For $f\in [0,1]^n$ with $\sum_i f_i = 1$, define the self-convolution $g=f*f$ with $$g_i = \sum_{j,k \mid j+k \equiv 0 \bmod n} f_j \cdot f_k$$ and define the L2-norm as $$||f|| = \sqrt{\sum_i f_i^2}.$$

My question is it true that $$||g||\le ||f||^2 ?$$

I arrived here by translating into the frequency domain where the convolution becomes multiplication as far as I understand. I also tried to get here with some facts about convolutions and Lp norms https://en.wikipedia.org/wiki/Young%27s_convolution_inequality but I got a much weaker bound.