论文标题
涉及Riemann Zeta功能的功能的属性和对数凸的增加
Increasing property and logarithmic convexity of functions involving Riemann zeta function
论文作者
论文摘要
令$α> 0 $为常数，令$ \ ell \ ge0 $为整数，让$γ（z）$表示经典的Euler gamma函数。借助Riemann Zeta函数$ζ（z）$的积分表示，借助单调性规则，其比率是两个具有参数的积分的比率，并且通过完整的单调性和函数的另一个属性$ \ frac {1} {1} {e^t-1} $及其衍生物，作者及其衍生物，呈现的作者呈现，该属性呈现，该属性呈现，该功能，该属性，该属性，该功能，该属性，该属性，该属性，该属性，该函数的衍生物属性 (1) for $\ell\ge0$, the function \begin{equation*} x\mapsto\binom{x+α+\ell}α\frac{ζ(x+α)}{ζ(x)} \end{equation*} is increasing from $(1,\infty)$ onto $(0,\infty)$, where $ \ binom {z} {w} $表示扩展的二项式系数; （2）对于$ \ ell \ ge1 $，函数$ x \mapstoγ（x+\ ell）ζ（x）$是$（1，\ infty）$的对数凸上。
Let $α>0$ be a constant, let $\ell\ge0$ be an integer, and let $Γ(z)$ denote the classical Euler gamma function. With the help of the integral representation for the Riemann zeta function $ζ(z)$, 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+α+\ell}α\frac{ζ(x+α)}{ζ(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 $x\mapstoΓ(x+\ell)ζ(x)$ is logarithmically convex on $(1,\infty)$.
