I am reading a paper using some Fourier analysis over finite abelian groups. They mention an inequality $|\langle f,g\rangle|\leq \|f\|_{p} \|g\|_{\infty}^{1/p}$, and they say this is an application of Holder's inequality. This is part of a long inequality and later they mention that $p$ needs to be sufficiently large, and I don't know if it is related to this inequality.
I don't see why it is true because what I can get is $|\langle f,g\rangle|\leq \|f\|_{p}\|g\|_{\frac{p}{p-1}}$.
Consider constant functions $f=a, g=b$ on $(0,1)$. The inequality says $|ab| \leq |a||b|^{1/p}$. This does not hold for all $a$ and $b$ whatever $p$ is. I will let you figure out why.