Sharpening the gap between L¹ and L² norms

We refine the classical Cauchy--Schwartz inequality \|X\|₁ ≤ \|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 ≤ 1 - C (\|X\|ₚᵖ - 1)q-2/q-p(\|X\|qq - 1)2-p/q-p holds true for all Borel measurable random variables X with \|X\|₂=1 and \|X\|ₚ<∞. We illustrate two applications of this result: one for biased Rademacher sums and another for exponential sums.