Strengthening $L^p$ Interpolation Bound

Question

Suppose that $f$ is a function such that $|f|\ge 1$ on $\text{supp}(f)$ so that $$\int |f|^p$$ is increasing with respect to $p$. By standard $L^p$ interpolation we know that $$\Big(\int |f|^{(p+q)/2}\Big)^{2}\le \int |f|^q\int |f|^p.$$ However suppose that $p\neq q$ and that $|f|$ is not constant a.e. on $\text{supp}(f)$, then the above is not an equality (via the conditions under which Holder gives equality). Combining this with the fact that $\int |f|^p$ is increasing with respect to $p$ in our scenario, this tells us that there is some $\epsilon >0$ such that $$\Big(\int |f|^{(p+q)/2+\epsilon}\Big)^{2}\le \int |f|^q\int |f|^p.$$ Is there any way to find one such possible $\epsilon$ (expressible using any of the known entities)? Most likely it will depend on something along the lines of $$\int_{\text{supp}(f)}\sup|f| - |f|$$ since this in a way measures the degree to which $f$ fails to be constant on its support, and thus the degree to which Holder fails to be equality.