Univalence criteria and quasiconformal extension of a general integral operator

E. Deniz; S. Kanas; H. Orhan

doi:10.37863/umzh.v74i1.1148

E. Deniz Kafkas Univ., Kars, Turkey https://orcid.org/0000-0002-9570-8583
S. Kanas Univ. Rzeszуw, Poland
H. Orhan Atatürk Univ., Erzurum, Turkey

DOI: https://doi.org/10.37863/umzh.v74i1.1148

Анотація

УДК 517.5

Унiвалентнi критерiї та квазiконформне розширення iнтегрального оператора загального вигляду

Запропоновано достатнi умови аналiтичностi та унiвалентностi для функцiй, що визначаються деяким iнтегральним оператором. Цей результат зводиться до критерiю квазiконформного розширення за допомогою методу Бекера. Далi отримано новi критерiї унiвалентностi та вказано важливi зв’язки з iншими результатами. Також з основного результату при рiзних значеннях параметрiв, якi задiянi у формулюваннi цього результату, випливають деякi вже вiдомi умови унiвалентностi.

Посилання

L. V. Ahlfors, Sufficient conditions for quasiconformal extension, Ann. Math. Stud., 79, 23 – 29 (1974). DOI: https://doi.org/10.1515/9781400881642-004

J. M. Anderson, A. Hinkkanen, Univalence criteria and quasiconformal extensions, Trans. Amer. Math. Soc., 324, № 2, 823 – 842 (1991); https://doi.org/10.2307/2001743 DOI: https://doi.org/10.1090/S0002-9947-1991-0994162-4

J. Becker, L¨ownersche Differential gleichung und quasikonform fortsetzbare schlichte Funktionen, J. reine angew. und Math., 255, 23 – 43 (1972); https://doi.org/10.1515/crll.1972.255.23 DOI: https://doi.org/10.1515/crll.1972.255.23

J. Becker, Über die Lösungsstruktur einer Differentialgleichung in der konformen Abbildung (German) , J. reine und angew. Math., 285, 66 – 74 (1976); https://doi.org/10.1515/crll.1976.285.66 DOI: https://doi.org/10.1515/crll.1976.285.66

J. Becker, Conformal mappings with quasiconformal extensions, Aspects of Contemporary Complex Analysis, Acad.Press (1980), p. 37 – 77.

Th. Betker, L¨oewner chains and quasiconformal extensions, Complex Var. Theory and Appl., 20, № 1- 4, 107 – 111 (1992); https://doi.org/10.1080/17476939208814591 DOI: https://doi.org/10.1080/17476939208814591

M. Çağlar, H. Orhan, Sufficient conditions for univalence and quasiconformal extensions, Indian J. Pure and Appl. Math., 46, № 1, 41 – 50 (2015); https://doi.org/10.1007/s13226-015-0106-y DOI: https://doi.org/10.1007/s13226-015-0106-y

E. Deniz, H. Orhan, Some notes on extensions of basic univalence criteria, J. Korean Math. Soc., 48, № 1, 179 – 189 (2011); https://doi.org/10.4134/JKMS.2011.48.1.179 DOI: https://doi.org/10.4134/JKMS.2011.48.1.179

E. Deniz, H. Orhan, Loewner chains and univalence criteria related with Ruscheweyh and Sгlгgean derivatives, J. Appl. Anal. and Comput., 5, № 3, 465 – 478 (2015); https://doi.org/10.1007/s40435-015-0212-z DOI: https://doi.org/10.11948/2015036

V. Ya. Gutljanskiĭ, The stratification of the class of univalent analytic functions, Dokl. Akad. Nauk SSSR, 196, 498 – 501 (1971); Engl. transl.: Soviet Math. Dokl., 12, 155 – 159 (1971).

V. Ya. Gutljanskiĭ, V. I. Ryazanov, Geometric and topological theory of functions and mappings, Naukova Dumka, Kyiv (2011).

I. Hotta, L¨owner chains with complex leading coefficient, Monatsh. Math., 163, № 3, 315 – 325 (2011); https://doi.org/10.1007/s00605-010-0200-5 DOI: https://doi.org/10.1007/s00605-010-0200-5

I. Hotta, Explicit quasiconformal extensions and L¨oewner chains, Proc. Japan Acad. Ser. A. Math. Sci., 85, № 8, 108 – 111 (2009); https://doi.org/10.3792/pjaa.85.108 DOI: https://doi.org/10.3792/pjaa.85.108

S. Kanas, A. Lecko, Univalence criteria connected with arithmetic and geometric means, I, Folia Sci. Univ. Tech. Resov., 20, 49 – 59 (1996).

S. Kanas, A. Lecko, Univalence criteria connected with arithmetic and geometric means, II, Proc. Second Int. Workshop of Transform Methods and Special Functions, Varna’96, Bulgar. Acad. Sci (Sofia), p. 201 – 209 (1996). DOI: https://doi.org/10.1080/17476939608814851

B. A. Klishchuk, R. R. Salimov, Lower bounds for the area of the image of a circle, Ufa Math. J., 9, № 2, 55 – 61 (2017); https://doi.org/10.13108/2017-9-2-55 DOI: https://doi.org/10.13108/2017-9-2-55

J. G. Krz´yz, Convolution and quasiconformal extension, Comment. Math. Helv., 51, № 1,

– 104 (1976), https://doi.org/10.1007/BF02568145 DOI: https://doi.org/10.1007/BF02568145

Z. Lewandowski, On a univalence criterion, Bull. Acad. Polon. Sci. Ser. Sci. Math., 29, № 3-4, 123 – 126 (1981).

P. T. Mocanu, Une propriété de convexité généralisée dans la théorie de la représentation conforme. (French), Stud. Univ. Babe¸s-Bolyai Math., 11(34), 127 – 133 (1969).

S. Moldoveanu, N. N. Pascu, A new generalization of Becker’s univalence criterion (I), Stud. Univ. Babes-Bolyai Math., 31(54), № 2, 153 – 157 (1989).

S. Moldoveanu, N. N. Pascu, Integral operators which preserve the univalence, Stud. Univ. Babes-Bolyai Math., 32 (55), № 2, 159 – 166 (1990).

P. Montel, Families normales de fonctions analytiques, Gauthier-Villars, Paris (1927).

H. Ovesea, A generalization of Ruscheweyh’s univalence criterion, J. Math. Anal. amd Appl., 258, № 1, 102 – 109 (2001); https://doi.org/10.1006/jmaa.2000.7362 DOI: https://doi.org/10.1006/jmaa.2000.7362

N. N. Pascu, On a univalence criterion, II, Itinerant seminar on functional equations approximation and convexity, Cluj-Napoca, 153--154, Preprint, 85-6, Univ. "Babeş-Bolyai'', (1985).

V. Pescar, A new generalization of Ahlfor’s and Becker’s criterion of univalence, Bull. Malays. Math. Sci. Soc., 19, № 2, 53 – 54 (1996).

J. A. Pfaltzgraff, $k$-quasiconformal extension criteria in the disk, Complex Var. Theory and Appl., 21, № 3-4, 293 – 301 (1993), https://doi.org/10.1080/17476939308814638 DOI: https://doi.org/10.1080/17476939308814638

Ch. Pommerenke, Über die Subordination analytischer Funktionen, J. reine und angew. Math., 218, 159 – 173 (1965); https://doi.org/10.1515/crll.1965.218.159 DOI: https://doi.org/10.1515/crll.1965.218.159

Ch. Pommerenke, Univalent functions, Vandenhoeck Ruprecht in G¨ottingen, 376 pp., 1975.

D. Raducanu, H. Orhan, E. Deniz, On some sufficient conditions for univalence, An. Ştiinţ. Univ. "Ovidius'' Constanţa Ser. Mat., 18, № 2, 217 – 222 (2010).

S. Ruscheweyh, An extension of Becker’s univalence condition, Math. Ann., 220, № 3, 285 – 290 (1976); https://doi.org/10.1007/BF01431098 DOI: https://doi.org/10.1007/BF01431098

E. A. Sevost’yanov, S. A. Skvortsov, On the convergence of mappings in metric spaces with direct and inverse modulus conditions, Ukr. Math. J., 70, № 7, 1097 – 1114 (2018). DOI: https://doi.org/10.1007/s11253-018-1554-4

E. Sevost’yanov, A. Markysh, On Sokhotski – Casorati –Weierstrass theorem on metric spaces, Complex Var. and Elliptic Equat., 64, № 12, 1973 – 1993 (2019), https://doi.org/10.1080/17476933.2018.1557155 DOI: https://doi.org/10.1080/17476933.2018.1557155

V. Singh, P. N. Chichra, An extension of Becker’s criterion for univalence, J. Indian Math. Soc., 41, № 3-4, 353 – 361 (1977).

H. M. Srivastava, S. Owa (Editors), Current topics in analytic function theory, World Sci. Publ. Co., Singapore ect. (1992); https://doi.org/10.1142/1628 DOI: https://doi.org/10.1142/1628