Increasing property and logarithmic convexity of functions involving Riemann zeta function

Abstract: Let $\alpha>0$ be a constant, let $\ell\ge0$ be an integer, and let $\Gamma(z)$ denote the classical Euler gamma function. With the help of the integral representation for the Riemann zeta function $\zeta(z)$, by virtue of a monotonicity rule for the ratio of two integrals with a parameter, and by means of complete monotonicity and another property of the function $\frac{1}{e^t-1}$ and its derivatives, the authors present that, (1) for $\ell\ge0$, the function \begin{equation*} x\mapsto\binom{x+\alpha+\ell}{\alpha}\frac{\zeta(x+\alpha)}{\zeta(x)} \end{equation*} is increasing from $(1,\infty)$ onto $(0,\infty)$, where $\binom{z}{w}$ denotes the extended binomial coefficient; (2) for $\ell\ge1$, the function $x\mapsto\Gamma(x+\ell)\zeta(x)$ is logarithmically convex on $(1,\infty)$.