Inequality between Fourier coefficients implies inequality for $L^p$ norms on the circle

Question

Given two functions from $L_p [-\pi,\pi]$, where $p\geq 2$, $p$ is an even integer, and $f_n>|g_n|$ for every $n$ (where $f_n$ is the $n$th Fourier coefficient), I need to prove that $\|f\|_{L_p}\geq \|g\|_{L_p}.$ I have ran out of ideas, and would appreciate any help or advice. Preferably using tools available for an introductory harmonic analysis course. Thanks. Note: $f_n$ is not in $|\ |$ on purpose, implying $f_n$ are real and non-negative.

Answer 1

Let's write $p=2k$. The goal is to prove that $\|f^k\|_{L_2}\ge \|g^k\|_{L_2}$. By Parseval's identity, we can work with Fourier coefficients of $f^k$ and $g^k$. These are obtained by convolving the coefficients repeatedly:
$$ \widehat{f^k}(n) = \sum_{i_1+i_2+\dots +i_k=n} \hat f(i_1)\hat f(i_2)\cdots \hat f(i_k) $$ and similarly for $\widehat{g^k}(n)$. By the triangle inequality, $$ |\widehat{g^k}(n)|\le \sum_{i_1+i_2+\dots +i_k=n} |\hat g(i_1)\hat g(i_2)\cdots \hat g(i_k)| \le \widehat{f^k}(n) $$ and the conclusion follows.