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Title On Hankel Determinant of the inverse of q-Bounded Turning Functions.

Author(s) Zia Mahmood Razwi

Abstract The aim of this thesis is to investigate the Hankel determinants of the inverse functions of a subclass of univalent functions known as ˇq-bounded turning functions. This class, which generalizes classical bounded turning functions by incorporating the parameter ˇq, has attracted attention due to its connections with fractional analysis and ˇq-calculus. In this work, we focus on estimating the third Hankel determinant for the inverse functions associated with this class. By leveraging the analytical properties of ˇq-Carathéodory functions and the relationships between a function’s coefficients and those of its inverse, we derive an exact inequality for the determinant. Using tools from ˇq-calculus, subordination theory, and coefficient bounds, we obtain new sharp bounds that explicitly illustrate how the parameter ˇq influences the determinant’s value. The results not only generalize known outcomes for classical bounded turning functions but also provide new insights into the analytic structure and geometric behavior of inverse ˇqbounded turning functions in the open unit disc. Furthermore, we discuss geometric properties of the inverses and explore applications to extremal problems, thereby extending the understanding of coefficient problems in Geometric Function Theory. A minor graphical analysis is also performed to validate the new results against the classical literature.

Type Thesis/Dissertation MS

Faculty Engineering and Computer Science

Department Mathematics

Language English

Publication Date 2025-12-31

Subject Mathematics

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