Monotonicity and absolute convexity of two functions involving Riemann zeta function

Abstract

Let $ρ>0$ be a constant, let $j\ge0$ be an integer, and let $Γ(z)$ denote the Euler gamma function. With the aid of the integral representation for the Riemann zeta function $ζ(z)$, 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 $t\mapsto\binom{t+ρ+j}ρ\frac{ζ(t+ρ)}{ζ(t)}$, study the absolute convexity and logarithmic convexity of the function $t\mapstoΓ(t+j)ζ(t)$, 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 $B_{2n}$, where $\binom{z}{z}$ denotes the extended binomial coefficient.