Properties on subclasses of analytic functions mapped onto various domains and their applications in image enhancement

Abstract

This thesis delves into theoretical advances in Geometric Function Theory (GFT) by analyzing various subclasses of analytic and univalent functions. It begins with a detailed exploration of the subclass of bounded turning functions, denoted as BT and#1009;and#8722;1,L and convex functions Kand#1009;and#8722;1,L, associated with domains bounded by an epicycloid with and#1009; and#8722; 1 cusps. Fundamental function properties, such as coefficient inequalities and determinant evaluations, specifically the second and third Hankel determinants, are extensively investigated to establish new theoretical results. Furthermore, this research introduces a subclass of Sakaguchi-type functions associated with petal-shaped domain, namely SCt,and#961;, and examines coefficient bounds, Fekete Szeg¨o inequality, and Toeplitz determinants T2(2) and T3(1). Extending classical function theory, the study defines new subclasses of Sakaguchi-type functions based on the Mittag Leffler-type Poisson distribution series, formulating the class p and#8722; and#934;Sand#8727;(t,µ,and#957;,J,K). Several key function properties, including necessary and sufficient conditions, convex combinations, growth and distortion bounds, and partial sums are rigorously analyzed to provide deeper in sight into the structural behavior of these function classes. Next, this thesis explores the application of the mathematical framework developed in im proving retinal images and edge-detected images. A new enhancement approach is proposed, utilizing GFT-based coefficient bounds from p and#8722; and#934;Sand#8727;(t,µ,and#957;,J,K), where these coefficients are convoluted with input images to enhance visual quality. Performance metrics, including the Peak Signal-to-Noise Ratio (PSNR), the Structural Similarity Index Measure (SSIM), the Contrast Improvement Ratio (CIR) and the Absolute Mean Brightness Error (AMBE), are employed to evaluate the effectiveness of the proposed method. Additionally, a new approach for enhancing edge detection is introduced through SCt,and#961;, leveraging conformal mappings and coefficient constraints to define an optimized function class for dedicated co