About a $2\pi$ periodical function
Posted: Thu Jul 14, 2016 6:56 pm
Let \( \displaystyle f \) be a \( \displaystyle C^1 , 2\pi \)-periodical function. If \[ \displaystyle \int_{0}^{2\pi}f(x)\mathrm{d}x = 0 \]show that
\[ \displaystyle \int_{0}^{2\pi} \left( f^{\prime}(x) \right)^{2} \mathrm{d}x \geq \int_{0}^{2\pi} \left( f(x) \right)^{2} \mathrm{d}x \]
and the equality holds if and only if \( \displaystyle f(x) = a\cos(x) +b\sin(x) \) for some constants \( \displaystyle a,b \in \mathbb{R} \).
\[ \displaystyle \int_{0}^{2\pi} \left( f^{\prime}(x) \right)^{2} \mathrm{d}x \geq \int_{0}^{2\pi} \left( f(x) \right)^{2} \mathrm{d}x \]
and the equality holds if and only if \( \displaystyle f(x) = a\cos(x) +b\sin(x) \) for some constants \( \displaystyle a,b \in \mathbb{R} \).