Functions with real poles and zeros
Raymond M. Redheffer · Pacific Journal of Mathematics · 1968
Throughout this paper {λ n } is a real sequence with λ n Φθ and λ n ^ λ n +i 9 -°° oo.Our objective is to obtain conditions on the growth of F{z) from conditions on the function Λ(u) = Λ^u) -Λ 2 (u).Denoting the even part of Λ(u) by A β (u\ we can state our first result as follows: Suppose Λ(u) = O(u) and lim ur r/u ) for each u Φ 0. Then log I F(re ίθ ) \ = πΛ(r) | sin θ \ -2A e (r)θ sin d + r cos θ Γ • apart from an error term ηr log | 2 esc θ |, where η -> 0 uniformly in 0< 1^1 < π as r-^oo.This improves theorems of Pfluger, Kahane and Rubel, Cartwright, and others, in that we do not assume existence of lim A(u)/u, we do not assume that F is entire or even, and the error term has a convergent integral with respect to θ.Similar theorems for functions with negative poles and zeros, given later in the paper, generalize other familiar results.Here the error term involves log (2 sec hθ).Another kind of result is briefly described as follows: Let R -R(x) and S = S(x) be positive functions such that the ratios x/R, R/x, x/S, S/x are bounded as | x \ -» oo.Then for many purposes the function[ can be replaced by the function Λ*(z;R,S)= -R Z -U This remark is given content by detailed estimates of the error and of A*. (For simplicity of statement the text takes R -S but the form mentioned here is sometimes more con-