Is there an $\epsilon>0$ so that for every nonnegative integrable function $f:{\mathbb R}\to{\mathbb R}_{\geq 0}$,

$$\frac{\| f \ast f \|_\infty \| f \ast f \|_1}{\|f \ast f \|_2^2} > 1+\epsilon?$$

Of course, we want to assume that all of the norms in use are finite. The applications I have in mind have $f$ being the indicator function of a compact set.

A larger framework for considering this problem follows. Set $N_f(x):=\log(\| f \|_{1/x})$. Holder's Inequality, usually stated as $$\| fg \|_1 \leq \|f\|_p \|g\|_q$$ for $p,q$ conjugate exponents, becomes (with $f=g$) $N_f(1/2+x)+N_f(1/2-x)\geq 2N_f(1/2)$. In other words, Holder's inequality implies that $N_f$ is convex at $x=1/2$. The generalized Holder's Inequality gives convexity on $[0,1]$.

It is possible for $N_f$ to be linear, but only if $f$ is a multiple of an indicator function. What I am asking for is a quantitative expression of the properness of the convexity when $f$ is an autocorrelation.