On growth order of solutions of differential equations in a neighborhood of a branch point

Let $M_k$ be {the} set of $k$-valued meromorphic in $G={zcolon r_0leqslant |z|}$ functions with {a}~branch point of order $k-1$ {at} $infty$; let $E_ast$ be a set of circles {with finite} sum of radii. Denote $ M_ast(r,f)=max|f(z)|, zin{te^{itheta}colon 0leqslantthetaleqslant2kpi, r_0leqslant tleqslant r}setminus E_ast, f!in! M_k; $ $m(r,f)=frac{1}{2pi k}int_0^{2pi k}!ln^+!|f(re^{itheta})|dtheta$. If $fin M_k$ is a solution of the equation $P(z,f,f')=0$ and $P$ is a polynomial in all variables then either $|f(re^{itheta})|le r^nu,$ $re^{itheta}in Gsetminus E_ast, nu>0$ or $m(r,f)$ has growth order $rhogeqslantfrac{1}{2k}$, and the following equality holds $ln M_ast(r,f)=(c+o(1))r^rho,$ $cneq0,$ $rto+infty.$