Abstract
We consider functions f(z), z ∈ D ⊂ ℂ, determining the mappings w = f(z) that, at the points ζ of the domain D, have the same dilatation ratio along the three pairwise noncollinear rays issuing from ζ. Under an additional condition on the disposition of rays, the Trokhimchuk generalization of Men'shov's theorem on the holomorphy of such functions can be extended to functions for which the assumption that they are continuous is replaced by the assumption that (\log+|f(z)|)p is integrable with respect to the plane Lebesgue measure for each positive p< 2.
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REFERENCES
D. Menchoff, "Sur les fonctions monog`enes,"Bull. Soc. Math. France 59 (1931),141–182.
H. Bohr,"Ueber streckentreue und konforme Abbildung,"Math. Z.,1 (1918).
Yu.Yu. Trokhimchuk,Continuous Mappings and Conditions of Monogeneity [in Russian ],Fizmatgiz, Moscow,1963.
M.T. Brodovich,"Conditions of monogeneity for noncontinuous mappings,"in:Function Theory, Functional Analysis, and Their Applications. no. 12 [in Russian ],Kharkov State Univ., Kharkov, 1970,pp.94–103.
D.S. Telyakovskii,"Generalization of a heorem of Men shov on monogenic functions,"in:Abstracts of Papers, Tenth Saratov Winter School [in Russian ],Saratov State Univ., Saratov,2000,pp.135–136.
M.M. Dzhrbashyan,Integral Transforms and Representations of Functions in the Complex Domain [in Russian ],Nauka, Moscow,1966.
S. Saks,Theory of the Integral Stechert,New York,1937.
D.S. Telyakovskii,"Generalization of a theorem of Men shov on monogenic functions,"Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],53 (1989),no.4,886–896.
G.M. Goluzin,Geometric Theory of Functions of a Complex Variable [in Russian ],Nauka, Moscow, 1966.
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Telyakovskii, D.S. A Generalization of Men'shov's Theorem on Functions Satisfying Condition K″. Mathematical Notes 76, 534–545 (2004). https://doi.org/10.1023/B:MATN.0000043483.90707.4d
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DOI: https://doi.org/10.1023/B:MATN.0000043483.90707.4d