Higher-Order Schippers-Schwarzian Derivatives for Non-Circular Starlike Functions

Pradip Das; Paweł Zaprawa; Nabadwip Sarkar

Higher-Order Schippers-Schwarzian Derivatives for Non-Circular Starlike Functions

Complex Variables 2026-07-28 v1

Authors: Pradip Das , Paweł Zaprawa , Nabadwip Sarkar

Abstract

We find sharp upper bounds for the initial higher-order Schippers-Schwarzian derivatives $\sigma_3(f)(0)$ and $\sigma_4(f)(0)$ for several subclasses of univalent and starlike functions in the unit disk. The functions in these classes are defined by subordination to domains bounded by an exponential-sine curve, a cardioid, or a petal-shaped curve. We determine the sharp bounds and find the corresponding extremal functions for each invariant in these subclasses. As an application, we also obtain sharp bounds for the initial Grunsky coefficients $\omega_{1,1}$ and $\omega_{1,2}$ .

Keywords

Schippers-Schwarzian derivatives starlike functions univalent functions sharp bounds subordination Grunsky coefficients

Cite

@article{arxiv.2607.26268,
  title  = {Higher-Order Schippers-Schwarzian Derivatives for Non-Circular Starlike Functions},
  author = {Pradip Das and Paweł Zaprawa and Nabadwip Sarkar},
  journal= {arXiv preprint arXiv:2607.26268},
  year   = {2026}
}