A log-convex generalization of Alzer–Fonseca–Kovačec type inequalities and applications

Abstract

We develop a multiplicative/logarithmic counterpart of the convexanalytic scheme of Phung–Huy [9] for Alzer–Fonseca–Kovačec type inequalities. At the scalar level, we establish comparison inequalities for strictly increasing log-convex functions on the logarithmic scale. We also record a derivative-weighted estimate expressed via the logarithmic derivative $r = (\log \varphi)'$, and we clarify parameter regimes where this derivative factor can improve the universal constant. Concrete applications are presented for the function $t^t$ and the Gamma function $\Gamma$. We then extend the scalar comparison to commuting positive definite matrices under unitarily invariant norms with the universal constant $\frac{v(1−v)}{\tau(1− \tau)}$. For the non-commuting case, we obtain “envelope” bounds via spectral pinching onto finite-dimensional abelian subalgebras, and we include a Heinz-centered quantitative estimate in the pinched abelian setting, combining a Lipschitz control for $g = \log \circ \varphi$ with the Heinz norm inequality under unitarily invariant norms.