I know the following is a well-known result.
Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. Show that $$ 2|f^{'}(0)| \le \sup_{z, w \in D} |f(z)-f(w)|$$ Furthermore, there is equality if and only if $f$ is linear.
I need some reference about the second part, i.e. there is equality if and only if $f$ is linear.
This was first proved by Landau and Toeplitz in 1907. A reference for the proof (and for generalizations) is the paper Area, capacity and diameter versions of Schwarz's lemma by Burckel, Marshall, Minda, Poggi-Corradini and Ransford.
See Theorem 1.3 here
Edit.
Let us first prove the inequality: $$\sup_{z,w}|f(z)-f(w)|\geq\sup_z|f(z)-f(-z)|=\sup|g(z)|\geq|g'(0)|=2|f'(0)|,$$ where the Schwarz Lemma was applied to $g(z)=f(z)-f(-z)$. Equality in Schwarz lemma can happen only if $g(z)=kz$, thus $f(z)=kz+\phi(z)$, where $\phi$ is even.
Now let us see when equality is possible in the first inequality. We must have $$\sup_{|z|<1,|w|<1}|k(z-w)+\phi(z)-\phi(w)|=2|k|,$$ which an even function $\phi$ analytic in the unit disc. We have to derive from here that $\phi$ is constant.
In the reference http://www.math.wustl.edu/~geknese/schwarzpoly.pdf, where extremal functions for the Schwarz lemma in the polydisc are described. But those functions are extremal at every point. And our function is extremal at only one point, the origin.
I can prove that $\phi$ is constant only under the additonal restriction that $\phi$ is differentiable in the closed disc. WLOG $k=1$. Let $|z|=1$ and put $w=-ze^{it}$, where $t$ is small. Then, neglecting the high powers of $t$, we must have $$|2+it-\phi'(z)it|\leq 2$$ This implies that $\phi'(z)$ must be real for all $z$ on the unit circle. But such function must be constant, and as $\phi$ is even, we conclude that $\phi$ is constant.
To get rid of the additional assumption of differentiability, one may combine this and and this.
