Horvath's inequality unifies recent refinements of the Cauchy-Schwarz inequality

This paper shows how a classical inequality in inner product spaces for any orthogonal projection by Horvath [1, Infinite Dimensional Analysis, Quantum Probability and Related Topics 21 (2018), 1850016] can serve as a unified tool to derive recent statistical inequalities from Scarlatti [3, Statistical Papers 65 (2024), 5931-5940] and Lupu and Tanase [2, Statistics and Probability Letters (2026) 234 110695].