Monotonicity and inequalities for ratios of Bernoulli polynomials and numbers

Abstract

In the work, the authors establish the monotonicity of the ratios $$\frac{B_{2\ell -1}(s)}{B_{2\ell +1}(s)},\frac{B_{2\ell }(s)}{B_{2\ell +1}(s)},\frac{B_{2m}(s)}{B_{2\ell }(s)},\frac{B_{2\ell }(s)}{B_{2\ell -1}(s)},$$ where $B_{\ell }(s)$ denotes the Bernoulli polynomials. Using these newly established monotonicity properties, the authors derive several new inequalities and also recover a number of known inequalities involving the Bernoulli polynomials $B_{\ell }(s)$, the Bernoulli numbers $B_{2\ell }$, and their ratios such as $\frac{B_{2\ell +2}}{B_{2\ell }}$.

26 pages