A local mean-value theorem for analytic functions with smooth boundary values
Glasgow mathematical journal, Tome 15 (1974) no. 1, pp. 27-29
Voir la notice de l'article provenant de la source Cambridge University Press
Let f be an analytic function on a connected open set Ώ in the complex plane. Then, for given a, b ∈ Ώ, the equationneed not have a solution z ∈ Ώ. As a matter of fact, this would happen with each locally one-to-one analytic function which is not one-to-one on Ώ. But if we fix a a ∈ Ώ, then, for all b sufficiently close to a, (1) is solvable for z. This is an easy consequence of the Open Mapping Theorem applied to f'. For, assuming that f' is non-constant (otherwise, (1) holds for all a, b, z ∈Ώ), the Open Mapping Theorem tells us that f'(Ώ), the image under f' of Ώ, is an open neighbourhood of f'(a); so it is a direct consequence of the definition of f'(a) that there exists δ > 0 such that 0 < |b – a| < δ implies (f(b) – f(a))/(b – a) ∈f'(Ώ). A stronger statement has been obtained by J. M. Robertson [1, p. 329], who has shown thatand, if f''(a) ≠ 0, then.
Novinger, W. P. A local mean-value theorem for analytic functions with smooth boundary values. Glasgow mathematical journal, Tome 15 (1974) no. 1, pp. 27-29. doi: 10.1017/S0017089500002068
@article{10_1017_S0017089500002068,
author = {Novinger, W. P.},
title = {A local mean-value theorem for analytic functions with smooth boundary values},
journal = {Glasgow mathematical journal},
pages = {27--29},
year = {1974},
volume = {15},
number = {1},
doi = {10.1017/S0017089500002068},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500002068/}
}
TY - JOUR AU - Novinger, W. P. TI - A local mean-value theorem for analytic functions with smooth boundary values JO - Glasgow mathematical journal PY - 1974 SP - 27 EP - 29 VL - 15 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500002068/ DO - 10.1017/S0017089500002068 ID - 10_1017_S0017089500002068 ER -
%0 Journal Article %A Novinger, W. P. %T A local mean-value theorem for analytic functions with smooth boundary values %J Glasgow mathematical journal %D 1974 %P 27-29 %V 15 %N 1 %U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500002068/ %R 10.1017/S0017089500002068 %F 10_1017_S0017089500002068
[1] 1.Robertson, J. M., A local mean value theorem for the complex plane, Proc. Edinburgh Math. Soc. (21) 16 (1968/1969), 329–331. Google Scholar | DOI
[2] 2.Rudin, Walter, Real and Complex Analysis (New York, 1966). Google Scholar
[3] 3.Samuelsson, Åke, A local mean value theorem for analytic functions, Amer. Math. Monthly 30 (1973), 45. Google Scholar | DOI
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