Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 18 (2023), 259 -- 272

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

GEOMETRIC DISTRIBUTION SERIES CONNECTED WITH CERTAIN SUBCLASSES OF UNIVALENT FUNCTIONS

Masoumeh Taliyan, Shahram Najafzadeh and Mohammad Reza Azimi

Abstract. In this paper, we consider the class of normalized analytic functions of the form f(z)=z+∑n=2∞ anzn. Following this functions, we define the functions whose coefficients are probabilities of the geometric distribution series and other special modes of this series. Also, we consider different special classes of f(z). In the following we consider some lemmas that make connection between defined special classes with the function f(z). Follower of this topic we will consider the theorems that make connection between defined classes with the functions whose coefficients are probabilities of geometric distribution series. Also we define Alexander-type integral operator and find the necessary and sufficient conditions for being this operator to defined general classes.

2020 Mathematics Subject Classification: 30C45; 30C50
Keywords: Analytic functions, Convolution product, Subordinate, Geometric distribution series

Full text

References

  1. M. K. Aouf, On certain subclass of analytic p-valent functions of order alpha, Rend. Math. Appl. (7)8(1988), 89--104. Zbl 0686.30012. MR 0986230.

  2. R. Bharati, R. Parvatham and A. Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math. 28(1997), 17--32. Zbl 0898.30010. MR 1457247.

  3. T. Bulboaca, Differential subordinations and superordinations: Recent results, House of Scientific Book Publ, Cluj-Napoca, 2005.

  4. T. R. Caplinger and W. M. Causey, A class of univalent functions, Proc. Amer. Math. Soc. 39(1973), 357--361. MR0320294. Zbl 0267.30010.

  5. A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56(1)(1991), 87--92. MR1097287. Zbl 0744.30010.

  6. A. W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl. 155(1991), 364--370. MR1097287. Zbl 0726.30013.

  7. O. P. Juneja and M. L. Mogra, A class of univalent0419.30014 functions, Bull. Sci. Math. 103(4)(1979), 435--447. MR0548919. Zbl 0419.30014.

  8. S. Kanas and H. M. Srivastava, Linear operators associated with K-uniformly convex functions, Integral Transforms Spec. Funct. \bf 9(2)(2000), 121--132. MR1784495. Zbl 0959.30007.

  9. S. Kanas and A. Wisniowska, Conic regions and K-uniform convexity, J. Comput. Appl, Math. 105(1-2)(1999), 327--336. MR1690599. Zbl 0944.30008.

  10. S. Kanas and A. Wisniowska, Conic domains and starlike functions, Rev. Roumaine Math. Pures Appl. 45(2000), 647--657. MR1836295. Zbl 0990.30010.

  11. K. Löwner, Untersuchungen über die verzerrung bei Kanformen Abbidungen des Einheitskreises |z|<1, die durch Funktionen mit nichtverschwindender Ableitung geliefert werden, Leipzig Ber. 69(1917), 89--106.

  12. S. S. Miller and P. T. Mocanu, Differential subordination: Theory and applications, Monographs and Textbooks in Pure and Applied Mathematics 225, Marcel Dekker Inc., New York and Basel, 2000. MR1760285.

  13. R. H. Nevanlinna, Über die konforme Abbildung von sterngebieten, Helsingfors Centraltr, 1921.

  14. F. Rφnning, On starlike functions associated with parabolic regions, Ann. Univ. Mariae Curies-Sklodowska Sect. A. 45(1991), 117--122. MR1322145. Zbl 0769.30011.

  15. F. Ronning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Soc. 118(1993), 189--196. MR1128729. Zbl 0805.30012.

  16. S. Shams, S. R. Kulkarni and J. M. Jahangiri, Classes of uniformly starlike and convex functions, Int. J. Math. Sci. 55(2004), 2959--2961. MR2145534. Zbl 1067.30033. 0801679S. L. Shukla and Dashrath, On a class of univalent functions, Soochow J. Math. 10(1984), 117--126. MR0801679. Zbl 0578.30008.



M. Taliyan
Department of Mathematics
Faculty of Sciences, University of Maragheh, 55181-83111, Golshahr, Maragheh, Iran
e-mail: taliyan.ma2018@gmail.com
Sh. Najafzadeh
Department of Mathematics
Payame Noor University, P. O. Box 19395-3697, Tehran, Iran
e-mail: najafzadeh1234@yahoo.ie, shnajafzadeh44@pnu.ac.ir
M. R. Azimi
Department of Mathematics
Faculty of Sciences, University of Maragheh, 55181-83111, Golshahr, Maragheh, Iran
e-mail: mhreza.azimi@gmail.com

http://www.utgjiu.ro/math/sma