Let for a matrix A $\sigma(A)=(\sigma_1(A) \ldots \sigma_n(A))$ be a sequence of it singular values. The p-th Schatten norm is defined as $$ \|A\|_{S_p}=\|\sigma(A)\|_p, \quad 1\leq p \leq \infty. $$ Let $q, k$ are $1\leq q<p<k\leq \infty$.
Is there are any relations/ inequalities between $\|A\|_{S_p}, \|A\|_{S_q}, \|A\|_{S_k}$?
For $p$-norms, we have the standard inequalities for $q < p$, $\|x\|_p \le \|x\|_q \le n^{1/q-1/p}\|x\|_p$ (see this). These inequalities can be proven from Hölder's inequality. Thus, for $q<p$, it follows that $\|A\|_{S_p} \le \|A\|_{S_q} \le n^{1/q-1/p}\|A\|_{S_p}$. Moreover, these inequalities are tight, which follows from the tightness of the vector $p$-norms. ($\sigma(A) = (1,0,\ldots,0)$ and $\sigma(A) = (1,\ldots,1)$ should make the inequalities strict, if I remember correctly.)