Estimates of the minimum of absolute value of an analytic function in terms of the maximum of absolute value on a greater circle

The talk will be devoted to a lower estimate of the value of $\max\limits_{qR\leqslant r\leqslant R}m(f,r)$ in terms of value of $M^{-d}(f,aR)$, $a>1$, $d>0$, $q<1$, where $m(f,r)=\min\limits_{|z|=r}|f(z)|$, $M(f,t)=\max\limits_{|z|=t}|f(z)|$.