Abstract. Let $F_{\alpha,\beta }(x)=\beta E_{\beta } (x^{\beta })-\alpha E_{\alpha }(x^{\alpha })$, where $E_{\alpha }$ denotes the Mittag--Leffler function. We prove that if $\alpha, \beta\in (0,1]$, then $F_{\alpha,\beta }$ is completely monotonic on $(0,\infty )$ if and only if $\alpha\leq \beta $. This extends a result of T. Simon, who proved in 2015 that $F_{\alpha,1}$ is completely monotonic on $(0,\infty )$ if $\alpha\in (0,1]$. Moreover, we apply our monotonicity theorem to obtain some functional inequalities involving $F_{\alpha,\beta }$.

DOI: 10.14232/actasm-018-263-5

AMS Subject Classification (1991): 26A48, 33E12

Keyword(s): Mittag--Leffler function, completely monotonic, functional inequalities

Received January 17, 2018. (Registered under 13/2018.)