Meromorphic functions of lower order less than one [MatSciRep:64]

Abstract

The two properties of the polynomials, viz., 1) A polynomial takes on every complex value the same number of times; 2) On large circles |z| = r, the absolute value of a polynomial p(z) is large and "Limit r tends to Infinity{|p(r*e^(ialpha))| / |p(r*e^(ibeta))|} = 1" , uniformly in Alpha and Beta. The example of the exponential function shows that neither of these two properties subsists for entire functions. These lectures discuss the problem of finding analogues for the properties 1 and 2 for the entire and meromorphic functions of lower grade. Some auxiliary results are given in sections 1 and 2; Analogues of property 2 are discussed in sections 3 to 5 of these lectures, while analogues of property 1 are discussed in sections 6 to 8. A knowledge of the fundamentals of Nevanlinna Theory is assumed such as it can be found in W.K. Hayman's Meromorphic functions, chapters 1 and 2.