Hankel inequalities of a novel subclass of biunivalent functions and their applications in image enhancement

Abstract

This paper introduces a new subclass of bi-univalent functions and establishes sharp bounds for their second- and third-order Hankel determinants. These theoretical results extend earlier work in geometric function theory and contribute to the study of coefficient problems for analytic and bi-univalent functions. Building on these findings, we develop an image enhancement algorithm that incorporates Hankel determinants into a convolution-based framework. The proposed method is evaluated using standard quality metrics such as Peak Signal to Noise Ratio, Structural Similarity Index Measure, Mean Squared Error, and Pearson Correlation Coefficient, and further analyzed through histogram plots and mesh visualizations. Experimental results on natural images, flower datasets, and brain stroke Computed Tomography (CT) scans demonstrate that the algorithm produces clearer, more detailed, and structurally preserved images compared to existing coefficient-based methods. The study highlights the dual significance of Hankel determinants, both as a theoretical tool in function theory and as a practical mechanism for improving digital image quality, thereby bridging mathematical analysis with real-world applications in medical imaging and computer vision.