A new class of log-convexity inequalities and applications

In this paper, we develop a new class of refined inequalities for log convex functions, motivated by and extending the classical Young-type inequality. Our method generalizes the recent refinements established by Hu, as well as their subsequent extensions by Ighachane et al., through a piecewise refinement technique. The resulting inequalities provide sharp multiple-term improvements for log-convex functions on $[0,1]$ and further yield two-weight versions that explicitly capture the dependence on pairs of points $0 < x \le y < 1$. We further extend our results to general intervals and to weighted power means, obtaining new refined and reverse estimates for the p-power mean interpolation for $p\le 0$. As applications, we derive strengthened Young-type inequalities for unitarily invariant norms, and related results for positive definite matrices. Finally, by exploiting the log-convexity of numerical radius mappings under unitarily invariant norms, we obtain refined Young-type bounds for the numerical radius and its generalized forms. Our results unify and significantly sharpen several known inequalities in convexity theory, operator means, and matrix analysis.