On certain general integral operators of analytic functions

B. A. Frasin

Abstract


In this paper, we obtain new sufficient conditions for the operators \(F_{\alpha_1,\alpha_2,...,\alpha_n,\beta}(z)\) and \(G_{\alpha_1,\alpha_2,...,\alpha_n,\beta}(z)\) to be univalent in the open unit disc \(\mathcal{U}\), where the functions \(f_1, f_2,..., f_n\) belong to the classes \(S^*(a, b)\) and \(\mathcal{K}(a, b)\). The order of convexity for the operators \(F_{\alpha_1,\alpha_2,...,\alpha_n,\beta}(z)\) and \(G_{\alpha_1,\alpha_2,...,\alpha_n,\beta}(z)\) is also determined. Furthermore, and for \(\beta= 1\), we obtain sufficient conditions for the operators \(F_n(z)\) and \(G_n(z)\) to be in the class \(\mathcal{K}(a, b)\). Several corollaries and consequences of the main results are also considered.

Keywords


Analytic functions; starlike and convex functions; integral operator

Full Text:

PDF

References


Ahlfors, L. V., Sufficient conditions for quasiconformal extension, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), pp. 23-29. Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, N.J., 1974.

Becker, J., Lownersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen, J. Reine Angew. Math. 255 (1972), 23-43.

Becker, J., Lownersche Differentialgleichung und Schlichtheitskriterien, Math. Ann. 202 (1973), 321-335.

Breaz, D., Univalence properties for a general integral operator, Bull. Korean Math. Soc. 46 (2009), no. 3, 439-446.

Breaz, D., Breaz, N., Two integral operators, Studia Universitatis Babes-Bolyai Math., 47 (2002), no. 3, 13-19.

Breaz, D., Breaz, N., Univalence conditions for certain integral operators, Studia Universitatis Babes-Bolyai, Mathematica, 47 (2002), no. 2, 9-15.

Breaz, D., Owa, S., Some extensions of univalent conditions for certain integral operators, Math. Inequal. Appl., 10 (2007), no. 2, 321-325.

Bulut, S., Univalence preserving integral operators defined by generalized Al- Oboudi differential operators, An. S¸tiint¸. Univ. ”Ovidius” Constant¸a Ser. Mat. 17 (2009), no. 1, 37-50.

Eenigenburg, P., Miller, S. S., Mocanu, P. T. and Reade, M. O., On a Briot–Bouquet differential subordination, General inequalities, 3 (Oberwolfach, 1981), 339-348, Internat. Schriftenreihe Numer. Math., 64, Birkhauser, Basel, 1983.

Frasin, B. A., General integral operator defined by Hadamard product, Mat. Vesnik 62 (2010), no. 2, 127-136.

Frasin. B. A., Aouf, M. K., Univalence conditions for a new general integral operator, Hacet. J. Math. Stat. 39 (2010), no. 4, 567-575.

Jabkubowski, Z. J., On the coefficients of starlike functions of some classes, Ann. Polon. Math. 26 (1972), 305-313.

Pascu, N., An improvement of Becker’s univalence criterion, Proceedings of the Commemorative Session: Simion Stoılow (Brasov, 1987), 43-48, Univ. Brasov, Brasov, 1987.

Pescar, V., A new generalization of Ahlfor’s and Becker’s criterion of univalence, Bull. Malaysian Math. Soc. (2) 19 (1996), no. 2, 53-54.

Seenivasagan, N., Sufficient conditions for univalence, Applied Math. E-Notes, 8 (2008), 30-35.

Seenivasagan, N., Breaz, D., Certain sufficient conditions for univalence, Gen. Math. 15 (2007), no. 4, 7-15.




DOI: http://dx.doi.org/10.2478/v10062-012-0003-3
Date of publication: 2016-07-24 20:22:24
Date of submission: 2016-07-24 13:59:08


Statistics


Total abstract view - 700
Downloads (from 2020-06-17) - PDF - 316

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2012 B. A. Frasin