Sharp Bounds for Higher-Order Schippers Functionals Associated with Lune and Bean Domains
Abstract
We obtain sharp bounds for the third- and fourth-order Schippers functionals, |σ3(f)(0)| and |σ4(f)(0)|, for subclasses of univalent functions associated with non-classical geometric domains. In particular, we investigate the lune-starlike class 𝒮leftmoon* and the lune-convex class 𝒞leftmoon determined by the subordination zf'(z)/f(z) ≺ z+√(1+z2), 1+zf''(z)/f'(z) ≺ z+√(1+z2), respectively, together with the bean-domain class ℬ𝒯B associated with B(z)=√(1+tanh z). Using Carathéodory coefficient parametrizations and extremal optimization techniques, we derive exact estimates for the higher-order Schwarzian derivatives at the origin and identify the corresponding extremal functions. In addition, geometric descriptions of the associated extremal image domains are provided to illustrate the sharpness phenomena. The obtained results further yield sharp bounds for the initial Grunsky coefficients g1,1 and g1,2. These findings provide a precise description of higher-order Schwarzian structures for univalent functions related to lune and bean shaped domains.
Related
- Sharp bounds for higher mixed Steklov-Robin eigenvalues on domains with holes
- Sharp Bounds for Schippers–Schwarzian Derivatives via the Schur-Parameter Method
- Lower bounds and Poincaré inequality for the Brezis-Seeger-Van Schaftingen-Yung functionals
- Variance lower bounds for geometric functionals of rotationally invariant log-concave random polytopes
- Accumulated-Variance Functionals of Hidden Markov and Semi-Markov Processes: Finite-Sample Upper and Lower Bounds
BibTeX
@misc{srivastava2026sharp,
title = {{Sharp Bounds for Higher-Order Schippers Functionals Associated with Lune and Bean Domains}},
author = {Hari M. Srivastava and Pradip Das and Firdoshi Parveen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.06667},
eprint = {2608.06667},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.06667},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Bounds for Higher-Order Schippers Functionals Associated with Lune and Bean Domains
AU - Hari M. Srivastava
AU - Pradip Das
AU - Firdoshi Parveen
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2608.06667
UR - https://arxiv.org/abs/2608.06667
AB - We obtain sharp bounds for the third- and fourth-order Schippers functionals, |σ_3(f)(0)| and |σ_4(f)(0)|, for subclasses of univalent functions associated with non-classical geometric domains. In particular, we investigate the lune-starlike class 𝒮_leftmoon^* and the lune-convex class 𝒞_leftmoon determined by the subordination zf'(z)/f(z) ≺ z+√(1+z^2), 1+zf''(z)/f'(z) ≺ z+√(1+z^2), respectively, together with the bean-domain class ℬ𝒯_B associated with B(z)=√(1+tanh z). Using Carathéodory coefficient parametrizations and extremal optimization techniques, we derive exact estimates for the higher-order Schwarzian derivatives at the origin and identify the corresponding extremal functions. In addition, geometric descriptions of the associated extremal image domains are provided to illustrate the sharpness phenomena. The obtained results further yield sharp bounds for the initial Grunsky coefficients g_1,1 and g_1,2. These findings provide a precise description of higher-order Schwarzian structures for univalent functions related to lune and bean shaped domains.
ER -