Alpha-convex functions in complex Banach spaces

Abstract

Let 𝔹 be the unit ball in a complex Banach space 𝕏. We study biholomorphic mappings F: 𝔹 → 𝕏 of the form F(z) = f(z)z for z ∈ 𝔹, where f: 𝔹 → ℂ is a holomorphic function satisfying f(0) = 1. Our main contributions are as follows: Inclusion Results: We establish inclusion conditions for ℳ_α(𝔹), class of α-convex functions in 𝔹 to belong to the classes of starlike mappings or quasi-convex mappings of type B on 𝔹 for specified ranges of α. Characterization: We prove that F(z) = f(l_u(z))/l_u(z)z ∈ ℳ_α(𝔹) if and only if f ∈ ℳ_α(𝕌), where f ∈ 𝒮(𝕌), u ∈ 𝕏 with |u| = 1, and l_u ∈ T(u). Duality: We present a duality relationship between the classes 𝒮^*(𝕌) and ℳ_α(𝔹) for α≥ 0. Hankel Determinant Bounds: We determine sharp upper bounds for the second Hankel determinant of mappings in ℳ_α(𝔹) when α≥ 0. The results established here directly extend the previous results to the broader setting of complex Banach spaces.

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BibTeX RIS

BibTeX

@misc{sokol2026alphaconvex,
  title = {{Alpha-convex functions in complex Banach spaces}},
  author = {Janusz Sokol and Sujoy Majumder and Goutam Halder},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21260},
  eprint = {2610.21260},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21260},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Alpha-convex functions in complex Banach spaces
AU  - Janusz Sokol
AU  - Sujoy Majumder
AU  - Goutam Halder
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21260
UR  - https://arxiv.org/abs/2610.21260
AB  - Let 𝔹 be the unit ball in a complex Banach space 𝕏. We study biholomorphic mappings F: 𝔹 → 𝕏 of the form F(z) = f(z)z for z ∈ 𝔹, where f: 𝔹 → ℂ is a holomorphic function satisfying f(0) = 1. Our main contributions are as follows: Inclusion Results: We establish inclusion conditions for ℳ_α(𝔹), class of α-convex functions in 𝔹 to belong to the classes of starlike mappings or quasi-convex mappings of type B on 𝔹 for specified ranges of α. Characterization: We prove that F(z) = f(l_u(z))/l_u(z)z ∈ ℳ_α(𝔹) if and only if f ∈ ℳ_α(𝕌), where f ∈ 𝒮(𝕌), u ∈ 𝕏 with |u| = 1, and l_u ∈ T(u). Duality: We present a duality relationship between the classes 𝒮^*(𝕌) and ℳ_α(𝔹) for α≥ 0. Hankel Determinant Bounds: We determine sharp upper bounds for the second Hankel determinant of mappings in ℳ_α(𝔹) when α≥ 0. The results established here directly extend the previous results to the broader setting of complex Banach spaces.
ER  -

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