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

  • W. K Hayman and E. L Ortiz

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.

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References

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1Erdélyi, A, Magnus, W, Oberhettinger, F and Tricomi, F. G. Higher Transcendental Functions, I (New York: McGraw-Hill, 1953).
2Friedland, S. and Hayman, W. K.. Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions. Comment. Math. Helv., to appear.
3Luke, Y. L.. The Special Functions and their Approximation, I and II (New York: Academic Press, 1969).
4Ortiz, E. L.. The Tau Method. SIAM J. Numer. Anal. 6 (1969), 480492.
5Ortiz, E. L.. A recursive method for the approximate expansion of functions in a series of polynomials. Comput. Physics Comm. 4 (1972), 151156.
6Szegö, G.. Orthogonal polynomials, 3rd edn. Amer. Math. Soc. Colloquium Publications (Providence: Amer. Math. Soc, 1967).
7Wilkinson, J. H.. Rounding Errors in Algebraic Processes (London: H.M.S.O., 1963).

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