TITLE: Proving $|\int_{C}f(z)\, dz|\leq10M$ when $|f(z)|\leq M$ on $D=\{|z|=50\}$
QUESTION [6 upvotes]: I am trying to solve a question from my complex analysis test that
I didn't manage to do during the test in order to practice for the
next exam.
The problem is as follows: 

Let $f:\,\mathbb{C}\to\mathbb{C}$ be analytic function. Assume that
  $|f(z)|\leq M$ for every $z$ s.t $|z|=50$.
Consider the upper half of the ellipse $C=\{2x^{2}+y^{2}=50|\,
 y\geq0\}$. 
Prove $|\int_{C}f(z)\, dz|\leq10M$

What I tried:
Since $f$ is analytic, if $|C|\leq L$ then if $|f(z)|\leq M$ on
$C$ we get that $|\int_{C}f(z)\, dz|\leq ML$.
We also know that $$\max_{z\leq50}|f(z)|=\max_{z=50}|f(z)|\leq M$$
If $$2x^{2}+y^{2}=50$$ then $$x^{2}+y^{2}=50-x^{2}\leq50$$ hence for
every $z$ on $C$ we have $|f(z)|\leq M$.
From here I believe it only remains to show $|C|\leq10$ which I am
unable to do.
Can someone please help me out ? I guess that I am on the right track,
but I couldn't bound $|C|$ from above by $10$

REPLY [2 votes]: Let $\gamma(t) := t$ for $t \in [-5,5]$. $C + \gamma$ is a closed path, hence
 $$\int_{C} f(z) \, dz + \int_{\gamma} f(z) \, dz = 0$$ by Cauchy's integral theorem. Thus $$\left| \int_C f(z) \, dz \right| = \left|\int_{\gamma} f(z) \, dz \right| \leq M \cdot |\gamma| = 10 M$$

Actually, it's impossible to prove $|C| \leq 10$. To find a lower bound for $|C|$ calculate the length of the red path: 
$\hspace{80pt}$ 
By Pythagorean theorem, this length is equal to $$2 \cdot \sqrt{5^2+\sqrt{50}^2} = 2 \cdot \sqrt{75} \approx 17,32$$ 
Hence in particular $|C| >17$.