Sharp coefficient, logarithmic and Schwarzian estimates for α-convex functions on the real line
Abstract
The class Mα of α-convex functions, introduced by Mocanu in 1969, interpolates between starlike (α=0) and convex (α=1) mappings and has traditionally been studied only for α≥0. We show that several of its sharp invariant estimates in fact extend to the entire real line. Our main result characterizes Mα for every real α≠0 through an explicit pointwise inequality, generalizing a theorem of Chuaqui, Duren and Osgood from the convex case, and we determine the sharp value of β for which Mα⊂ Cβ when α≥1; we also show that the reciprocal inclusion Cβ⊂ Mα fails for every α>1. Building on this, we obtain a sharp Fekete--Szegő inequality, valid for every α∉{0,−1,−1/2}, and sharp bounds for the first two logarithmic coefficients of Mα, which recover the classical values for starlike and convex functions as special cases. We also determine the order of Mα, establish a piecewise sharp bound for its Schwarzian norm together with an explicit quasiconformal extension criterion via the Ahlfors--Weill theorem, and compute the Schwarzian norm of the α-Koebe function explicitly for α=1/n, n∈ℕ, verified for n≤9 and numerically supported for n<50, leading to a conjecture for the general case. Finally, we propose a conjectural sharp bound for the second Hankel determinant of Mα, consistent with the known values $1$ and 1/8 for the starlike and convex subclasses.
Related
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BibTeX
@misc{bravo2026sharp,
title = {{Sharp coefficient, logarithmic and Schwarzian estimates for $α$-convex functions on the real line}},
author = {Víctor Bravo and Pablo Carrasco and Rodrigo Hernández and Osvaldo Venegas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.21574},
eprint = {2606.21574},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.21574},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp coefficient, logarithmic and Schwarzian estimates for α-convex functions on the real line
AU - Víctor Bravo
AU - Pablo Carrasco
AU - Rodrigo Hernández
AU - Osvaldo Venegas
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2606.21574
UR - https://arxiv.org/abs/2606.21574
AB - The class M_α of α-convex functions, introduced by Mocanu in 1969, interpolates between starlike (α=0) and convex (α=1) mappings and has traditionally been studied only for α≥0. We show that several of its sharp invariant estimates in fact extend to the entire real line. Our main result characterizes M_α for every real α≠0 through an explicit pointwise inequality, generalizing a theorem of Chuaqui, Duren and Osgood from the convex case, and we determine the sharp value of β for which M_α⊂ C_β when α≥1; we also show that the reciprocal inclusion C_β⊂ M_α fails for every α>1. Building on this, we obtain a sharp Fekete--Szegő inequality, valid for every α∉{0,−1,−1/2}, and sharp bounds for the first two logarithmic coefficients of M_α, which recover the classical values for starlike and convex functions as special cases. We also determine the order of M_α, establish a piecewise sharp bound for its Schwarzian norm together with an explicit quasiconformal extension criterion via the Ahlfors--Weill theorem, and compute the Schwarzian norm of the α-Koebe function explicitly for α=1/n, n∈ℕ, verified for n≤9 and numerically supported for n<50, leading to a conjecture for the general case. Finally, we propose a conjectural sharp bound for the second Hankel determinant of M_α, consistent with the known values $1$ and 1/8 for the starlike and convex subclasses.
ER -