Hölder-like inequality $\|f\|_\pi\|g\|_\infty\le\frac{1}{16}\|f\|_\pi^2+4\|g\|_\infty^2$

Question

I don't know how the second bound occurs. It's not Hölder, Cauchy-Schwarz or Minkowsi.

Of course we have:

It appears in the following paper on page 531: https://projecteuclid.org/journals/bernoulli/volume-13/issue-2/Penalized-nonparametric-mean-square-estimation-of-the-coefficients-of-diffusion/10.3150/07-BEJ5173.full

Thank you.

Answer 1

It looks like the Peter-Paul inequality: $$ ab \le \frac{a^2}{2\epsilon} + \frac{\epsilon b^2}{2} $$ with $\epsilon = 8$.