In the second paragraph on Page 71 of the book Matrix Analysis by Bhatia, 1997, it says ``as a consequence of (III.12) we have Theorem III 4.4''. How can one get the inequality in Theorem III 4.4 from (III.12) for $\Phi\left(x_{1},\cdots,x_{n}\right)=\leftx_{1}\right+\cdots+\leftx_{n}\right$?

Inequality III.12 is the famous Lidskii majorization for Hermitian matrices $A$ and $B$, which says that $$\lambda^\downarrow(A)  \lambda^\downarrow(B) \prec \lambda(AB) \prec \lambda^\downarrow(A)  \lambda^\uparrow(B).$$ Now, recall the following simple but crucial fact: Fact. $x \prec y \implies$ $x\quad \prec_w\quad y$, where $\prec_w$ denotes weak majorization, and $x$ denotes the vector obtained from $x$ by taking elementwise absolute values. Now Problem II.5.11(vi) asks you to prove that $x\ \prec_w\ y\ $ iff $\Phi(x) \le \Phi(y)$ for every symmetric gauge function $\Phi$. Once you have proved / believed this, the inequality that you allude to follows after invoking the abovementioned "fact". 

