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Convexity Preservation of Inverse Euler Operators and a Problem of S. Miller

Convexity Preservation of Inverse Euler Operators and a Problem of S. Miller CONVEXITY PRESERVATION OF INVERSE EULER OPERATORS AND A PROBLEM OF S. MILLER M. GOLDSTEIN, R. R. HALL, T. SHEIL-SMALL AND H. L. SMITH 1. The following conjecture of S. Miller appears as Problem 5.61 in [1, p. 554]. If /(z ) is analytic in \z\ < 1 and if where n is a given natural number, then The case /(z ) = 1 shows that no better bound is possible in (2). The conjecture was stated with the additional hypothesis that /(0 ) = 0. The case f(z) = jz suggests that in this case the correct inequality implied by (1) is The estimates (2) and (3) follow as the case B = k\ (0 ^ k ^ n) of the following theorem. THEOREM 1. Let 1 = B ^ B ^ ... ^ B be a finite, non-decreasing sequence o x n and for f(z) analytic in \z\ < 1 define n D (a) If E {f){z) maps {\z\ < 1} into a convex domain D, then f(z) maps {\z\ < 1} into D. (b) If f{z) has a zero of order N at z — 0 and if (5) then (6) \f( / A We adopt the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Convexity Preservation of Inverse Euler Operators and a Problem of S. Miller

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/14.6.537
Publisher site
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Abstract

CONVEXITY PRESERVATION OF INVERSE EULER OPERATORS AND A PROBLEM OF S. MILLER M. GOLDSTEIN, R. R. HALL, T. SHEIL-SMALL AND H. L. SMITH 1. The following conjecture of S. Miller appears as Problem 5.61 in [1, p. 554]. If /(z ) is analytic in \z\ < 1 and if where n is a given natural number, then The case /(z ) = 1 shows that no better bound is possible in (2). The conjecture was stated with the additional hypothesis that /(0 ) = 0. The case f(z) = jz suggests that in this case the correct inequality implied by (1) is The estimates (2) and (3) follow as the case B = k\ (0 ^ k ^ n) of the following theorem. THEOREM 1. Let 1 = B ^ B ^ ... ^ B be a finite, non-decreasing sequence o x n and for f(z) analytic in \z\ < 1 define n D (a) If E {f){z) maps {\z\ < 1} into a convex domain D, then f(z) maps {\z\ < 1} into D. (b) If f{z) has a zero of order N at z — 0 and if (5) then (6) \f( / A We adopt the

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Nov 1, 1982

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