Vol. 44, No. 1, 1973

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Differential inequalities and local valency

Walter Kurt Hayman

Vol. 44 (1973), No. 1, 117–137
Abstract

An entire function f(z) is said to have bounded value distribution (b.v.d.) if there exist constants p,R such that the equation f(z) = w never has more than p roots in any disk of radius R. It is shown that this is the case for a particular p and some R > 0 if and only if there is a constant C > 0 such that for all z

|f(p+1)(z)| ≦ C  max  |f(ν)(z)|,
ν=1 to p

so that f(z) has bounded index in the sense of Lepson.

Mathematical Subject Classification 2000
Primary: 30A32
Secondary: 34A40, 34C10, 30A66
Milestones
Received: 27 August 1971
Published: 1 January 1973
Authors
Walter Kurt Hayman