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

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

$$ N_{(p,q)} (A) := \inf \left\lbrace \epsilon > 0 :\left(\frac{a}{\epsilon}\right)^p + \left(\frac{b}{\epsilon}\right)^q \leq 1 \right\rbrace.$$

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