Integral inequalities using signed measures corresponding to majorization

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.


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 s := (s 1 , . . . , s n ) ∈ R n and t := (t 1 , . . . , t n ) ∈ R n .
(a) We say that s is weakly majorized by t, written s ≺ w t, if where s [1] ≥ s [2] ≥ · · · ≥ s [n] and t [1] ≥ t [2] ≥ · · · ≥ t [n] are the entries of s and t, respectively, in decreasing order. (b) We say that s is majorized by t, written s ≺ t, if (1) holds, and in addition Theorem 1 Let C ⊂ R be an interval, and let s := (s 1 , . . . , s n ) ∈ C n and t := (t 1 , . . . , t n ) ∈ C n .
(a) Hardy-Littlewood-Pólya inequality (see Hardy, Littlewood and Pólya [6] and Niculescu and Persson [14]) If s ≺ t, then for every convex function f : C → R. Conversely, if inequality (2) holds for every convex function f : C → R, then s ≺ t. (b) (see Niculescu and Persson [14]) If f : C → R is an increasing and convex function and s ≺ w t, then (2) also holds.
Let (X , A) be a measurable space. The unit mass at x ∈ X (the Dirac measure at x) is denoted by ε x . The set of all subsets of X is denoted by P (X ).
By N + we denote the set of positive integers. The previous result can be reformulated by using discrete measures. Let s := (s 1 , . . . , s n ) ∈ C n and t := (t 1 , . . . , t n ) ∈ C n , and consider the discrete measures Among the weighted versions of Hardy-Littlewood-Pólya inequality, we highlight the following inequality by Fuchs [5].
Theorem 2 Let C ⊂ R be an interval, and let f : C → R be a convex function. If (s 1 , . . . , s n ) ∈ C n , (t 1 , . . . , t n ) ∈ C n and q 1 , . . . , q n are real numbers such that (a) s 1 ≥ · · · ≥ s n and t 1 ≥ · · · ≥ t n , Of course, as with the Hardy-Littlewood-Pólya inequality, (3) can be rewritten in the equivalent form where μ and ν are discrete signed measures on P (C).
The above two inequalities and their applicability alone justify the investigation of the following problem: Let (X , A, μ) and (Y , B, ν) be measure spaces, where μ and ν are finite signed measures. Let C ⊂ R be an interval with nonempty interior, and let ϕ : X → C, ψ : Y → C be μ-integrable and ν-integrable functions, respectively. Under which conditions does inequality 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 μ and ν are measures and C = 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 ϕ and ψ are defined on a compact interval X = Y = [a, b] (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 to be valid, where μ and ν are finite signed measures on a σ -algebra containing the Borel sets of [a, b].
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 i∈X f (s i ) p i ≤ j∈Y f t j q j to be satisfied for any convex function f : C → R, where X is either {1, . . . , m} for some m ≥ 1 or N + , Y is either {1, . . . , n} for some n ≥ 1 or N + , (s i ) i∈X and t j j∈Y are sequences from C and ( p i ) i∈X and q j j∈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 C f dμ 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 R. Our last application consists of two parts. We give necessary and sufficient conditions for the satisfaction of inequalities Let C ⊂ R be an interval with nonempty interior. The following notations are introduced for some special functions defined on C: We begin with a simple but essential statement.
Proof We only prove that (b) follows from (a), the converse statement can be handled similarly.
It is easy to check that Thus the conditions μ (X ) = ν (X ), X ϕdμ = X ψdν and (6) imply that The proof is complete.
The next result contains integral inequalities for some special functions.

Lemma 2
Let (X , A) be a measurable space, and let μ and ν be finite measures on A. Assume ϕ ∈ L (μ) and ψ ∈ L (ν) such that (a) If the function f is either id R or p R,w for some w ∈ R, then and X ϕdμ = X ψdν (10) are also satisfied. If f = n R,w for some w ∈ R, then inequality (8) holds too.
Proof (a) Let the functions ϕ n , ψ n : X → R be defined by Then (ϕ n ) n∈N + is a decreasing sequence of μ-integrable functions on X such that ϕ n → ϕ pointwise on X . By the monotone convergence theorem, Since n∈N + {ϕ ≥ −n} = X , ψ n → ψ pointwise on X . It follows from the definition of ψ n that |ψ n | ≤ |ψ| , n ∈ N + .
Based on our previous two findings, the dominated convergence theorem implies that By applying (11), (12) and (7), we obtain that and therefore the result holds for f = id R . Now assume that f = p R,w for some w ∈ R. In this case If {ϕ ≥ w} = ∅, inequality (8) trivially follows from (13), and thus it can be supposed that {ϕ ≥ w} = ∅. Then by the first part of (13) and (7), and therefore it follows from the definition of the set {ψ ≥ w} and from the second part of (13) that The proof is complete.
The following result is a natural complement to the previous statement. (9) and (10) are also satisfied. If f = p C,w for some w ∈ R, then inequality (15) holds too.

Lemma 3 Let (X , A) be a measurable space, and let μ and ν be finite measures on
Proof (a) Under the conditions −ψ ∈ L (ν), −ϕ ∈ L (μ), and and therefore Lemma 2 (a) implies that where f is either id R or p R,w for some w ∈ R. This gives the result by using that Since (−w) − = w + , Lemma 2 (b) can be applied. The proof is complete.

Approximation results
The interior of an interval C ⊂ R is denoted by C • .
We need some approximation results from Horváth [7].
Theorem 3 Let C ⊂ R be an interval with nonempty interior, and let f : C → R be a continuous convex function. Remark 1 Let C ⊂ R be an interval with nonempty interior, and let f : C → 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 For nonnegative convex functions, we can formulate a more precise result than Theorem 3.
Theorem 4 Let C ⊂ R be an interval with nonempty interior, and let f : C → R be a nonnegative, continuous and convex function.
(a) Theorem 3 (a) is satisfied by a sequence whose elements have the form is satisfied by a sequence whose elements have the form is satisfied by a sequence whose elements have the form Since f is increasing and nonnegative, the right-hand limit p of f exists at the left-hand endpoint a ∈ [−∞, ∞[ of C and p ≥ 0. It can be supposed that there exists be the equation of the left-hand tangent line to the graph of f at t 1 . Obviously, f − (t 1 ) > 0, and hence there is exactly one solution t 0 of the equation It is easy to think that either a < t 0 < t 1 or t 0 = a. In the first case while in the second case 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 f Then f 1 and f 2 are nonnegative, continuous and convex function, f 1 is increasing, f 2 is decreasing and and therefore the result follows from (b) and (c).
The proof is complete.

Some recent results
The σ -algebra of Borel sets on an interval C ⊂ R is denoted by B C . The Lebesgue measure on B C is denoted by λ 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 ⊂ R be an interval with nonempty interior, and let f : (a) Suppose that one of the following two conditions is met: is satisfied, then are satisfied, then inequality (17) holds too. (b) Suppose that one of the following two conditions is met:  [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.

Main results
We start by introducing some special function sets: Let C ⊂ R be an interval with nonempty interior, and let (X , A, μ) and (Y , B, ν) be measure spaces, where μ and ν are finite signed measures. Furthermore, let ϕ : X → C, ψ : Y → C be functions such that ϕ ∈ L (μ) and ψ ∈ L (ν).
(a) Let denote F i C the set of all increasing and convex functions on C. We define and F C (ϕ, ψ). With these preliminaries, we can state and prove our main result.

Theorem 6
Let (X , A, μ) and (Y , B, ν) be measure spaces, where μ and ν are finite signed measures. Let C ⊂ R be an interval with nonempty interior, and let ϕ : X → C, ψ : Y → C be functions such that ϕ ∈ L (μ) and ψ ∈ L (ν). Then holds if and only if μ (X ) = ν (Y ) and it is satisfied in the following special cases: the function f is either id C or p C,w (w ∈ C • ).

ψ) inequality (20) holds if and only if μ (X ) ≤ ν (Y )
and it is satisfied in the following special cases: f = n C,w (w ∈ C • ).

(c 1 ) For every f ∈ F C (ϕ, ψ) inequality (20) holds if and only if μ (X ) = ν (Y ) and it is
satisfied in the following special cases: the function f is either id C or −id C or p C,w (w ∈ C • ).

(c 2 ) For every nonnegative f ∈ F C (ϕ, ψ) inequality (20) holds if and only if μ (X ) ≤ ν (Y )
and it is satisfied in the following special cases: the function f is either p C,w or n C,w (w ∈ C • ).
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 ( f n ) is such a sequence, then ( f n • ϕ) is also increasing and converges pointwise to f • ϕ on X . Similarly, ( f n • ψ) is increasing too, and converges pointwise to f • ψ on Y .
By Remark 1 (b), if g is a piecewise linear, increasing and convex function on C, then g is of the form for suitable points t 1 < t 2 < · · · < t k in the interior of C, α ≥ 0, β ∈ R and γ i > 0 (i = 1, . . . , k). Since ϕ ∈ L (μ) and μ is finite, g • ϕ ∈ L (μ). Similarly, g • ψ ∈ L (ν), and hence g ∈ F i C (ϕ, ψ). By applying B. Levi's theorem, we obtain that 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 ( f n ) ∞ n=1 from F i C (ϕ, ψ) such that f n is continuous (n ∈ N + ) and ( f n ) converges pointwise to f on C. In this case the sequences ( f n • ϕ) and ( f n • ψ) 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 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.

Remark 3
The previous result extends Theorem 10 in Horváth [7], where only the following cases have been considered: 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. (X , A) be a measurable space, and let μ and ν be finite measures on A such that μ (X ) = ν (X ). Let C ⊂ R be an interval with nonempty interior, and let ϕ, ψ : X → C be functions such that ϕ ∈ L (μ) and ψ ∈ L (ν).
The proof is complete.

Applications
We first state Theorem 6 for discrete measures.
Theorem 8 Let the set X denote either {1, . . . , m} for some m ≥ 1 or N + , and let the set Y denote either {1, . . . , n} for some n ≥ 1 or N + . Assume ( p i ) i∈X and q j j∈Y are real sequences such that i∈X | p i | < ∞, j∈Y q j < ∞.
Let C ⊂ R be an interval with nonempty interior, and let (s i ) i∈X and t j j∈Y be sequences from C such that i∈X | p i s i | < ∞, j∈Y q j t j < ∞. Then are satisfied.  Then ϕ ∈ L (μ) and ψ ∈ L (ν). Now the result is an immediate consequence of Theorem 6, since ψ). The proof is complete.
When X and Y are finite sets, we can obtain the following important and interesting consequence of the previous theorem.   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 ≤ u l for some 1 ≤ l < k then and Moreover, which imply (26) for all u l+1 < w ≤ u l . The proof is complete.

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 ≥ · · · ≥ s n , t 1 ≥ · · · ≥ t n and p i = q i (i = 1, . . . , n). 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 i C , F d C and F C and their subsets of nonnegative functions.
Our next result is related to Steffensen-Popoviciu measures.
Theorem 10 Let (X , A) be a measurable space, and let μ be a finite signed measure on A. Let C ⊂ R be an interval with nonempty interior, and let ψ : X → C be a μ-integrable function.
holds if and only if  Proof (a) Since μ is a finite signed measure and ψ ∈ L (μ), p C,w • ψ is also μ-integrable for every w ∈ C • . Moreover, f 1 : C → R, f 1 (t) := 1 and p C,w (w ∈ C • ) are nonnegative elements of F i C , 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.
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 ⊂ R be an interval with nonempty interior, and let μ be a finite signed measure on B C such that id C is μ-integrable. Proof Apply Theorem 10 to the measure space (C, B C , μ). The proof is complete.
Our last application is the extension of the classical integral Jensen and Lah-Ribarič inequalities to signed measures.
where μ − denotes the negative part of μ. Since ψ > a, It follows from (50) that 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.
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