Caputo-hybrid Hermite-Hadamard and Newton’s type inequalities in multiplicative calculus with applications

The study addresses the need for a comprehensive mathematical framework for multiplicative (geometric) P-convex functions, specifically in the context of Pcap (Proportional Caputo-Hybrid) operators. The research identifies a significant gap in existing literature concerning the formulation of specific inequal-ities and their applications to this class of functions. This study aims to fill the gap by developing and presenting new Hr Hd (Hermite-Hadamard) type inequalities tailored to multiplicative P-convex functions using Pcap operators. The lack of a detailed understanding and established results in this area underscores the importance of the study. The main contributions include the derivation of novel inequalities that ex-tend the traditional concepts of convexity into a multiplicative framework. Additionally, a new Newton?s type identity applicable to multiplicatively P-differentiable functions is introduced, which provides fresh insights and tools for analysis in this domain. The practical implications of the findings are demonstrated through applications to special means and type-1 modified Bessel functions. These applications not only validate the theoretical results, but also highlight their versatility and relevance in broader mathemat-ical contexts. Research significantly advances the theoretical understanding of multiplicative P-convex functions, offering new analytical tools and frameworks. This advancement has potential implications for various mathematical and applied fields, including optimization and numerical analysis. The study suggests that future research could explore additional applications and extend the theoretical framework to other types of functions and operators, thereby broadening the scope and impact of the findings in this emerging area of study.