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PreprintCC BY 4.0arXiv

Some properties of completely convex functions

  • Hugo Arbeláez
  • Rodrigo Hernández
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Abstract

We study the class CCα of completely convex functions of order α, defined through a two-point analogous of the classical convexity condition. We give several equivalent characterizations of CCα, together with representation formulas in terms of two auxiliary Schwarz functions linked by a compatibility equation. From these representations, we derive coefficient bounds, a subordination result zf'/f, and a two-point Taylor expansion in which the Schwarzian derivative Sf arises naturally as the coefficient of the linear term. This yields a pointwise bound for Sf and a bound for the pre-Schwarzian norm |Pf|CCα, improving the classical Yamashita bound when $α\leq 1-1/\sqrt2 $. We also show that, unlike the class Cα of convex functions of order α, every f∈ CCα is bounded when α>0.

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Cite

BibTeXRIS
BibTeX
@misc{arbelaez2026some,
  title = {{Some properties of completely convex functions}},
  author = {Hugo Arbeláez and Rodrigo Hernández},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40889},
  eprint = {2610.40889},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40889},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Some properties of completely convex functions
AU  - Hugo Arbeláez
AU  - Rodrigo Hernández
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40889
UR  - https://arxiv.org/abs/2610.40889
AB  - We study the class CC_α of completely convex functions of order α, defined through a two-point analogous of the classical convexity condition. We give several equivalent characterizations of CC_α, together with representation formulas in terms of two auxiliary Schwarz functions linked by a compatibility equation. From these representations, we derive coefficient bounds, a subordination result zf'/f, and a two-point Taylor expansion in which the Schwarzian derivative S_f arises naturally as the coefficient of the linear term. This yields a pointwise bound for S_f and a bound for the pre-Schwarzian norm |P_f|_CC_α, improving the classical Yamashita bound when $α\leq 1-1/\sqrt2 $. We also show that, unlike the class C_α of convex functions of order α, every f∈ CC_α is bounded when α>0.
ER  -