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Sharp Tilting Inequalities
Abstract
For positive semidefinite matrices A,B and 1/2≤α≤1, we establish Tr [A Dlog(Aα+Bα)(Bα)] ≤ ss(1−s)1−sTr[AαB1−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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BibTeX
@misc{cheng2026sharp,
title = {{Sharp Tilting Inequalities}},
author = {Hao-Chung Cheng and Po-Chieh Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39598},
eprint = {2610.39598},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39598},
}
RIS
TY - UNPB
TI - Sharp Tilting Inequalities
AU - Hao-Chung Cheng
AU - Po-Chieh Liu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39598
UR - https://arxiv.org/abs/2610.39598
AB - For positive semidefinite matrices A,B and 1/2≤α≤1, we establish Tr [A Dlog(A^α+B^α)(B^α)] ≤ s^s(1−s)^1−sTr[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.
ER -