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Growth, Distortion, and Schwarzian Norm Estimates for Exponentially Convex Functions
Abstract
In this paper, we investigate the growth, distortion, pre-Schwarzian and Schwarzian norms of functions in the exponentially convex class 𝒞eλ, 0<λ≤π/2, defined by 1+zf''(z)/f'(z)≺ eλz. By representing the associated Schwarz function explicitly, we derive parameter-dependent estimates for f, f', and the pre-Schwarzian derivative. We further obtain Schwarzian norm estimates under both the general normalization and the additional condition f''(0)=0. The corresponding extremal problems are analyzed through suitable Schwarz functions, and the dependence of the resulting bounds on the exponential parameter is made explicit.
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BibTeX
@misc{ahamed2026growth,
title = {{Growth, Distortion, and Schwarzian Norm Estimates for Exponentially Convex Functions}},
author = {Molla Basir Ahamed and Rajesh Hossain},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27563},
eprint = {2609.27563},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27563},
}
RIS
TY - UNPB
TI - Growth, Distortion, and Schwarzian Norm Estimates for Exponentially Convex Functions
AU - Molla Basir Ahamed
AU - Rajesh Hossain
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27563
UR - https://arxiv.org/abs/2609.27563
AB - In this paper, we investigate the growth, distortion, pre-Schwarzian and Schwarzian norms of functions in the exponentially convex class 𝒞_e^λ, 0<λ≤π/2, defined by 1+zf''(z)/f'(z)≺ e^λz. By representing the associated Schwarz function explicitly, we derive parameter-dependent estimates for f, f', and the pre-Schwarzian derivative. We further obtain Schwarzian norm estimates under both the general normalization and the additional condition f''(0)=0. The corresponding extremal problems are analyzed through suitable Schwarz functions, and the dependence of the resulting bounds on the exponential parameter is made explicit.
ER -