Abstract
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.
Highlights
We generalize inequalities (1.4), (1.7), and (1.8) for r-convex functions, and we present refinements of these inequalities
We prove some interesting identities and inequalities for r-convex functions by using Lemma 1.7 and the following Fink’s identity
We present inequality (1.4) for r-convex functions as follows
Summary
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]). We present some interesting results by using the following Čebyšev functional and Grüss type inequalities (see [2] and [3]): Let Lp[s, t] (1 ≤ p < ∞) and L∞[s, t] denote the space of p-power integrable functions and the space of essentially bounded functions defined on [s, t] respectively together with the norms ζ p=. Theorem 1.10 Suppose that ζ1 : [s, t] → R is a Lebesgue integrable function and ζ2 : [s, t] → R is an absolutely continuous function such that (· – s)(t – ·)(ζ2)2 ∈ L1[s, t].
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