# A complex integral with boundary on the norm

I was asked this question. I couldn't even manage to find a starting point on the question. Any help is welcomed.

P.S: I know it is from a book but I do not know from which one. If you have any idea which book it may be, I can search online through the book's name as well.

• Do you know that $\left|\int f(z)dz\right|\le \int \left|f(z)\right| dz$? – user247327 Apr 17 at 14:22

Note that\begin{align}\left\lvert x+i\sqrt{1-x^2}\cos\theta\right\rvert&=\sqrt{x^2+(1-x^2)\cos^2\theta}\\&\leqslant\sqrt{x^2+1-x^2}=1.\end{align}Therefore,\begin{align}\bigl\lvert P(x)\bigr\rvert&\leqslant\frac1\pi\int_0^\pi\left\lvert x+i\sqrt{1-x^2}\cos\theta\right\rvert^n\,\mathrm dx\\&\leqslant\frac\pi\pi\\&=1.\end{align}