STUDY OF GROWTH PROPERTIES AND SPACES OF ENTIRE AND ANALYTIC FUNCTIONS OF TWO COMPLEX VARIABLES

Abstract

INTRODUCTION: An entire function is a function f:C C, which is regular in every finite region of the complex plane. The general theory of entire functions has found its origin in the work of Weierstrass [65]. His work was developed, primarily by Picard, Borel, Poincare, Hadamard and others. The beginning of the twentieth century saw the introduction of many new concepts by eminent mathematicians such as Valiron, Lindelof, Leven, Wiman, Nevanlinna and Hardy etc. Since then, Whittakar, Haymen, Boas, Shah, Clunie and others have contributed richly to the theory of entire functions. 0.2 ENTIRE FUNCTIONS: Let f(z), z = reie, be an entire function. Let (0.2.1) M(r) = M(r,f) = max If(z)1. lzkr Then M(r) is said to be the maximum modulus of f(z) for Izi = r. It has been established that the maximum absolute value of an 1 entire function over a closed disc coincides with the maximum absolute value of that function over, its boundary. Blumenthal [7] showed that M(r) is a steadily increasing continuous function of r and is differentiable in adjacent intervals. Further, log M(r) is a convex function of log r and has the representation [63..