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Transactions of the American Mathematical Society

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Slowly varying functions in the complex plane

Author: Monique Vuilleumier
Journal: Trans. Amer. Math. Soc. 218 (1976), 343-348
MSC: Primary 30A84
MathSciNet review: 0399479
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Abstract: Let f be analytic and have no zeros in $ \vert\arg z\vert < \alpha \leqslant \pi $; f is called slowly varying if, for every $ \lambda > 0,f(\lambda z)/f(z) \to 1$ uniformly in $ \vert\arg z\vert \leqslant \beta < \alpha $, when $ \vert z\vert \to \infty $. One shows that f is slowly varying if and only if $ zf'(z)/f(z) \to 0$ uniformly in $ \vert\arg z\vert \leqslant \beta < \alpha $, when $ \vert z\vert \to \infty $.

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

  • [1] J. Karamata, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj) 4 (1930), 38-53.
  • [2] B. Bajšanski and J. Karamata, Regularly varying functions and the principle of equicontinuity, Publ. Ramanujan Institute no. 1 (1968/69), 235-246. MR 42 #3222. MR 0268323 (42:3222)
  • [3] J. Karamata, Some theorems concerning slowly varying functions, Math. Res. Center, U.S. Army, Tech. Sum. Report No. 432, Madison, Wisconsin, 1963.
  • [4] M. L. Cartwright, Integral functions, Cambridge Tracts in Math. and Math. Phys., no. 44, Cambridge Univ. Press, Cambridge, 1956. MR 17, 1067. MR 0077622 (17:1067c)

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Keywords: Slowly varying functions
Article copyright: © Copyright 1976 American Mathematical Society

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