Monotonicity and absolute convexity of two functions involving Riemann zeta function

Abstract

Let $\rho >0$ be a constant, let $j\ge 0$ be an integer, and let $\Gamma (z)$ denote the Euler gamma function. With the aid of the integral representation for the Riemann zeta function $\zeta (z)$, by virtue of a monotonicity rule, and by means of some properties of the function $\frac{1}{e^t-1}$ and its derivatives, the authors discuss the increasing monotonicity of the function $t\mapsto \left(\genfrac{}{}{0ex}{}{t+\rho +j}{\rho }\right)\frac{\zeta (t+\rho )}{\zeta (t)}$, where $\left(\genfrac{}{}{0ex}{}{z}{w}\right)$ denotes the extended binomial coefficient, study the absolute convexity and logarithmic convexity of the function $t\mapsto \Gamma (t+j)\zeta (t)$, and derive the increasing monotonicity and inequalities of some sequences involving the ratios $\left|\frac{B_{2n+2}}{B_{2n}}\right|$ of the Bernoulli numbers $B_{2n}$.

arXiv comment9 pages