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17.—An Upper Bound for the Largest Zero of Hermite's Function with Applications to Subharmonic Functions

Published online by Cambridge University Press:  14 February 2012

W. K Hayman  and
E. L Ortiz
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Synopsis

Let

be Hermite's function of order λ and let h = h(λ) be the largest real zero of Hλ(t). Set

In this paper we establish the inequality

where

Equality holds for S = ½. The result is also fairly accurate as S→0 and S→1. The proof is analytical except in the ranges −1·1 ≦ h ≦ −0·1 and where the argument is concluded by means of a computer.

The following deduction is made elsewhere [2, Theorem A]. If u(x) is subharmonic in Rm(m ≧ 2) and the set E where u(x) > 0 has at least k components, where k ≧ 2, then the order ρ of u(x) is at least ϕ(1/k). In particular, if ρ < 1, E is connected. This result fails for ρ = 1.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 75 , Issue 3 , 1976 , pp. 183 - 197
Copyright
Copyright © Royal Society of Edinburgh 1976

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References

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