On Hardy spaces, univalent functions and the second coefficient
Martin Chuaqui, Iason Efraimidis, Rodrigo Hernández
TL;DR
This work analyzes how the second Taylor coefficient $a_2$ of normalized univalent functions governs Hardy-space membership for the function $f$ and its derivative, with a focus on convex and other geometric subclasses. It establishes sharp Hardy-space bounds $f'\in H^p$ and $f\in H^q$ in terms of $|a_2|$ (and a refined lower order $\beta$), and connects these analytic properties to geometric features such as the angle at infinity via the sector bound aperture $|a_2|\pi$. The paper also derives sharp asymptotic growth for the coefficients $a_n$, boundary smoothness results, and extends Hardy-space estimates across several subclasses (starlike, close-to-convex, convex in one direction, typically real). In addition, it demonstrates that derivative pathologies akin to Lohwater–Piranian–Rudin occur for every prescribed $a_2$, and provides an appendix-based measure-theoretic perspective for convex functions with $a_2=0$, strengthening the structural understanding of these coefficient-driven phenomena.
Abstract
We consider normalized univalent functions with prescribed second Taylor coefficient $a_2$. For convex functions $f$ we study the Hardy spaces to which $f$ and $f'$ belong, refining in particular on a theorem of Eenigenburg and Keogh, and give a sharp asymptotic estimate and an explicit uniform bound for their coefficients. Relating the lower order of a convex function to the angle at infinity of its range we deduce that its range lies always in some sector of aperture $|a_2|π$. We give sharp smoothness conditions on the boundary for convex functions with prescribed second coefficient. We find the sharp Hardy space estimates for $f$ and $f'$ when $f$ belongs to other geometric subclasses, such as those of starlike, close-to-convex, convex in one direction, convex in the positive direction and typically real funtions. We extend a theorem of Lohwater, Piranian and Rudin, in which a univalent function whose derivative has radial limits almost nowhere is constructed, by showing that this pathological behavior can be obtained for any prescribed value of the second coefficient, in particular, manifesting itself arbitrarily close to the Koebe function.