Abstract
In this paper, we study certain classes of analytic functions which satisfy a subordination condition and are associated with the crescent-shaped regions. We first give certain integral representations for the functions belonging to these classes and also present a relevant example. Making use of some known lemmas, we derive sufficient conditions for the functions to be in these classes. Some results on coefficient estimates are also obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. K. Aouf, J. Dziok and J. Sokół, “On a subclass of strongly starlike functions”, Appl. Math. Letters, 24, 27–32, 2011.
J. Dziok, R. K. Raina and J. Sokół, “On alpha–convex functions related to shell–like functions connected with Fibonacci numbers”, Appl.Math. Comput., 218, 966–1002, 2011.
J. Dziok, R. K. Raina and J. Sokół, “Certain results for a class of convex functions related to a shell–like curve connected with Fibonacci numbers”, Comput. Math. Appl., 61 (9), 2605–2613, 2011.
J. Dziok, R. K. Raina and J. Sokół, “On a class of starlike functions related to a shell–like curve connected with Fibonacci numbers”, Math. Comput.Modelling, 57, 1203–1211, 2013.
J. Dziok, R. K. Raina and J. Sokół, “Differential subordinations and alpha–convex functions”, Acta Math. Scientia, 33B, 609–620, 2013.
P. L. Duren, Univalent Functions (Springer, New York, 1983).
S. Kanas, “Alternative characterization of the class k–UCV and related classes of univalent functions”, SerdicaMath. J., 25, 341–350, 1999.
S. Kanas and A. Wisniowska, “Conic regions and k–uniform convexity”, J. Comput. Appl.Math., 105, 327–336, 1999.
S. Kanas and H. M. Srivastava, “Linear operators associated with k–uniformly convex functions”, Integral Transform. Spec. Funct., 9, 121–132, 2000.
S. Kanas and A. Wisniowska, “Conic domains and starlike functions”, Rev. Roumaine Math. Pures Appl., 45, 647–657, 2000.
F. R. Keogh and E. P. Merkes, “A coefficient inequality for certain classes of analytic functions”, Proc.Amer. Math. Soc., 20, 8–12, 1969.
W. Ma and D. A. Minda, “Unified treatment of some special classes of univalent functions”, Proceedings of Conf. on Complex Analysis, Eds. Z.Li et al, S. Int. Press, 157–169, 1994.
S. S. Miller and P. T. Mocanu, “On some classes of first order differential subordinations”, Michigan Math. J., 32, 185–195, 1985.
S. S. Miller, P. T. Mocanu, Differential Subordinations, Theory and Applications (Dekker, Basel 2000).
Z. Nehari, ConformalMapping,McGraw–Hill, New York, 1952.
R. K. Raina and J. Sokół, “Some properties related to a certain class of starlike functions”, C. R. Acad. Sci. Paris, Ser. I, 353, 973–978, 2015.
J. Sokół, “Coeffcient estimates in a class of strongly starlike functions”, Kyungpook Math. J., 49, 349–353, 2009.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © R. K. Raina, P. Sharma, J. Sokół, 2018, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2018, No. 6, pp. 83–93.
About this article
Cite this article
Raina, R.K., Sharma, P. & Sokół, J. Certain Classes of Analytic Functions Related to the Crescent-Shaped Regions. J. Contemp. Mathemat. Anal. 53, 355–362 (2018). https://doi.org/10.3103/S1068362318060067
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1068362318060067
Keywords
- Analytic functions
- convex functions
- starlike functions
- subordination
- k-uniformly convex
- k-uniformly starlike