Accuracy of approximation of subharmonic functions by logarithms of moduli of analytic functions in the Chebyshev metrics

Abstract

It is known that a subharmonic function of finite order ρ can be approximated by the logarithm of the modulus of an entire function at a point z outside an exceptional set up to C log ¦z¦. In this paper, we prove that if such an approximation becomes more precise, i.e., the constant C decreases, then, beginning with C = ρ/4, the size of the exceptional set enlarges substantially. Similar results are proved for subharmonic functions of infinite order and for functions that are subharmonic in the unit disk. These theorems improve and complement a result by Yulmukhametov. Bibliography: 20 titles.