The Log-convexity of Some Sequences Related to Cauchy Numbers of Two Kinds

Abstract

For Cauchy numbers of the first kind \( \{a_n\}{n \geq 0} \) and Cauchy numbers of the second kind \( \{b_n\}{n \geq 0} \), this paper focuses on the log-convexity of some sequences related to \( \{a_n\}{n \geq 0} \) and \( \{b_n\}{n \geq 0} \). For example, we discuss log-convexity of \( \{n|a_n| – |a_{n+1}|\}{n \geq 1} \), \( \{b{n+1} – nb_n\}{n \geq 1} \), \( \{n|a_n|\}{n \geq 1} \), and \( \{(n + 1)b_n\}_{n \geq 0} \). In addition, we investigate log-balancedness of some sequences involving \( a_n \) (or \( b_n \)).