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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 $|\arg z| < \alpha \leqslant \pi$; f is called slowly varying if, for every $\lambda > 0,f(\lambda z)/f(z) \to 1$ uniformly in $|\arg z| \leqslant \beta < \alpha$, when $|z| \to \infty$. One shows that f is slowly varying if and only if $zf’(z)/f(z) \to 0$ uniformly in $|\arg z| \leqslant \beta < \alpha$, when $|z| \to \infty$.

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

    J. Karamata, Sur un mode de croissance régulière des fonctions, Mathematica (Cluj) 4 (1930), 38-53.
  • B. Bajšanski and J. Karamata, Regularly varying functions and the principle of equi-continuity, Publ. Ramanujan Inst. 1 (1968/69), 235–246. (1 plate). MR 268323
  • J. Karamata, Some theorems concerning slowly varying functions, Math. Res. Center, U.S. Army, Tech. Sum. Report No. 432, Madison, Wisconsin, 1963.
  • M. L. Cartwright, Integral functions, Cambridge Tracts in Mathematics and Mathematical Physics, No. 44, Cambridge, at the University Press, 1956. MR 0077622

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