Log-concavity of two sequences related to Cauchy numbers of two kinds

Abstract

For Cauchy numbers of the first kind $\{a_n{\}}_{n\ge 0}$ and Cauchy numbers of the second kind $\{b_n{\}}_{n\ge 0}$, we prove that two sequences ${\left\{\sqrt[n]{|a_n|}\right\}}_{n\ge 2}$ and ${\left\{\sqrt[n]{b_n}\right\}}_{n\ge 1}$ are log-concave. In addition, we show that two sequences ${\left\{\frac{1}{\sqrt[n]{|a_n|}}\right\}}_{n\ge 2}$ and ${\left\{\frac{1}{\sqrt[n]{b_n}}\right\}}_{n\ge 1}$ are log-balanced.