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

Abstract: We refine the classical Cauchy--Schwartz inequality $|X|{1} \leq |X|{2}$ 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 $|X|{2}=1$ and $|X|_{p}<\infty$. We illustrate two applications of this result: one for biased Rademacher sums and another for exponential sums.