Lower bound for the minimum modulus of an entire function of genus zero on a frequent sequence of circles

We consider the class of entire functions of genus zero (as is known, such a class contains all entire functions of order $<1$) with positive roots. It is shown that for any function $f$ from this class and an arbitrary $\delta>0$ there is a sequence $r_n\uparrow+\infty$ such that the sequence of ratios $r_{n+1}/r_n$ is bounded and an estimate from below for the minimum modulus of $f$ on a circle $|z|=r_n$ through a negative (equil to $-1-\delta$) power of the maximum modulus of $f$ on the same circle is valid. In the case of small values of the upper limit of the ratios $r_{n+1}/r_n$ estimates for exponent of the maximum modulus that are close to optimal were found.