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Let $\mathbb{C_+}=\{z\in\mathbb{C} :\text{Im}(z)>0\}$. If $f: \mathbb{C_+} \to \mathbb{C_+}$ is analytic and $a\in \mathbb{C_+}$, is $|f'(a)| \leq \frac{\text{Im} (f(a))}{\text{Im}(a)}$?

This seems like it would be related to Schwarz's lemma, but we are in $\mathbb{C_+}$ instead of in $\mathbb{D}$.

Tiberio
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  • The upper half plane ${z\in\mathbb{C} : \text{Im}(z)>0}$. – Tiberio Jan 03 '20 at 03:00
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    Yes, related to the Schwarz lemma, and can be proved by finding suitable conformal maps between the upper halfplane and the unit disk. And yes, I just noticed that $\mathbb{C}_+$ is defined as the upper halfplane... – Lukas Geyer Jan 03 '20 at 03:01
  • Ok, so I should find $g:\mathbb{C_+} \to \mathbb{D}$ and then consider $g(f(z))$? – Tiberio Jan 03 '20 at 03:05
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    If you want to apply the Schwarz lemma directly, find two conformal maps $g$ and $h$ from the upper halfplane to the unit disk, with $g(a) = 0$ and $h(f(a))=0$, then consider $h(f(g^{-1}(z)))$. – Lukas Geyer Jan 03 '20 at 03:18
  • That makes sense, thank you! – Tiberio Jan 03 '20 at 03:25
  • Also: https://math.stackexchange.com/q/2735030/42969. – Martin R Jan 03 '20 at 05:12

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