Meromorphic functions in $m$-punctured complex planes

M.O.Hanyak; A.A.Kondratyuk

An extension of the Nevanlinna value distribution theory for meromorphic functions in $m$-punctured complex planes is proposed. The main characteristics are one-parameter and possess the same properties as in the classical case. Analogs of the Jensen and of the First Fundamental Theorem for $m$-punctured complex planes are obtained. We propose proofs of Decomposition Lemma, Main Lemma on logarithmic derivative and Second Fundamental Theorem for meromorphic functions in $m$-punctured planes.

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	Meromorphic functions in $m$-punctured complex planes
Author	M.O.Hanyak, A.A.Kondratyuk hanyakm@mail.ru, kond@franko.lviv.ua Department of Mathematical and Functional Analysis,Faculty of Mechanics and Mathematics,Ivan Franko National University of Lviv,
Abstract	An extension of the Nevanlinna value distribution theory for meromorphic functions in $m$-punctured complex planes is proposed. The main characteristics are one-parameter and possess the same properties as in the classical case. Analogs of the Jensen and of the First Fundamental Theorem for $m$-punctured complex planes are obtained. We propose proofs of Decomposition Lemma, Main Lemma on logarithmic derivative and Second Fundamental Theorem for meromorphic functions in $m$-punctured planes.
Keywords	meromorphic function, value distribution theory, m-punctured complex plane
DOI	doi:10.30970/ms.27.1.53-69
Reference	1. Maтeвoсян Г. У. Oб одной факторизации функций, мероморфных в многосвязных областях, и о некоторых её приложениях // Изв. АН Aрм. ССР. -- 1974. -- IX, 194 5, 387--408. 2. Korhonen R. Nevanlinna Theory in an annulus. In G. Barsegian, I. Laine and C. C. Yang, editors Value Distribution Theory and Related Topics, Advances in Complex Analysis and Its Applications. Kluwer Academic Publishers, Boston -- Dordrecht -- New York -- London, 2004 -- 167--179. 3. Zmorovich V. A. On some classes of analytic functions on annuli, Math. Sb. 40,(1956), 225--238. 4. Krystiyanyn A. Ya., Kondratyuk A.~A., On the Nevanlinna theory for meromorphic functions on annuli I, Matematychni Studii, 23, No 1 (2005), 19--30. 5. Krystiyanyn A. Ya., Kondratyuk A.~A., On the Nevanlinna theory for meromorphic functions on annuli II, Matematychni Studii, 24, No 2 (2005), 57--68. 6. Jensen J. L. W. V. Sur un nouvel et important théorème de la théorie des fonctions // Acta Math. -- 1899. -- 22, 219--251. 7. Ahlfors L. V. Complex Analysis 2d ed., McGraw-Hill Book Company, New York, 1966. 8. Rudin W. Real and Complex Analysis, McGraw-Hill, New York, 1974. 9. Кроуэлл Р., Фокс Р. Введение в теорию узлов, Москва, 1967, 265~с. 10. Hayman W. K. Meromorphic functions, Oxford, Clarendon Press, 1964, 191~pp. 11. Гольдберг A. A., Островский И. В. Распределение значений мероморфных функций. -- M.:Наука, 1970. -- 591~c. 12. Rubel L. A. Entire and meromorphic functions, New York: Springer, 1996, 2000~p.
Pages	53-69
Volume	27
Issue	1
Year	2007
Journal	Matematychni Studii
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