A sharp Mittag-Leffler bound for series with separated exponents

Abstract

Let $\eta \ge 0$ and $\delta >0$, and let $(r_j{)}_{j\ge 0}$ be a sequence of real numbers satisfying $\eta \le r_0<r_1<r_2<\cdots$ and $r_{j+1}-r_j\ge \delta$ for every $j\ge 0$. We prove that, for $\delta \le 1$ and every $x>0$, \[ \sum_{j=0}^\infty \frac{x^{r_j}}{\G(1+r_j)} \leq \sum_{j=0}^\infty \frac{x^{η+jδ}}{\G(1+η+jδ)} = x^ηE_{δ,1+η}(x^δ), \] where $E_{\delta ,1+\eta }$ denotes the two-parameter Mittag-Leffler function. Moreover, equality holds for some $x>0$ if and only if $r_j=\eta +j\delta$ for every $j$. The proof combines a finite compression argument for unimodal functions with a shift inequality for the Mittag-Leffler function. The upper bound $\delta \le 1$ is sharp, since counterexamples exist for every $1<\delta <2$.