Let $z$ and $w$ be complex numbers. Than it is claimed that $$ |\sqrt{z+w} - \sqrt{z}| \leq \frac{|w|}{\sqrt{|z| + |w|}} $$ or $$ |\sqrt{z+w} + \sqrt{z}| \leq \frac{|w|}{\sqrt{|z| + |w|}} $$ I am hoping someone can show me how to perform such a perturbation analysis in complex analysis. I am able to prove this for real numbers via Taylor expansion.
2026-03-25 06:08:59.1774418939
Perturbation of complex square root function
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This claim is absolutely wrong. Choose $w=-1,z=1$ therefore$$|\sqrt{z+w}-\sqrt z|=1\\|\sqrt{z+w}+\sqrt z|=1$$while $${|w|\over \sqrt{|w|+|z|}}={1\over\sqrt 2}$$but $$1\not\le {1\over \sqrt 2}$$