Improved Hermite-Hadamard bounds for (v,w)-strongly convex functions with applications to energy storage optimization

This paper develops improved Hermite–Hadamard integral inequalities for a generalized class of strongly convex functions defined by pairs of auxiliary functions and quantified by a convexity modulus parameter. Introducing this function class enables extensions to traditional convex analysis, yielding refined integral bounds that provide tighter approximations while containing classical results as special cases. The convexity modulus governs error quantification in these approximations, creating a theoretically rigorous framework. Implemented in energy storage optimization, this approach demonstrates 12–18% reduction in battery degradation costs, 8–15% improvement in state-of-charge management efficiency, 4–6% tighter capacity fade estimation bounds, and enhanced stability for renewable-integrated grids. The convexity modulus provides quantitative control, bridging theoretical approximations and practical performance, with specific applications in lithium-ion battery optimization (where it correlates with degradation rate), voltage regulation through storage control, and degradation-aware scheduling. By unifying classical strong convexity with function-pair-defined convexity, these results offer robust tools for energy storage optimization under variable operating conditions, advancing both mathematical theory and power systems engineering.