Vol. 44, No. 1, 1973

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Two primary factor inequalities

John H. E. Cohn

Vol. 44 (1973), No. 1, 81–92
Abstract

In the theory of integral functions, the expressions

                   ∑p zr
E(z,p) = (1− z)exp {  -r },p = 1,2,⋅⋅⋅
1
(1)

called primary factors, are of some importance, and it is of interest to find upper bounds for |E(z,p)|. Clearly E(z,p) = 0 only for z = 1, and so for other values, define f(z,p) = log |E(z,p)|. It is known that for suitable constants ap,bp the inequalities

f(z,p) ap|z|p,|z|1,z1 (2)
f(z,p) bp|z|p+1,|z|1,z1 (3)
are satisfied; for instance Hille has shown that one may take ap = 1+ 1p1∕r 2+log p and bp = 1.

In this paper, the smallest values of both ap and bp are determined, the latter in closed form.

Mathematical Subject Classification
Primary: 30A04
Milestones
Received: 11 May 1970
Revised: 22 December 1970
Published: 1 January 1973
Authors
John H. E. Cohn