A sharp Mittag-Leffler bound for series with separated exponents

Let $η\geq 0$ and $δ>0$, and let $(r_j)_{j\geq 0}$ be a sequence of real numbers satisfying $η\leq r_0 < r_1 < r_2 < \cdots$ and $r_{j+1}-r_j \geq δ$ for every $j\geq 0$. We prove that, for $δ\leq 1$ 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 $E_{δ,1+η}$ denotes the two-parameter Mittag-Leffler function. Moreover, equality holds for some $x>0$ if and only if $r_j = η+ jδ$ 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 $δ\leq 1$ is sharp, since counterexamples exist for every $1 < δ< 2$.