Monotonicity and inequalities for the ratios of two Bernoulli polynomials

TL;DR

This paper investigates the monotonicity of ratios of Bernoulli polynomials and derives new inequalities involving Bernoulli polynomials, Bernoulli numbers, and their ratios, contributing to the mathematical understanding of these special functions.

Contribution

The authors establish the monotonicity of specific ratios of Bernoulli polynomials and derive new inequalities, expanding the theoretical framework of Bernoulli-related functions.

Findings

Proved monotonicity of ratios of Bernoulli polynomials.

Derived new inequalities for Bernoulli polynomials and numbers.

Connected inequalities to known properties of Bernoulli functions.

Abstract

In the article, the authors establish the monotonicity of the ratios \begin{equation*} \frac{B_{2n-1}(t)}{B_{2n+1}(t)}, \quad \frac{B_{2n}(t)}{B_{2n+1}(t)},\quad \frac{B_{2m}(t)}{B_{2n}(t)},\quad \frac{B_{2n}(t)}{B_{2n-1}(t)} \end{equation*} and derive some known and new inequalities of the Bernoulli polynomials $B_n(t)$, the Bernoulli numbers $B_{2n}$, and their ratios such as $\frac{B_{2n+2}}{B_{2n}}$.