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.

中文翻译：

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通过格林函数和芬克的恒等式及相关结果完善一些哈代–利特伍德–波利亚类型不等式

在本文中，首先我们给出了一些与Green函数和Fink的恒等式相关的有趣恒等式，然后进一步介绍了r凸函数的一些有趣的不等式。我们还提出了一些Hardy–Littlewood–Pólya型不等式的提法，并将其应用于Shannon熵。此外，我们使用Čebyšev泛函和Grüss型不等式，并给出了获得的恒等式中其余部分的界线。最后，我们将获得的恒等式与Hölder不等式一起用于积分，并给出Ostrowski型不等式。