Generalizations of Lindelöf conditions for distribution of zeros of entire functions

Abstract

The Lindelöf condition is the first example of a nonradial condition for the distribution of zeros of entire functions of finite integer order. Its further development is used in the classical Rubel — Taylor theorem. It also involves negative integer powers of the complex variable. We generalize the Lindelöf condition by replacing the power test functions by arbitrary harmonic functions on concentric annuli. In particular, from this generalization, we easily deduce the necessity of the Lindelöf conditions in Rubel — Taylor theorem.