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 alpha > 1 of the radii of neighbouring circumferences. We introduce the notion of the optimal exponent d(alpha) 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 alpha = 9/4 and for alpha is an element of (1, 9/8]. We find an asymptotics for d(alpha) as alpha -> 1. 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 is an element of [0, 1] is estimated by a power of the maximum of its modulus.