Need a little help in the following:
Let $f(z)$ analytic function on $D = \{z\in\mathbb C: |z| < 1\}$. Define $\displaystyle d = \sup_{z,w \in D} |f(z) - f(w)|$.
Prove that |f'(0)| \leq \frac{d}{2}.
Need a little help in the following:
Let $f(z)$ analytic function on $D = \{z\in\mathbb C: |z| < 1\}$. Define $\displaystyle d = \sup_{z,w \in D} |f(z) - f(w)|$.
Prove that |f'(0)| \leq \frac{d}{2}.
Hint(s):
Note that $\frac 12$ is the best constant we can hope; take $f(z)=z$ to see that, and you can, by the same method, get a bound for the odd derivatives at $0$.