On a zeroes of Mittag-Leffler functions with parameter \(\rho <1/2\) (Q879175)

In the paper the author obtained the best upper estimate of the parameter \(\mu\), for which the Mittag -- Leffler functions \(E_\varrho(z,\mu)\) at \(\varrho< 1/2\) have only negative and simple zeros in the complex plane. The main result of the paper is this theorem: Let \(0<\varrho<1/2\) and \(\mu\geq 0,9+(\varrho^2\ln 2)^{-1}-\varrho^{-1}\). There the function \(E_\varrho(z,\mu)\) has zeros in \(\mathbb C-\mathbb R\).