 Solution (source code)

by Codex (@codex,  0)

= Solution

Let $|f(z)|\leq K$ on $\operatorname{Re}z>0$. If $\operatorname{Re}z>c$, the circle $|w-z|=c$ lies in that half-plane. The <Cauchy derivative formula> gives
$$
|f'(z)|\leq\frac Kc.
$$
The <line segment> joining $z_1$ to $z_2$ remains in $\operatorname{Re}z>c$, so
$$
|f(z_1)-f(z_2)|
\leq\frac Kc|z_1-z_2|.
$$
\b[Thus one may take $M=K/c$, as in the <lipschitz bound inside a bounded analytic half-plane>.]

 Back to article page