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Schwarzian norm estimates for some classes of analytic and harmonic mappings

  • Molla Basir Ahamed
  • Rajesh Hossain
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Abstract

Let π’œ be the normalized class of analytic functions f in the unit disc 𝔻 := {z ∈ β„‚ : |z| < 1}. For Ξ²> 1, let 𝒩(Ξ²) denote the subclass of π’œ satisfying Re{1 + z f''(z)/f'(z)} < Ξ² for z ∈ 𝔻. The main purpose of this paper is to establish sharp bounds for the pre-Schwarzian norm β€–Pfβ€– and Schwarzian norm β€–Sfβ€– for functions f ∈ 𝒩(Ξ²), parametrized by f''(0), with special emphasis on the case f''(0) = 0. In addition, sharp growth, distortion, and radius results (convexity and concavity) for 𝒩(Ξ²) are obtained. As an application, we determine the sharp pre-Schwarzian norm estimate for harmonic mappings f = h + gΜ… whose analytic part h belongs to 𝒩(Ξ²).

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Cite

BibTeXRIS
BibTeX
@misc{ahamed2026schwarzian,
  title = {{Schwarzian norm estimates for some classes of analytic and harmonic mappings}},
  author = {Molla Basir Ahamed and Rajesh Hossain},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30595},
  eprint = {2608.30595},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30595},
}
RIS
TY  - UNPB
TI  - Schwarzian norm estimates for some classes of analytic and harmonic mappings
AU  - Molla Basir Ahamed
AU  - Rajesh Hossain
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30595
UR  - https://arxiv.org/abs/2608.30595
AB  - Let π’œ be the normalized class of analytic functions f in the unit disc 𝔻 := {z ∈ β„‚ : |z| < 1}. For Ξ²> 1, let 𝒩(Ξ²) denote the subclass of π’œ satisfying Re{1 + z f''(z)/f'(z)} < Ξ² for z ∈ 𝔻. The main purpose of this paper is to establish sharp bounds for the pre-Schwarzian norm β€–P_fβ€– and Schwarzian norm β€–S_fβ€– for functions f ∈ 𝒩(Ξ²), parametrized by f''(0), with special emphasis on the case f''(0) = 0. In addition, sharp growth, distortion, and radius results (convexity and concavity) for 𝒩(Ξ²) are obtained. As an application, we determine the sharp pre-Schwarzian norm estimate for harmonic mappings f = h + gΜ… whose analytic part h belongs to 𝒩(Ξ²).
ER  -