Throughout this paper {λ_{n}} is
a real sequence with λ_{n}≠0 and λ_{n} ≦ λ_{n+1},−∞ < n < ∞. The counting function
Λ_{1}(u) is the number of λ_{n} between 0 and u, counted negatively for negative u.
Similarly μ_{n} is a real sequence with μ_{n}≠0 and with counting function Λ_{2}(u).
In this summary (which corresponds to the case p = 1 of the paper) we
define
by taking all factors for which λ_{n} and ∕x_{n} lie on the interval (−R,R) and then
letting R →∞. Our objective is to obtain conditions on the growth of F(z) from
conditions on the function Λ(u) = Λ_{1}(u) − Λ_{2}(u).
Denoting the even part of Λ(u) by Λ_{e}(u), we can state our first result as follows:
Suppose Λ(u) = O(u) and
for each u≠0. Then
apart from an error term ηr log 2csc𝜃, where η → 0 uniformly in 0 < 𝜃 < π as
r →∞. This improves theorems of Pfluger, Kahane and Rubel, Cartwright, and
others, in that we do not assume existence of limΛ(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(2sec𝜃).
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→∞. Then for many purposes the function
can be replaced by the function
This remark is given content by detailed estimates of the error and of Λ^{∗}. (For
simplicity of statement the text takes R = S but the form mentioned here is
sometimes more convenient, and is allowed by the proofs.) The development leads to
a simple, systematic method of proving variety of growth theorems. Formulation of
such a method is a principal goal of this paper, and our specific examples are
intended only for illustration.
