$\underset{f\in \mathcal{F}_{\alpha,\beta}}{\sup} |f'(\alpha)|$ of $f$ holomorphic such that $|f(z)|\leqslant 1$ and $f(\alpha)=\beta$

Question

Question

Let $\mathcal{F}_{\alpha,\beta}$ be the set of all holomorphic functions $f$ on the open unit disk $\mathbb{D}$ such that $|f(z)|\leqslant 1$ for all $z\in \mathbb{D}$ and $f(\alpha)=\beta$. What is $$\underset{f\in \mathcal{F}_{\alpha,\beta}}{\sup} |f'(\alpha)|$$

My attempt I thought of using Cauchy equality, and this gives me $\beta$ as a bound. But then I can't manage to find a $f$ corresponding. I thing this is equivalent to searching $f$ such that $\int_0^{2\pi} f(re^{it})dt=\int_0^{2\pi} f(re^{it})e^{-it}dt$ but I'm not sure.

Could someone help ?

Answer 1

Hints for you:

Use the following link to find bijective holomorphic functions $g,h:\mathbb{D}\to \mathbb{D}$ such that $g(0)=\alpha$ and $h(\beta)=0$.

Can we characterize the Möbius transformations that maps the unit disk into itself?

Now, let $j=h\circ f\circ g$.

Notice that $j(0)=0$. Therefore, the Schwarz lemma tells us that $|j'(0)|\leq 1$, with equality attained if and only if $j(z)=az$ for some complex $a$ with modulus $1$.

But also, $j'(0)=h'(f\circ g(0))\times(f\circ g)'(0)=h'(\beta)\times f'(g(0))\times g'(0)=h'(\beta)\times f'(\alpha)\times g'(0)$.