Lower bounds for entire functions

Abstract

We study lower bounds for entire functions of proximate order and of completely regular growth. We introduce the notion of the index condensation for sequences of complex numbers of proximate order. This notion generalizes that of the index of condensation for sequences of order one. We also introduce a properly balanced set, which is a properly distributed set with a zero condensation index. We show that a regular set is properly balanced and we prove that the properly balanced property of the zero set of an entire function is a necessary and sufficient condition for the existence of family of pairwise disjoint circles with the centers at its zeros and with relatively small radii. Outside these circles, the absolute value of the function admits lower bounds asymptotically coinciding with its upper bounds in the entire plane. Thus, we show that the notion of a properly balanced set naturally generalizes the notion of a regular set in the case of arbitrary sequences including multiples sequences. A method for constructing of an exceptional set consisting of circles with centers at zeroes of entire function is also provided. In some cases, we can make the sum of the radii of these circles arbitrarily small.