On some generalized number theoretic functions and Ighachanea-Akkouchia Holder's inequalities

TL;DR

This paper generalizes certain number theoretic functions and extends Holder's inequalities to these new functions, building on recent refinements involving binomial coefficients and their applications to special functions.

Contribution

It introduces new generalizations of number theoretic functions and derives Holder's inequalities for these functions, expanding their applicability to special functions.

Findings

Generalized number theoretic functions are established.

Holder's inequalities are extended to these generalized functions.

Applications to special functions like the beta and gamma functions are discussed.

Abstract

Recently, it has been shown by Ighachanea and Akkouchia \cite{0.1} that using binomial coefficients, one can derive some new refinements of Holder's inequalities. This inequalities then can be applied to a wide class of special functions such as the Nielsen's beta function and some extended gamma functions. In this paper, we have derived some generalizations of previously known number theoretic functions. Furthermore, based on the results of Ighachanea and Akkouchia, Holder's inequalities for the derived generalized functions are established.