Monotonicity and absolute convexity of two functions involving Riemann zeta function

Abstract

Let $\rho>0$ be a constant, let $j\ge0$ 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 $\frac{1}{\operatorname{e}^t-1}$ and its derivatives, the authors discuss the increasing monotonicity of the function $t\mapsto\binom{t+\rho+j}{\rho}\frac{\zeta(t+\rho)}{\zeta(t)}$, where $\binom{z}{w}$ 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 $\bigl|\frac{B_{2n+2}} {B_{2n}}\bigr|$ of the Bernoulli numbers $B_{2n}$.