Monotonicity and absolute convexity of two functions involving Riemann zeta function

2022/01/12 by Bai-Ni Guo, Bai‐Ni Guo, Feng Qi +2
Mathematics · #11M41 #26A48 #26A51 #33B15 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Inequalities and Applications #Number Theory (math.NT) #Primary 11M06 #Secondary 11B73 #math.CA #math.NT #msc:11B73 #msc:11M06 #msc:11M41 #msc:26A48 #msc:26A51 #msc:33B15

paper · pdf · doi:10.48550/arxiv.2201.06970

9 pages

openalex publication_date 2022/01/12 · openalex created_date 2025/10/10 · arxiv created 2026/07/30 · openalex updated_date 2026/08/01 · arxiv updated 2026/08/03

Abstract

Let ρ>0 be a constant, let j≥0 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 (1)/(e^t-1) and its derivatives, the authors discuss the increasing monotonicity of the function t↦\binomt+ρ+jρ(ζ(t+ρ))/(ζ(t)), study the absolute convexity and logarithmic convexity of the function t↦Γ(t+j)ζ(t), and derive the increasing monotonicity and inequalities of some sequences involving the ratios |\fracB_2n+2 B_2n| of the Bernoulli numbers B_2n, where \binomzz denotes the extended binomial coefficient.

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