m\)), then conditions (49) and (50) imply that conditions (2) and (3) hold. So, using these substitutions in (38), we get (51). □

Theorem 12

Let \(g:I\rightarrow \mathbb{R}\)be a function. If, for \(f(x):=xg(x)\), \(x\in I\), all the conditions of Theorem 11hold, then

$$\begin{aligned} \widehat{I}_{g}(\mathbf {q}, \mathbf {w}):={} &\sum _{i=1}^{m}q_{i}g \biggl( \frac{q_{i}}{w_{i}} \biggr) \\ \geq {}&\widehat{I}_{g}(\mathbf {r}, \mathbf {w})+ \sum_{k=1}^{n-1} \frac{k}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \bigl( \widetilde{I}_{G_{d}}( \mathbf {q}, \mathbf {w}, s)-\widetilde{I}_{G_{d}}( \mathbf {r}, \mathbf {w}, s) \bigr) \\ &{} \times \frac{(xg)^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-(xg)^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}- \zeta _{1}}\,ds. \end{aligned}$$
(52)

Proof

Following the proof of Theorem 11 for \(f(x):=xg(x)\), we get (52). □

The notion of entropic measure of disorder and the theory of majorization are closely related. Next we present two special cases for majorization relations with the connection to entropic inequalities.

In the first case we discuss a generalized majorization-type inequality with the entropy of a discrete probability distribution.

Definition 6

Let \(\mathbf {r}= (r_{1},\ldots, r_{m} )\) be a positive probability distribution. Then the Shannon entropy of r is defined as follows:

$$ S(\mathbf {r}):=-\sum_{i=1}^{m}r_{i} \log r_{i}. $$

Note that the definition does not provide any problem for the zero probability case, because \(\lim_{x\rightarrow 0}x\log x=0\).

Corollary 2

Let \(\mathbf {q}= (q_{1},\ldots, q_{m} )\), \(\mathbf {r}= (r_{1},\ldots, r_{m} )\in \mathbb{R}_{+}^{m}\), and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\)be a positive probability distribution such that conditions (49) and (50) hold with \(\frac{q_{i}}{w_{i}}, \frac{r_{i}}{w_{i}}\in I\) (\(i=1, 2,\ldots, m\)). If log has base b greater than 1 and \(\frac{\mathbf {q}}{\mathbf {w}}\)and \(\frac{\mathbf {r}}{\mathbf {w}}\)are decreasing, then for the Shannon entropy of w, the following estimate holds:

$$\begin{aligned} S(\mathbf {w}) &\leq \sum_{i=1}^{m}w_{i} \log \biggl(\frac{r_{i}}{w_{i}} \biggr)- \sum_{k=1}^{n-1} \frac{k}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \bigl(\widetilde{I}_{G_{d}}( \mathbf {q}, \mathbf {w}, s)-\widetilde{I}_{G_{d}}( \mathbf {r}, \mathbf {w}, s) \bigr) \\ &\quad {} \times \frac{1}{\zeta _{2}-\zeta _{1}} \biggl( \frac{(-1)^{k}(k-1)!}{\zeta _{1}^{k}\ln b}(s-\zeta _{1})^{k-1}- \frac{(-1)^{k}(k-1)!}{\zeta _{2}^{k}\ln b} (s-\zeta _{2})^{k-1} \biggr)\,ds. \end{aligned}$$
(53)

If log has base b between 0 and 1, then inequality (53) is reversed.

Proof

Take \(f(x):=-\log x\), which is an n-convex function for \(n=\mathrm{even}\) (\(n>3\)) and \(q_{i}=1\) (\(i=1, 2,\ldots, m\)). Then, by using Theorem 11, we get (53). Moreover, for \(n=\mathrm{odd}\) (\(n>3\)), the inequality in (53) is reversed. □

Corollary 3

Let \(\mathbf {q}= (q_{1},\ldots, q_{m} )\)and \(\mathbf {r}= (r_{1},\ldots, r_{m} )\)be two positive probability distributions such that conditions (49) and (50) hold with \(q_{i}\), \(r_{i}\in I\) (\(i=1, 2,\ldots, m\)). If log has base b greater than 1 and q and r are decreasing, then the relation between the Shannon entropies of q and r is given by the following estimate:

$$\begin{aligned} S(\mathbf {q}) &\leq S(\mathbf {r})- \sum_{k=1}^{n-1} \frac{k}{(k-1)!} \int _{\zeta _{1}}^{ \zeta _{2}} \bigl(\widetilde{I}_{G_{d}}( \mathbf {q}, \mathbf {w}, s)- \widetilde{I}_{G_{d}}(\mathbf {r}, \mathbf {w}, s) \bigr) \\ &\quad {} \times \frac{(x\log x)^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-(x\log x)^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}}\,ds, \end{aligned}$$
(54)

where for \(u=1, 2\), \((x\log x)'(\zeta _{u})=\frac{1}{\ln b}(1+\ln \zeta _{u})\)and \((x\log x)^{(k)}(\zeta _{u})= \frac{(-1)^{k}(k-2)!}{\zeta _{u}^{k-1}\ln b}\), \(k\geq 2\). If log has base b between 0 and 1, then inequality (54) is reversed.

Proof

Take \(g(x):=\log x\) so that \(xg(x):=x\log x\) is an n-convex function for \(n=\mathrm{even}\) (\(n>3\)) and \(w_{i}=1 \) (\(i=1, 2,\ldots, m\)). Then, by using Theorem 12, we get (54). Moreover, for \(n=\mathrm{odd}\) (\(n>3\)), the inequality in (54) is reversed. □

In the second case we study a generalized majorization-type inequality in terms of the K–L distance or relative entropy between two probability distributions.

Definition 7

Let \(\mathbf {r}= (r_{1},\ldots, r_{m} )\) and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\) be two positive probability distributions. Then the K–L distance between them is defined by

$$ L(\mathbf {r}, \mathbf {w}):=\sum_{i=1}^{m}r_{i} \log \biggl( \frac{r_{i}}{w_{i}} \biggr). $$

Corollary 4

Let \(\mathbf {q}= (q_{1},\ldots, q_{m} )\), \(\mathbf {r}= (r_{1},\ldots, r_{m} )\), \(\mathbf {w}= (w_{1},\ldots, w_{m} ) \in \mathbb{R}_{+}^{m}\)such that conditions (49) and (50) hold with \(\frac{q_{i}}{w_{i}}, \frac{r_{i}}{w_{i}}\in I \) (\(i=1, 2,\ldots, m\)). If log has base b greater than 1 and \(\frac{\mathbf {q}}{\mathbf {w}}\)and \(\frac{\mathbf {r}}{\mathbf {w}}\)are decreasing, then

$$\begin{aligned} & \sum_{i=1}^{m}w_{i}\log \biggl( \frac{q_{i}}{w_{i}} \biggr) \\ &\quad \leq \sum_{i=1}^{m}w_{i} \log \biggl(\frac{r_{i}}{w_{i}} \biggr)- \sum_{k=1}^{n-1} \frac{k}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \bigl(\widetilde{I}_{G_{d}}( \mathbf {q}, \mathbf {w}, s)-\widetilde{I}_{G_{d}}(\mathbf {r}, \mathbf {w}, s) \bigr) \\ &\quad \quad {} \times \frac{1}{\zeta _{2}-\zeta _{1}} \biggl( \frac{(-1)^{k}(k-1)!}{\zeta _{1}^{k}\ln b}(s-\zeta _{1})^{k-1}- \frac{(-1)^{k}(k-1)!}{\zeta _{2}^{k}\ln b} (s-\zeta _{2})^{k-1} \biggr)\,ds. \end{aligned}$$
(55)

If log has base b between 0 and 1, then inequality (55) is reversed.

Proof

Take \(f(x):=-\log x\), which is an n-convex function for \(n=\mathrm{even}\) (\(n>3\)). Then, by Theorem 11, we get (55). Moreover, for \(n=\mathrm{odd}\) (\(n>3\)), the inequality in (55) is reversed. □

Corollary 5

Let \(\mathbf {q}= (q_{1},\ldots, q_{m} )\), \(\mathbf {r}= (r_{1},\ldots, r_{m} )\), and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\)be positive probability distributions such that conditions (49) and (50) hold with \(\frac{q_{i}}{w_{i}}, \frac{r_{i}}{w_{i}}\in I\) (\(i=1, 2,\ldots, m\)). If log has base b greater than 1 and \(\frac{\mathbf {q}}{\mathbf {w}}\)and \(\frac{\mathbf {r}}{\mathbf {w}}\)are decreasing, then the relation between the K–L distance of \((\mathbf {r}, \mathbf {w})\)and \((\mathbf {q}, \mathbf {w})\)is given by the following estimate:

$$\begin{aligned} L(\mathbf {q}, \mathbf {w}) &\geq L(\mathbf {r}, \mathbf {w})+ \sum _{k=1}^{n-1}\frac{k}{(k-1)!} \int _{\zeta _{1}}^{\zeta _{2}} \bigl(\widetilde{I}_{G_{d}}( \mathbf {q}, \mathbf {w}, s)-\widetilde{I}_{G_{d}}(\mathbf {r}, \mathbf {w}, s) \bigr) \\ &\quad {} \times \frac{(x\log x)^{(k)}(\zeta _{1})(s-\zeta _{1})^{k-1}-(x\log x)^{(k)}(\zeta _{2})(s-\zeta _{2})^{k-1}}{\zeta _{2}-\zeta _{1}}\,ds, \end{aligned}$$
(56)

where for \(u=1, 2\), \((x\log x)'(\zeta _{u})=\frac{1}{\ln b}(1+\ln \zeta _{u})\)and \((x\log x)^{(k)}(\zeta _{u})= \frac{(-1)^{k}(k-2)!}{\zeta _{u}^{k-1}\ln b}\), \(k\geq 2\). If log has base b between 0 and 1, then inequality (56) is reversed.

Proof

Take \(g(x):=\log x\) so that \(xg(x):=x\log x\) is an n-convex function for \(n=\mathrm{even}\) (\(n>3\)). Then, by using Theorem 12, we get (56). Moreover, for \(n=\mathrm{odd}\) (\(n>3\)), the inequality in (56) is reversed. □

Remark 3

In Sect. 3, we use generalized majorization-type inequality (38) to obtain results in terms of the Shannon entropy and the K–L distance. Following the same way, we can also give all these results related to the Shannon entropy and the K–L distance by using the generalized majorization-type inequality given in (40).