Lower bound for minimum of modulus of entire function of genus zero with positive roots in terms of degree of maximal modulus at frequent sequence of points

We consider entire function of genus zero, the roots of which are located at a single ray. On the class of all such functions, we obtain close to optimal lower bounds for the minimum of the modulus on a sequence of the circumferences in terms of a negative power of the maximum of the modulus on the same circumferences under a restriction on the quotient $a>1$ of the radii of neighbouring circumferences. We introduce the notion of the optimal exponent $d(a)$ as an extremal exponent of the maximum of the modulus in this problem. We prove two-sided estimates for the optimal exponent for a “test” value $a=\tfrac{9}{4}$ and for $a\in(1,\tfrac{9}{8}]$. We find an asymptotics for $d(a)$ as $a\rightarrow1$. The obtained result differs principally from the classical $\cos(\pi\rho)$-theorem containing no restrictions for the frequencies of the radii of the circumferences, on which the minimum of the modulus of an entire function of order $\rho\in[0,1]$ is estimated by a power of the maximum of its modulus.