Let $z$ and $w$ denotes two complex numbers. I'm looking forward to finding a way to show that
$$ \min\{|\sqrt{z+w}+\sqrt{w}|,|\sqrt{z+w}-\sqrt{w}| \}\leq |z|(|z|+|w|)^{-1/2} $$ where the $\sqrt{\cdot}$ is taking value on the principal branch. Can someone give me a hint on proving this ?
The inequality does not hold true in general. For a counterexample, consider $\,z=1, w=-1\,$:
$$ \min\left\{\left|\sqrt{1+(-1)}+\sqrt{-1}\right|,\left|\sqrt{1+(-1)}-\sqrt{-1}\right| \right\} = |i| = 1 \,\color{red}{\gt}\, \frac{1}{\sqrt{2}}= |1|\,\big(|1|+|-1|\big)^{-1/2} $$