So I have the following question:
Let f be analytic in an open set which contains the closed unit disc $\overline{\mathbb{D}},$ and assume $M:=\max\{\textrm{Re}(f):|z|=1\}\geq0.$ Prove that for $z\in\mathbb{D},$ $$|f(z)|\leq \frac{1+|z|}{1-|z|}[M+|f(0)|].$$
So I am very inclined to believe this is an application of Schwarz's Lemma on some composition of conformal maps. However I don't feel like I have information regarding $f(\mathbb{D}).$ I wanted to shift $f$'s image into either the right or left half-plane, and Reimann's mapping Theorem guarantees such a conformal map, but I won't know what it looks like.
Also I know I draw conclusion about the range of conformal maps based on their behavior on the boundaries, but I only know that $f$ is analytic. So I can't be sure of what $f(\mathbb{D})$ looks like.
I tried looking at another post here, but they have stronger suppositions than I do, or at least so it seems to me.
As always, I appreciate any help or hints. Thanks.