$f: D(a,1)\rightarrow\mathbb{C}$, then $\left | f(z) - f(a) \right | \leq k\left | z-a \right |$

Question

$f: D(a,1)\rightarrow\mathbb{C}$ holomorphic with $\left | f(z) - f(a) \right | \leq k$ for all $z \ in D(a,1)$

Then $$\left | f(z) - f(a) \right | \leq k\left | z-a \right | $$ for all $z \in D(a,1)$.

I am not sure how to approach this problem and would be very thankful for any help.

Answer 1

Apply Schwarz Lemma to the function $g(z)=\frac 1 k (f(z+a)-f(a))$ defined for $|z|<1$.