Refinements of some Hardy–Littlewood–Pólya type inequalities via Green’s functions and Fink’s identity and related results

In this paper, first we present some interesting identities associated with Green’s functions and Fink’s identity, and further we present some interesting inequalities for r-convex functions. We also present refinements of some Hardy–Littlewood–Pólya type inequalities and give an application to the Shannon entropy. Furthermore, we use the Čebyšev functional and Grüss type inequalities and present the bounds for the remainder in the obtained identities. Finally, we use the obtained identities together with Hölder’s inequality for integrals and present Ostrowski type inequalities.


Introduction and preliminaries
The following inequality is given in the well-known book by Hardy, Littlewood, and Pólya holds. If p ∈ (0, 1), then (1.2) holds in the reversed direction (see [1]). In our work we use nonincreasing ( ) sequence in weighted mean (WM) and nondecreasing ( ) sequence in WM (see [11]) defined as follows.
where a i , q i ∈ R (i ∈ N) such that q k > 0 (1 ≤ k ≤ i) and Q i := i k=1 q k (i ∈ N) . If (1.3) holds in the reversed direction, then the sequence {a i } i∈N ⊂ R is called in WM.
If ϑ is concave, then (1.4) and (1.5) hold in the reversed direction. Definition 1.4 Let q = (q 1 , . . . , q m ) be a positive probability distribution. Then the Shannon entropy (see [8,9], and [22]) of q is defined by Khalid, Pečarić, and Pečarić presented the following interesting result associated with the Shannon entropy in [8].
In the second section, we generalize inequalities (1.4), (1.7), and (1.8) for r-convex functions, and we also present refinements of these inequalities.
For G δ (x,ũ) defined in (1.10)-(1.14) and for a twice differentiable function ϑ defined on [s, t], we can consider the following expression: Now if we use the values of G δ (x,ũ) over the intervals [s, x] and [x, t], then the following result is valid (see [4] and [13]).

Lemma 1.7
Let ϑ : [s, t] → R be a function such that ϑ ∈ C 2 ([s, t]), and let G δ be the Green's functions defined in (1.10)-(1.14). Then In the second section, we prove some interesting identities and inequalities for r-convex functions by using Lemma 1.7 and the following Fink's identity (see [5,10], and [12]). Theorem 1.8 Let s, t ∈ R, ϑ : [s, t] → R, r ≥ 1 and ϑ (r-1) be absolutely continuous on [s, t]. Then (1.21) Let ϑ be a real-valued function defined on [s, t]. A criterion to check the r-convexity (r ≥ 0) of a function ϑ is the following: Theorem 1. 9 If ϑ (r) exists, then ϑ is r-convex if and only if ϑ (r) ≥ 0.
In the third section, we present some interesting results by using the following Čebyšev functional and Grüss type inequalities (see [2] and [3]): denote the space of p-power integrable functions and the space of essentially bounded functions defined on [s, t] respectively together with the norms respectively. Suppose that ζ 1 , ζ 2 : [s, t] → R are two Lebesgue integrable functions. The Čebyšev functional is defined by ( 1.22) Cerone and Dragomir proved the next two results related to Grüss type inequalities in [2].
The organization of this paper is as follows: in the second section, we obtain some interesting identities related to Green's functions and Fink's result. Further, we use these identities and generalize inequalities of kind (1.4), (1.7), and (1.8) for r-convex functions. In addition, we also present refinements of these inequalities and give an application to the Shannon entropy. In the third section, we use the Čebyšev functional and Grüss type inequalities and find the new bounds for the remainder in the obtained identities. In the fourth section, we use the identities from section two together with Hölder's inequality for integrals and obtain Ostrowski type inequalities (see [20] and [21]).

Refinements of some Hardy-Littlewood-Pólya type inequalities and an application to the Shannon entropy
The first main theorem is related to the following identity which will play an important role in our paper.
By taking Fink's identity (1.20), it is obvious that Now interchange the integral and summation in the first term and apply Fubini's theorem in the second term, identity (2.1) is immediate for δ = 1.
We present inequality (1.4) for r-convex functions as follows.

Theorem 2.2 Let all the assumptions of Theorem 2.1 be satisfied, and let for r
If the reversed inequalities hold in (1.4) and (2.6), then statements (i) and (ii) are also equivalent.
Proof The idea of the proof is the same as given in [15].
Let statement (ii) be satisfied, and let ϑ be a twice differentiable convex function. As from Lemma 1.7 the function ϑ can be represented in the forms (1.15)-(1.19), it is easy to see that Now use inequality (2.6) together with ϑ (ũ) ≥ 0 for allũ ∈ [s, t] in (2.7), inequality (1.4) is immediate.
The differentiability condition can be eliminated here as it is possible to approximate uniformly a continuous convex function by convex polynomials (see [23, p. 172]).
We present refinement of inequality (1.4) as follows. (ii) Let (2.5) be satisfied, and let : [s, t] → R be a function defined by If is convex, then the RHS of (2.5) is nonnegative and we have hold for even r such that r > 3 and for odd r such that r ≥ 3 respectively. For u ≤ũ ≤ t, the inequality holds for r ≥ 3.
(i) Inequality (2.9) together with inequality (2.10) yields inequality (2.4) for even r such that r > 3. As ϑ is r-convex for even r such that r > 3, applying Theorem 2.2, we obtain (2.5). (ii) Clearly, inequality (2.5) can be written as (2.11) As the sequence {a i } m i=1 is in WM, replace ϑ by in Theorem 1.3 (i), the nonnegativity of the RHS of (2.11) is immediate, and we obtain (2.8).
An application to the Shannon entropy is the following: Corollary 2.5 Let all the assumptions of Theorem 2.1 be satisfied, and let q = (q 1 , . . . , q m ) be a positive probability distribution. Let r be even such that r > 3, and let ϑ : [s, t] → R be r-convex.

Proof
(i) Take a i = log 1 q i and use Theorem 2.4(i), (2.12) is immediate. (ii) Taking a i = -log 1 q i and following the proof of (i), we obtain (2.13).

Remark 2.6 Special cases when
, q i ≥ 1, also hold.