Digital image processing is essential in fields such as medical imaging, satellite imagery, and autonomous vehicles. Besides its applications, texture enhancement is vital for improving image visibility while preserving geometric structure. To improve visual clarity and image texture, we introduced a novel algorithm for texture enhancement. Initially, we determined coefficient inequalities of ${\mathcal{S}}^{\ast }({\phi }_H)$ which is a subclass of $\lambda$- generalized Sakaguchi type functions, and examined upper bounds of second and third order Hankel determinants and obtained sharp results. We proposed a texture enhancement algorithm that utilized convolution masks derived from Hankel determinants with the pixels of segmented image. This approach provided a mathematical tool for computer-aided dermatological analysis, which was used to enhance lesion boundaries and structural visibility in dermascopic images. Image quality was evaluated using different quality metrics like contrast, correlation, energy, homogeneity, and entropy. The experimental results demonstrated uniform texture enhancement and improved edge preservation in all directions. Comparative analysis showed the efficacy of our proposed algorithm compared to existing methods reported in this study, proving it suitable to enhance image quality.