Some Inequalities Arising from Analytic Summability of Functions

Analytic summability of functions was introduced by the second author in 2016. He utilized Bernoulli numbers and polynomials for a holomorphic function to construct analytic summability. The analytic summand function f(sigma) (if exists) satisfies the difference functional equation f(sigma) (z) = f (z) + f(sigma)(z - 1). 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 Bernoulli polynomials, sums of powers of natural numbers, (e.g., 1(p) + 2(p) + 3(p) + ... + r(p) <= 2p!/pi(p+1)(e(pi r) - 1)) and several inequalities for exponential, hyperbolic and trigonometric functions.