A preprint reports an exact answer to a question about finite matrix families: applying any nonnegative concave function to the relevant absolute values in a Lee-type Schatten norm inequality does not change the inequality’s sharp best constant. The linear and transformed problems have the same best constant for every finite matrix dimension, every finite number of matrices, and every exponent 1 ≤ p ≤ ∞.
The setup compares the linear baseline f(t)=t with the formulation using an arbitrary nonnegative concave function—one that remains nonnegative while bending downward. It ranges over families A₁,…,Aₘ of complex n×n matrices with m,n ≥ 1; cases with a zero denominator are excluded from the supremum. Here, “sharp” means best possible: the paper states that the linear best constant bounds the concave-function inequality and that the bound cannot be improved.
How the proof builds the transfer
The proof starts with a single cap function, hᵣ(t)=min{t,r}, which cuts an input off at a ceiling r. For r>0 and 1 ≤ p < ∞, the best constant for this clipped version already equals the linear best constant.
Next, the argument considers a nonzero finite nonnegative combination of cap functions. After taking the p-th power, that combination can be represented by a finite positive Borel measure, so the single-cap result can be assembled across the mixture.
The multiple-matrix step is noncommutative: interactions among different matrices are absorbed into a UCP map, meaning a unital completely positive map. The Araki–Lieb–Thirring inequality controls the remaining sandwich term, while commutativity is used only for functions of one matrix component at a time.
From caps to the full class
Those ingredients extend the inequality to every nonnegative concave function. The operator-norm endpoint is obtained by letting p tend to infinity, completing the stated range through p=∞. The conclusion is lossless: once the sharp constant for the linear problem is known, the same value transfers to the concave formulation.
What the theorem gives at key exponents
The theorem gives exact values at several important points. At p=1, the constant is 1 and best possible. At p=2, for fixed dimensions, it is √((1+√min{m,n})/2). At p=∞, it is √min{m,n}.
For p=2, the result also gives a dimension-uniform figure: sup over n ≥ 1 of the concave constant is √((1+√m)/2). In the two-matrix case, for 2 ≤ p < ∞, the all-dimensional constant is expressed through the number xₚ > 1 satisfying xₚᵖ = 2xₚ + 1: Cₚ = √(xₚ(xₚ + 1))/(xₚᵖ + 1)^(1/p).
What remains unresolved
The transfer theorem does not, by itself, fill the remaining gap in the underlying linear theory. The paper reports that a Tang–Zhang candidate formula fails throughout the specific interval 1 < p < 2, while the correct sharp linear constants in that interval remain undetermined. The concave problem is therefore reduced to the corresponding linear best-constant problem there; no replacement formula is supplied.
The scope is finite-dimensional. The formal optimization is over finite m and n, and the supplied text establishes no infinite-dimensional extension. It also does not give the missing fixed-dimensional or all-dimensional linear constants for 1 < p < 2.
The paper’s status
The work is listed as arXiv:2608.25989v1 [math.FA], dated 26 Aug 2026, with no journal listed in the supplied metadata. Its acknowledgments mention helpful correspondence and AI assistance but do not identify a funding source.
Paper data and sources
Original title: Sharp Concave-Function Transfer for Lee-Type Schatten Norm Inequalities
Authors: Xing Li
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text