We prove variable coefficient analogues of results in [5] on Hilbert transforms and maximal functions along convex curves in the plane.

We improve the current bounds for an inequality of Erd\H{o}s and Tur\'an from 1950 related to discrepancy angular equidistribution zeros a given polynomial. Building upon recent work Soundararajan, we establish novel connection between this extremal problem in Fourier analysis involving maxima Hilbert transforms, which provide complete solution. Prior Soundararajan (2019), refinements had been ...