Some estimates for Mittag-Leffler function in quantum calculus and applications

Abstract

The study of the Mittag-Leffler function and its various generalizations has become a very popular topic in mathematics and its applications. In the present paper we prove the following estimate for the $q$-Mittag-Leffler function: \begin{eqnarray*} \frac{1}{1+Γ_q\left(1-α\right)z}\leq e_{α,1}\left(-z;q\right)\leq\frac{1}{1+Γ_q\left(α+1\right)^{-1}z}. \end{eqnarray*} for all $0<\alpha <1$ and $z>0$. Moreover, we apply it to investigate the solvability results for direct and inverse problems for time-fractional pseudo-parabolic equations in quantum calculus for a large class of positive operators with discrete spectrum.