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Let $f \in \mathbb{H}(\mathbb{D})$(I mean$f$ is holomorphic on $\mathbb{D})$ be such that $\forall z \in \mathbb{D}, Re f(z) > 0$ and $f(0) = a > 0$. Prove that: $|f'(0)| \leq 2a$.

I tried to solve it with Cauchy integral formula but I can't. Is there a solution with this method?

It's the problem 1.14 in "A Course in Complex Analysis and Riemann Surfaces(Wilhelm Schlag)".

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    $\begingroup$ Map, with a conformal $g$, the upper half plane to the unit disc in such a way that $a$ maps to $0$. In the link, you would take $\beta=a$. Then use Schwarz's lemma with $g(f(z))$. $\endgroup$
    – conditionalMethod
    Commented Dec 3, 2019 at 12:59
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    $\begingroup$ Consider a holomorphic function $\phi:H^+\to \Bbb D$ and consider the composition $\phi\circ f$ and then apply problem 1.12, where $H^+$ is the open upper half-plane. $\endgroup$
    – Sumanta
    Commented Dec 3, 2019 at 13:03
  • $\begingroup$ Also: math.stackexchange.com/q/1684925 $\endgroup$
    – Martin R
    Commented Dec 3, 2019 at 13:39

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