ENTIRE FUNCTIONS WITH RADIALLY DISTRIBUTED ZEROS

Abstract

The main aim of this work is to study some results concerning a certain class of entire functions, namely those with radially distributed zeros. Entire functions, i.e. analytic functions which have no singularities in the finite plane are a subclass of meromorphic functions, i.e., functions whose only singularities in the finite plane are poles. Part A consists of the more classical and well-known results. In Chapter 1, we give some basic results concerning entire and meromorphic functions. We first introduce some elementary Nevanlinna theory by defining the characteristic function T(r, f) and the deficiency � (a, f) of a meromorphic function f(z). We next define the order of f(z) in terms of T(r, f) and show that for an entire function, T(r, f) is equivalent to ?n M(r, f) where M(r, f) is the maximum modulus of f(z) for lzl = r. Chapter 1 culminates with Hadamard' s Factorization Theorem which extends the fundamental theorem of algebra concerning the factorization of polynomials to meromorphic functions. In Chapter 2, we investigate the maximum term � (r) of an entire function f(z), i.e., the term of greatest modulus in the Taylor series expansion of f(z). We show that ?n �(r) is asymptotically equivalent to ?n M(r) if f(z) has finite order. We conclude the chapter by showing how to construct an entire function of given order, lower order and exponent of convergence. In Chapter 3, we state some classical results concerning rational approximation and uniform distribution. We first prove by an argument of Dirichlet a theorem concerning the simultaneous approximation of k numbers. We then state Weyl's theorem on uniform distribution and show that it is deeper form of Dirichlet's Theorem. Part A provides us with the basic knowledge to prove some modern results concerning entire functions with radially distributed zeros in Part B. In Chapter 4, we prove some relations between the order and the lower order for such functions. We first give a simpler proof of a theorem due to Abi-Khuzam that for an entire function with radially distributed zeros, the order and the lower order are cofinite. We next prove more general results due to Steinmetz. Lastly, we state a theorem due to Miles concerning a general result on the lower order of a meromorphic function with radially distributed zeros and poles. In Chapter 5, we consider relations for the deficiency of zero of an entire function f(z) with radially distributed zeros. We first prove the result due to Edrei, Fuchs and Hellerstein that � (0, f) > 0 for such functions if f(z) has a sufficiently large order. In Chapter 5.4, we rove the stronger result due to Hellerstein and Shea that in fact � (0, f) ? 1 as the order tends to infinity for such functions. Both these proofs require Weyl's theore1n on uniform distribution. Lastly, to conclude this work, we consider the case when f(z) is an entire function of infinite order with radially distributed zeros. In view of the results in Chapter 5 it is natural to ask if � (0, f) > 0 or even � (0, f) = l for such functions. In fact, the answer to this question is negative due to the results by Miles. For the sake of completeness, we also give the proofs of Lemma 5.10 and 5.12 due to Hellerstein and Williamson in the Appendix.