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

**My question** is it true that $$\|g\|\le \|f\|^2 \text{?}$$

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. 

EDIT: Sorry, my bound is definitely wrong. The L1-norm of $f,g$ is $1$, so the L2-norm will be at least $1/\sqrt{n}$. 
**My real question**: what bounds can we give for $||g||$ in terms of $||f||$?