New perspective in the field of growth properties of composite entire and meromorphic functions

Abstract

Suppose that f be a complex valued function of a complex variable newlinede...ned in the complex plane C. Then we say that the function f is entire if newlineit is analytic at all points in the complex plane C, and meromorphic if it is newlineanalytic at all points of C except for poles. The meromorphic function having newlinean essential singularity at the point at in...nity is called a transcendental newlinemeromorphic function. Nevanlinna theory (also called Value Distribution newlineTheory) was developed by Rolf Nevanlinna (1926), studies the distribution newlineof the roots of the equations such as newlinef (z) = a newlinewhere a is a complex number and f is an entire or meromorphic function. newlineThe oldest results in the value distribution theory is the Fundamental The- newlineory of Algebra (FTA) which states that If f is a polynomial of degree n newlinewith real or complex coe¢ cients, then the equation f (z) = 0 has at least newlineone root . The value distribution of polynomials is straightforward because newlineany two complex polynomials having the same roots can di¤er only by a newlineconstant multiple. In particular, the FTA can be restated as follows: A newlinenon-constant complex polynomial assumes every ...nite value an equal num- newlineber of times (counting multiplicities), which is determined by the order of newlinegrowth of maximum modulus of f on D r = fz : jzj lt rg as r ! 1: By newlineusing Picard s theorem we can say every transcendental entire function can newlineassume certain values an in...nite number of times. Also some entire func- newlinetions (like e z ) may not have any zeroes at all. Thus a direct generalization newlineof the FTA is not true. We can determine the distributions of values of newlinean entire function basically by their order. But for the meromorphic func- newlinetion we cannot measure the growth rate by using the maximum modulus, newlinesince a meromorphic function can hit the point at in...nity on a disc of ...nite newlineradius. In this situation, Nevanlinna introduced a real function (known as newlineNevanlinna characteristic function) by which we characterize the behavior of newlinemeromorphic functions. newline