Orders of Solutions of Fractional Differential Equation in Complex Domain

Abstract

We consider the fractional differential equation $^{c}D_{z}^{\alpha }f^{\prime }(z)+A(z)^{c}D_{z}^{\alpha }f(z)+B(z)f(z)=0$, where $^{c}D_{z}^{\alpha }$\ be the Caputo fractional derivative of orders $0<\alpha \leq 1$, and $z$\ is complex number, $A(z),B(z)$\ be entire functions. We will find conditions on $A(z),B(z)$\ which will guarantee that every solution $f\not\equiv 0$ of the equation will have infinite order.