Some theorems on the cos 𝜋𝜆 inequality

Abstract

In this paper we consider subharmonic functions u ≦ 1 u \leqq 1 in the unit disk whose minimum modulus and maximum modulus satisfy a certain inequality. We show the existence of an extremal member of this class with largest maximum modulus. We then obtain an upper bound for the maximum modulus of this function in terms of the logarithmic measure of a certain set. We use this upper bound to prove theorems about subharmonic functions in the plane.