Assume $A, B$ are self-ajoint compact operators. Is it true that $\|A+iB\|\le \|2A+iB\|$? Do we have a stronger inequality $\prod\_{k=1}^ns\_k(A+iB)\le \prod\_{k=1}^ns\_k(2A+iB)$ or even stronger one $s\_n(A+iB)\le s\_n(2A+iB)$, $n=1, 2, \ldots$, where $s\_n$ are s-numbers?



a monotone relation for s-numbers  
I just tried a few random matrices...
$$
\begin{align}
&A=\begin{bmatrix} -0.1 & -0.4\cr
-0.4&0\end{bmatrix},
\qquad
B=\begin{bmatrix} 1.5 & -0.5 + i\cr
-0.5 - i& 3.5\end{bmatrix},
\cr
&\|A+iB\| = \frac{7}{2}+\frac{\sqrt{61}}{10}\approx 4.28
\cr
&\|2A+iB\| = \frac{7}{2}+\frac{\sqrt{29}}{10}\approx 4.04
\end{align}
$$


