This study addresses two pivotal aspects. Firstly, it introduces a novel convexity class termed as geometrically $(s, P)$ -functions, detailing the characteristics of functions within this class and presenting the Hermite-Hadamard inequality (H-H I). Subsequently, this is leveraged to establish new theorems for trapezoidal inequalities. Secondly, this research focuses on the analysis of H-H I using simulations in the study of geometrically $(s, P)$ -functions. The study emphasizes the use of symbolic and numerical transformations of functions and their derivatives, combined with random parameter generation. Visualization techniques, including 3D scatter plots, are crucial in illustrating the complexities of these inequalities. The analytical approach adopted in this study not only sheds light on H-H I but also highlights the significant role of simulations in unraveling the intricacies of complicated mathematical functions.
