14 problems

Let $0<\alpha\leq\frac{1}{2}$ and define … Here $E_{\rho,\beta}$ denotes the Mittag-Leffler function with parameters $\rho$ and $\beta$. Complete-monotonicity conjecture. The funct…

Let $calpha ainleft(frac{3}{5}, 1right)$. The Mittag-Leffler function $E{alpha,1}(z)$ is defined by its standard power series, and the imaginary axis is the set…

The parameter-boundary conjecture. There exists an increasing convex function $f:[2,\infty)\to[3,\infty)$ with $f(2)=3$ such that, for every $\alpha\ge2$,

The reverse-inequality conjecture. For every $\alpha\in(1,2)$,

The non-real-zero criterion. For $alpha>2$, the absence of non-real zeros of $E_{alpha,beta}$ should be equivalent to the global reverse inequality.

Smallest-zero conjecture for $E_{\alpha,1}$. For any $\alpha\in(1,2]$, $E_{\alpha,1}(-z^\alpha)>0$ for all $z\in[0,1.559]$. Moreover, as $\alpha\to1+$,

Smallest-zero conjecture for $E_{\alpha,2}$. For each $\alpha\in[\alpha_0,2]$, the equation

Smallest-zero conjecture for $E_{\alpha,\alpha}$. For any $\alpha\in(1,2]$, $E_{\alpha,\alpha}(-z^\alpha)>0$ for all $z\in[0,2.93785]$. Moreover, as $\alpha\to1+$,

The Le Roy-type Mittag-Leffler function is associated, when the relevant complete-monotonicity condition holds, with an underlying positive random variable. Logarithmic infinite-di…

For $x\in\mathbb{R}$ and $\alpha\in[0,1]$, let $E_\alpha(x)$ denote the classical Mittag–Leffler function. Monotonicity conjecture. The mapping … is non-increasing on $[0,1]$ for e…

Let $\alpha\in(0,1]$, $m>0$, $l>m-1/\alpha$, and $x\geq 0$. Let $E_{\alpha,m,l}$ be the Kilbas–Saigo function. The conjectured two-sided bound. One has … The bounds are natural bec…

Let $\alpha\in(0,1]$, $m>0$, and $x\geq 0$. Define the Kilbas–Saigo function $E_{\alpha,m,l}$ by its usual power series, and let $G$ denote the Barnes-type function used in the sou…

Mainardi's conjecture. For any $t>0$ and fixed $\delta$, $0<\delta<1$, we get the conjectured formula from the source.

Mainardi's bounding conjecture. For every $t>0$ and every $\alpha\in(0,1)$, these two rational functions provide upper and lower bounds for $e_\alpha(t)$.