Sharp quasi-reverse Minkowski constant for exponents below two

Determine the sharp constant \(\widehat C_p\) for every exponent \(1<p<2\) such that, for all operators \(A,B\) on a finite-dimensional Hilbert space, \(\|A+B\|_p\leq\widehat C_p\bigl\||A|+|B|\bigr\|_p\).

Background

The paper proves that the proposed constant $C_p$, defined by $x_p^p=2x_p+1$ and $C_p=\sqrt{x_p(x_p+1)}/(x_p^p+1)^{1/p}$, is optimal for the two-summand Schatten-norm inequality when $p\geq2$. For $1

The authors first analyze a family of rank-one examples whose quotient is represented by $F_p(c,d)$, and show that its maximum exceeds the proposed value $C_p$. They then give a higher-dimensional numerical example, specifically for $p=6/5$, whose quotient exceeds the maximum of $F_p(c,d)$. Consequently, identifying the universal optimal constant $\widehat C_p$ for each $1