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arXiv · 2610.39598

Sharp Tilting Inequalities

Hao-Chung Cheng·Po-Chieh Liu
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Abstract

For positive semidefinite matrices $A,B$ and $1/2\leα\le1$, we establish $\operatorname{Tr}\!\left[A\,D\log(A^α+B^α)(B^α)\right] \le s^s(1-s)^{1-s}\operatorname{Tr}[A^αB^{1-α}]$, where $s=(1-α)/α$, with the optimal coefficient. The proof uses a superoperator representation of the directional derivative of the logarithm and an Araki-type estimate for the derivative of a fractional power. Both this inequality and its symmetric-quotient counterpart extend to positive trace-class $A$ and positive, possibly unbounded, self-adjoint $B$ on a separable Hilbert space.

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Hao-Chung Cheng, Po-Chieh Liu. 2026-10-29. Sharp Tilting Inequalities. https://arxiv.org/abs/2610.39598

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