Some sharp Schwarz type estimates and their applications in Banach spaces

Abstract

The primary objective of this paper is to develop methodologies for investigating Schwarz type lemmas and to present their applications in Banach spaces. First, we improve upon the main results obtained by Rogosinski [Jber. Dtsch. Math.-Verein 44: 258--261, 1934] %Osserman [Proc. Am. Math. Soc. 128: 3513-3517, 2000] and Chen et al. [J. Anal. Math. 152: 181-216, 2024]. Based on these sharp estimates, we then derive several sharp boundary Schwarz type lemmas (also known as Hopf type lemmas) for holomorphic mappings in Banach spaces, as well as for solutions to certain classes of elliptic partial differential equations on the Euclidean unit ball in ℂⁿ or on the unit disk in ℂ. Furthermore, we prove some sharp Schwarz type lemmas for holomorphic mappings that send a prescribed point to another prescribed point. Finally, these lemmas are applied to establish a sharp Minda type Schwarz inequality in Banach spaces and to provide a sharp refined bound on subballs of the unit ball.

Sharp Robertson-type coefficient estimates for radial starlike and quasi-convex of type B mappings of order α in complex Banach spaces
Sujoy Majumder, Abhijit Banerjee, Goutam Haldar
arXiv9 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv
Refined Bohr Inequalities and Bohr Radii Involving Schwarz Mappings for Holomorphic Functions in Banach Spaces and JB*-Triples
Nabadwip Sarkar, Pradip Das
arXiv2 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv
Some New Sharp Boundedness Result Concerning Linear functionals on Banach spaces
L. -E. Persson, V. Tsagareishvili, G. Tutberidze +1
arXiv1 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv
Sharp embeddings between quasi-Banach Besov spaces and shallow ReLU variation spaces
Yuwen Li, Yupeng Wang
arXiv2 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Estimates for a Parameterized Hankel Functional of Starlike Mappings of Order α and Quasi-Convex Mappings of Type B in Complex Banach Spaces
Sujoy Majumder, Abhijit Banerjee, Shantanu Panja +1
arXiv13 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv

Cite

BibTeX RIS

BibTeX

@misc{chen2026some,
  title = {{Some sharp Schwarz type estimates and their applications in Banach spaces}},
  author = {Shaolin Chen and Hidetaka Hamada and Megha Kundathil and Ramakrishnan Vijayakumar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.17237},
  eprint = {2605.17237},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.17237},
}

RIS

TY  - UNPB
TI  - Some sharp Schwarz type estimates and their applications in Banach spaces
AU  - Shaolin Chen
AU  - Hidetaka Hamada
AU  - Megha Kundathil
AU  - Ramakrishnan Vijayakumar
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2605.17237
UR  - https://arxiv.org/abs/2605.17237
AB  - The primary objective of this paper is to develop methodologies for investigating Schwarz type lemmas and to present their applications in Banach spaces. First, we improve upon the main results obtained by Rogosinski [Jber. Dtsch. Math.-Verein 44: 258--261, 1934] %Osserman [Proc. Am. Math. Soc. 128: 3513-3517, 2000] and Chen et al. [J. Anal. Math. 152: 181-216, 2024]. Based on these sharp estimates, we then derive several sharp boundary Schwarz type lemmas (also known as Hopf type lemmas) for holomorphic mappings in Banach spaces, as well as for solutions to certain classes of elliptic partial differential equations on the Euclidean unit ball in ℂ^n or on the unit disk in ℂ. Furthermore, we prove some sharp Schwarz type lemmas for holomorphic mappings that send a prescribed point to another prescribed point. Finally, these lemmas are applied to establish a sharp Minda type Schwarz inequality in Banach spaces and to provide a sharp refined bound on subballs of the unit ball.
ER  -

Abstract

Related

Cite