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

We refine the classical Cauchy–Schwartz inequality $\mathrm{\parallel }X{\mathrm{\parallel }}_1\le \mathrm{\parallel }X{\mathrm{\parallel }}_2$ 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 $\mathrm{\parallel }X{\mathrm{\parallel }}_2=1$ and $\mathrm{\parallel }X{\mathrm{\parallel }}_p<\mathrm{\infty }$. We illustrate two applications of this result: one for biased Rademacher sums and another for exponential sums.