Analytic Functions Expressed with q-Poisson Distribution Series

Abstract

Recently, the $q$- derivative operator has been used to investigate several subclasses of analytic functions in different ways with different perspectives by many researchers and their interesting results are too voluminous to discuss. The $q$- derivative operator are also used to construct some subclasses of analytic functions. In this study, we introduce certain subclasses of analytic and univalent functions in the open unit disk defined by $q$-derivative. Here, we give some conditions for an analytic and univalent function to belonging to these classes. Also, in the study, we define two functions using $q$-derivative and we aim to find the conditions for this functions to belonging to defined above subclasses of analytic functions.

Keywords

• q-poisson distribution
• analytic function
• univalent function
• q-derivative
• starlike function
• convex function

References

1. Altintaş O, Owa S. On subclasses of univalent functions with negative coefficients. Pusan Kyongnam Mathematical Journal. 4, 1988, 41-56.
2. Duren PL. Univalent Functions. Grundlehren der Mathematischen Wissenshaften, Bd. 259, New York, Springer-Verlag, Tokyo, 1983, 382p.
3. Goodman AW. Univalent Functions. Volume I, Polygonal, Washington, 1983, 246p.
4. Jackson FH. On $q$-functions and a certain difference operat\"{o}r. Trans. Roy. Soc. Edin. 46, 1908, 253-281.
5. Moustafa AO. A study on starlike and convex properties for hypergeometric functions. Journal of Inequalities in Pure and Applied Mathematics. 10(3), 2009, article 87, 1-16.
6. Mustafa N, Nezir V. On Subclasses of Analytic Functions Defined by $q$- Derivative and their Some Geometric Properties. $3^{rd} $ International Conference on Mathematical and Related Sciences: Current Trends and Developments, ICMRS -- 2020, 20-22 November, 2020, pp 129-1335.
7. Mustafa N, Korkmaz S. Analytic Functions Expressed with Poisson Distribution Series and their Some Properties. Journal of Contemporary Applied Mathematics, 2020 (submitted).
8. Porwal S, Dixit KK. An application of generalized Bessel functions on certain analytic functions. Acta Universitatis Matthiae Belii. Series Mathematics, 2013, 51-57.

9. Srivastava HM, Owa S. Current Topics in Analytic Function Theory. World Scientific, Singapore, 1992, 456p.