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A proof of Bernstein's theorem on regularly monotonic functions


Author: J. A. M. McHugh
Journal: Proc. Amer. Math. Soc. 47 (1975), 358-360
MSC: Primary 26A90
DOI: https://doi.org/10.1090/S0002-9939-1975-0354974-3
MathSciNet review: 0354974
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Abstract: A function is called ``regularly monotonic'' if it is of class $ {C^\infty }$ and each derivative is of a fixed sign (which may depend on the order of the derivative). We present a short proof of Bernstein's theorem on the analyticity of such functions.


References [Enhancements On Off] (What's this?)

  • [1] S. N. Bernštein, Leçons sur les propriétés extrémales et la meilleure approximations des fonctions analytiques d'une variable réelle, Gauthier-Villar s, Paris, 1926.
  • [2] R. P. Boas Jr., Signs of derivatives and analytic behavior, Amer. Math. Monthly 78 (1971), 1085–1093. MR 0296236, https://doi.org/10.2307/2316310
  • [3] Ralph P. Boas Jr., A primer of real functions, The Carus Mathematical Monographs, No. 13, Published by The Mathematical Association of America; distributed by John Wiley and Sons, Inc.; New York, 1960. MR 0118779

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0354974-3
Keywords: $ C$-infinity functions, analyticity, regularly monotonic functions
Article copyright: © Copyright 1975 American Mathematical Society

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