Sharpening the gap between $L^1$ and $L^2$ norms

Abstract

We refine the classical Cauchy--Schwartz inequality $\Vert X{\Vert }_1\le \Vert X{\Vert }_2$ by demonstrating that for any $p$ and $q$ with $q>p>2$, there exists a constant $C=C(p,q)$ such that $\|X\|_1 \leq 1 - C \Big{(}\|X\|_p^p - 1\Big{)}^{\frac{q-2}{q-p}}\Big{(}\|X\|_q^q - 1\Big{)}^{\frac{2-p}{q-p}}$ holds true for all Borel measurable random variables $X$ with $\Vert X{\Vert }_2=1$ and $\Vert X{\Vert }_p<\infty$. We illustrate two applications of this result: one for biased Rademacher sums and another for exponential sums.