Vol. 38, No. 1, 1971

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Whittaker constants for entire functions of several complex variables

Ken Shaw

Vol. 38 (1971), No. 1, 239–250
Abstract

Let f be an entire function of a single complex variable. The exponential type of f is given by

τ(f) = lim sup|f(n)(0)|1∕n
n→∞

The Whittaker constant W is defined to be the supremum of numbers c having the following property: if τ(f) < c and each of f,f,f′′, has a zero in the disc |z|1, then f 0. The Whittaker constant is known to lie between .7259 and.7378.

The present paper provides a definition and characterization of the Whittaker constant 𝒲n for n complex variables. The principle result of this characterization, which involves polynomial expansions of entire functions, is

W  > 𝒲2 ≧ 𝒲3  ≧ ⋅⋅⋅ .

Mathematical Subject Classification 2000
Primary: 32A15
Milestones
Received: 30 October 1970
Published: 1 July 1971
Authors
Ken Shaw