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An Improved Inequality between the von Neumann-Jordan and James Constants

  • Zhiyao Fang
  • Qi Liu
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Abstract

We provide a strengthened answer to the questions posed by Alonso, Martín, and Papini [Studia Math. 188 (2008), 135--150] concerning the comparison between the von Neumann--Jordan and James constants . The classical inequality CNJ(X)≤ J(X) between the von Neumann--Jordan and James constants was established by Takahashi and Kato [J. Math. Anal. Appl. 359 (2009), 602--609], Wang [Proc. Amer. Math. Soc. 138 (2010), 695--701], and Yang and Li [Appl. Math. Lett. 23 (2010), 277--281], while related sharp estimates were obtained by Kato and Takahashi [Rend. Circ. Mat. Palermo (2) Suppl. 82 (2010), 75--91]. We sharpen this inequality to CNJ(X)≤JB(X), where JB(X) is the Birkhoff--James orthogonal James constant introduced by Baronti and Papini [Constr. Math. Anal. 5 (2022), 37--45]. Our proof employs optimal distance ellipses and alternating contact directions.

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Cite

BibTeXRIS
BibTeX
@misc{fang2026improved,
  title = {{An Improved Inequality between the von Neumann-Jordan and James Constants}},
  author = {Zhiyao Fang and Qi Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13360},
  eprint = {2610.13360},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13360},
}
RIS
TY  - UNPB
TI  - An Improved Inequality between the von Neumann-Jordan and James Constants
AU  - Zhiyao Fang
AU  - Qi Liu
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.13360
UR  - https://arxiv.org/abs/2610.13360
AB  - We provide a strengthened answer to the questions posed by Alonso, Martín, and Papini [Studia Math. 188 (2008), 135--150] concerning the comparison between the von Neumann--Jordan and James constants . The classical inequality CNJ(X)≤ J(X) between the von Neumann--Jordan and James constants was established by Takahashi and Kato [J. Math. Anal. Appl. 359 (2009), 602--609], Wang [Proc. Amer. Math. Soc. 138 (2010), 695--701], and Yang and Li [Appl. Math. Lett. 23 (2010), 277--281], while related sharp estimates were obtained by Kato and Takahashi [Rend. Circ. Mat. Palermo (2) Suppl. 82 (2010), 75--91]. We sharpen this inequality to CNJ(X)≤JB(X), where JB(X) is the Birkhoff--James orthogonal James constant introduced by Baronti and Papini [Constr. Math. Anal. 5 (2022), 37--45]. Our proof employs optimal distance ellipses and alternating contact directions.
ER  -