Wiman-Valiron method for fractional derivatives and sharp growth estimates of $\alpha$-analytic solutions for linear fractional differential equations

We consider a fractional linear differential equation with successive derivatives given by $\mathbb{D}_\alpha^{n}y+ p_{n-1}(x) \mathbb{D}_\alpha^{n-1}y+ \dots +p_{1}(x)\mathbb{D}_\alpha y+p_0(x)y=0$, where $\mathbb{D}_\alpha^{j}$ is the $j$th iteration of the Caputo-Djrbashian fractional derivative of order $\alpha>0$, $p_j$ are $\alpha$-analytic functions for $0<x^\alpha <R$. Generalizing a result of Kilbas, Rivero Rodr\'iguez-Germ\'a and Trujillo, we prove the existence and uniqueness of the corresponding Cauchy problem in the class of $\alpha$-analytic functions. We establish an exact growth order for the solution when $p_j(x)=P_j(x^\alpha)$, where $P_j$ are polynomials, and $p_0$ dominates in some sense. This is the full counterpart of the classical case of ordinary differential equations. In particular, we demonstrate the sharpness of Kochubei's result and generalize it. To achieve this, we extend the Wiman-Valiron theory to analytic functions and the Djrbashian-Gelfond-Leontiev generalized fractional derivatives.