### Journals Information

**
Mathematics and Statistics Vol. 11(4), pp. 634 - 639 DOI: 10.13189/ms.2023.110403 Reprint (PDF) (598Kb) **

## Properties of Classes of Analytic Functions of Fractional Order

**K R Karthikeyan ^{*}, Senguttuvan Alagiriswamy **

Department of Applied Mathematics and Science, National University of Science & Technology, Oman

**ABSTRACT**

The study of Univalent Function Theory is very vast and complicated, so simplifying assumptions were necessary. In view of the Riemann Mapping theorem, the most apt thing would be to replace an analytic function defined on an arbitrary domain with an analytic function defined in the unit disc and having a Taylor's series expansion of the form . The powers of the series are usually integers, so all the prerequisite results also support the study of analytic functions having a series expansion with integers powers. The main deviation presented here is that we have defined a subclass of analytic functions using a Taylor's series whose powers are non-integers. To make this study more comprehensive, Janowski function which maps the unit disc onto a right half plane has been used in conjunction with two primary tools namely Subordination and Hadamard product. Motivated by the well-known class of λ-convex functions, here we have defined a fractional differential operator which is a convex combination of two analytic functions. Using the defined fractional differential operator, we introduce and study a new class of analytic functions involving a conic region impacted by the Janowski function. Necessary and sufficient conditions, coefficient estimates, growth and distortion bounds have been obtained for the defined function class. Since studies of various subclasses of analytic functions with fractional powers are rare, here we have pointed out several closely related studies by various authors. However, the superordinate function is a familiar function which has lots of applications.

**KEYWORDS**

Fractional Calculus, Differential Operator, Subordination, Starlike Function, Convex Function

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] K R Karthikeyan , Senguttuvan Alagiriswamy , "Properties of Classes of Analytic Functions of Fractional Order," Mathematics and Statistics, Vol. 11, No. 4, pp. 634 - 639, 2023. DOI: 10.13189/ms.2023.110403.

(b). APA Format:

K R Karthikeyan , Senguttuvan Alagiriswamy (2023). Properties of Classes of Analytic Functions of Fractional Order. Mathematics and Statistics, 11(4), 634 - 639. DOI: 10.13189/ms.2023.110403.