Inequality for the difference of square roots of complex numbers

Question

For $a, b \in \mathbb{C}$, I would like to know if there is a universal constant such that either $$ |\sqrt{a + b} - \sqrt{a}| \leq \frac{C |b|}{\sqrt{|a| + |b|}}. $$ or $$ |\sqrt{a + b} + \sqrt{a}| \leq \frac{C |b|}{\sqrt{|a| + |b|}}. $$ I have computed this sum on several examples and it seems to be true but I would like a formal proof.

Answer 1

Let $a=-1-\epsilon\, i$, $b=2\,\epsilon\, i$. Then $\sqrt{a+b}\sim i$ and $\sqrt{a}\sim-i$ hence $|\sqrt{a+b}-\sqrt{a}|\sim2$ as $\epsilon\to0$, while the right hand side is smaller than $2\,C\,\epsilon$.

For the inequality to hold you will have to take $a$ and $a+b$ in a region of the form $|\arg z|<\theta$ for some $\theta<\pi$.