  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 $(1,\infty)$.
