On Uniformly Starlike and Convex Univalent Functions

Document Type : Regular research papers

Authors

1 Mathematics and Computer Science Department, Faculty of Science, Beni-Suef University, Egypt

2 Higher Institute of Engineering and Technology New Damietta, Egypt.

3 Dept. of Math. and Computer Sci., Faculty of Science, Beni Suef University, Egypt.

10.21608/jfca.2025.370655.1167

Abstract

In (1908) Jackson generalized the ordinary derivative by introducing the q-difference derivative, which became an essential tool in the study of q-calculus. Later, (2013) Brahim and Sidomon, introduced a further generalization known as the symmetric q-derivative operator, which has significant applications in various mathematical fields. This paper's primary goal is to investigate how the symmetric q-derivative can be used to define a novel class of convex and uniformly starlike univalent functions inside the complex plane's open unit disk. Numerous intriguing geometrical and analytical features are present in this recently described class of functions. In this study, we establish a wide range of characterizations for these functions, such as coefficient estimates, distortion theorems, some radii of starlikeness, convexity, close -to- convexity. Furthermore,, we determine sharp lower bounds for the ratios of the functions in this class and its partial sums ℱ_{v} in the forms ((ℱ(z))/(ℱ_{v}(z))),((ℱ_{v}(z))/(ℱ(z))), ((ℱ′(z))/(ℱ′v(z))) and ((ℱ_{v}′(z))/(ℱ′(z))). The findings reported in this study provide a substantial contribution to the fields of q-calculus and geometric function theory by providing fresh viewpoints and possible uses in complex analysis.

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