Some subclasses of univalent functions associated with $k$-Ruscheweyh derivative operator

Keywords: Univalent functions, Pochhammer ksymbol, Differential subordination, Hadamard product, kRuscheweyh derivative operator

Abstract

UDC 517.5
The purpose of the present paper is to investigate some subordination, other properties and inclusion relations for functions in certain subclasses of univalent functions in the open unit disc which are defined by $k$-Ruscheweyh derivative operator.

 

References

T. Bulboaca, Differential subordinations and superordinations. Recent results, House Sci. Book Publ., Cluj-Napoca (2005).

J. H. Choi, M. Saigo, H. M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. and Appl., 276, № 1, 432 – 445 (2002), https://doi.org/10.1016/S0022-247X(02)00500-0 DOI: https://doi.org/10.1016/S0022-247X(02)00500-0

R. D´ıaz, E. Pariguan, On hypergeometric functions and Pochhammer $k$-symbol, Divulg. Mat., 15, № 2, 179 – 192 (2007).

D. J. Hallenbeck, St. Ruscheweyh, Subordination by convex functions, Proc. Amer. Math. Soc., 52, 191 – 195 (1975), https://doi.org/10.2307/2040127 DOI: https://doi.org/10.1090/S0002-9939-1975-0374403-3

M.-S. Liu, On certain sufficient condition for starlike functions, Soochow J. Math., 29, 407 – 412 (2003).

S. S. Miller, P. T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J., 28, № 2, 157 – 171 (1981). DOI: https://doi.org/10.1307/mmj/1029002507

S. S. Miller, P. T. Mocanu, Univalent solutions of Briot – Bouquet differential equations, J. Different. Equat., 58, 297 – 309 (1985), https://doi.org/10.1016/0022-0396(85)90082-8 DOI: https://doi.org/10.1016/0022-0396(85)90082-8

S. S. Miller, P. T. Mocanu, Differential subordination, theory and applications, Ser. Monographs and Textbooks in Pure and Appl. Math., vol. 225, Marcel Dekker Inc., New York, Basel (2000).

P. T. Mocanu, Gh. Oros, A sufficient condition for starlikeness of order $alpha$, Int. J. Math. and Math. Sci., 28, № 9, 557 – 560 (2001), https://doi.org/10.1155/S0161171201011656 DOI: https://doi.org/10.1155/S0161171201011656

St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49, 109 – 115 (1975), https://doi.org/10.2307/2039801 DOI: https://doi.org/10.1090/S0002-9939-1975-0367176-1

E. T. Whittaker, G. N. Watson, A course of modern analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, gourth ed., Cambridge Univ. Press, Cambridge (1927), https://doi.org/10.1017/CBO9780511608759 DOI: https://doi.org/10.1017/CBO9780511608759

D. R. Wilken, J. Feng, A remark on convex and starlike functions, J. London Math. Soc. (Ser. 2), 21, 287 – 290 (1980), https://doi.org/10.1112/jlms/s2-21.2.287 DOI: https://doi.org/10.1112/jlms/s2-21.2.287

Published
24.01.2022
How to Cite
Seoudy, T. M. “Some Subclasses of Univalent Functions Associated With $k$-Ruscheweyh Derivative Operator”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 1, Jan. 2022, pp. 122 -36, doi:10.37863/umzh.v74i1.2337.
Section
Research articles