涉及Riemann zeta函数的两个函数的单调性和绝对凸性
Monotonicity and absolute convexity of two functions involving Riemann zeta function
AI总结：
利用Riemann zeta函数的积分表示、单调性规则及函数1/(e^t-1)的性质，研究两个函数的单调性和凸性，并推导出Bernoulli数比值的单调性和不等式。
摘要：
设$\rho>0$为常数，$j\ge0$为整数，$\Gamma(z)$为Euler gamma函数。借助Riemann zeta函数$\zeta(z)$的积分表示，利用单调性规则以及函数$\frac{1}{e^t-1}$及其导数的某些性质，作者讨论了函数$t\mapsto\binom{t+\rho+j}{\rho}\frac{\zeta(t+\rho)}{\zeta(t)}$的递增单调性，研究了函数$t\mapsto\Gamma(t+j)\zeta(t)$的绝对凸性和对数凸性，并推导了涉及Bernoulli数$B_{2n}$的比值$\bigl|\frac{B_{2n+2}}{B_{2n}}\bigr|$的某些序列的递增单调性和不等式，其中$\binom{z}{z}$表示推广的二项式系数。
英文摘要：
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 $\frac{1}{\operatorname{e}^t-1}$ and its derivatives, the authors discuss the increasing monotonicity of the function $t\mapsto\binom{t+ρ+j}ρ\frac{ζ(t+ρ)}{ζ(t)}$, where $\binom{z}{w}$ denotes the extended binomial coefficient, 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 $B_{2n}$.
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