I am investigating a function defined in terms of the singular values of matrices. I initially simplified the problem by focusing on the eigenvalues of 2x2 Hermitian, positive-definite matrices.

For given $p,q \geq 1$, the function $N_{(p,q)} (A)$ is defined as follows:

$$N_{(p,q)} (A) = \inf \lbrace \epsilon>0 | (a/\epsilon)^p + (b/\epsilon)^q ≤ 1 \rbrace.$$

Here, $a$ and $b$ represent the singular values of A, with $a$ being the largest.

The main question is, how can one show that this function satisfies the triangle inequality?

I am particularly interested in connections to Horn's inequalities, as they concern the singular values of sums of matrices and seem highly relevant to the problem at hand. Furthermore, the underlying function $\Theta_{(p,q)} (a,b) = a^p + b^q$ is a convex modular functional, which is pertinent.

I am seeking guidance or references that could provide some insight into this problem. Has anyone encountered a similar problem or does anyone have suggestions on potential approaches?