Some inequalities arising from analytic summability of functions

Abstract

Analytic summability and  analytic summand functions were introduced  by M.H.Hooshmand in 2016. He utilized  Bernoulli  numbers and polynomials for a holomorphic function to construct analytic summability. The function $ f_\sigma $ (which is called analytic summand function) resembles to a (summable) holomorphic function $f$ on an open domain D and these two functions satisfy special difference functional equation. Moreover, some upper bounds for  $ f_\sigma $ and several inequalities between $f$ and  $ f_\sigma $ were presented by him. In this paper, by using Alzer's improved upper bound for Bernoulli numbers, we  improve those upper bounds and obtain some inequalities and new upper bounds .As some applications of the topic, we obtain several upper bounds for sums of powers of natural numbers, several inequalities for exponential, hyperbolic and trigonometric  functions.