Research Article
Year 2025,
Volume: 54 Issue: 2, 436 - 444, 28.04.2025
Abstract
We introduce a new geometric constant $C^{p}_{NJ}(\zeta,\eta,X)$ in Banach spaces, which is called the skew generalized von Neumann-Jordan constant. First, the upper and lower bounds of the new constant are given for any Banach space. Then we calculate the constant values for some particular spaces. On this basis, we discuss the relation between the constant $C^{p}_{NJ}(\zeta,\eta,X)$ and the convexity modules $\delta_X(\varepsilon)$, the James constant $J(X)$. Finally, some sufficient conditions for the uniform normal structure associated with the constant $C^{p}_{NJ}(\zeta,\eta,X)$ are established.
References
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- [5] Y. Cui, W. Huang, H. Hudzik and R. Kaczmarek, Generalized von Neumann-Jordan constant and its relationship to the fixed point property, Fixed Point Theory Appl 40, 1-11, 2015.
- [6] J. Gao and K.S. Lau, On the geometry of spheres in normed linear spaces, J. Austral. Math. Soc. Ser. A 48 (1), 101-112, 1990.
- [7] R.C. James, Uniformly non-square Banach spaces, Ann. of Math. 80, 542-550, 1964.
- [8] J. Lindenstrauss, On the modulus of smoothness and divergent series in Banach spaces, Michigan Math. J. 10, 241-252, 1963.
- [9] Q. Liu and Y. Li, On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces, Mathematics 9, 116, 2021.
- [10] Q. Liu, C. Zhou, M. Sarfraz and Y. Li, On new moduli related to the generalization of the Parallelogram law, Bull. Malays. Math. Sci. Soc. 45, 307-321, 2022.
- [11] E.M. Mazcuñán-Navarro, Banach space properties sufficient for normal structure, J. Math. Anal. Appl. 337 (1), 197-218, 2008.
- [12] H. Mizuguchi, A lower bound for the constant A1(X) in normed linear spaces, Beitr. Algebra Geom. 64, 535-543, 2023.
- [13] H. Mizuguchi, The von Neumann-Jordan and another constants in Radon planes, Monatsh. Math. 195, 307-322, 2021.
- [14] S. Saejung, Sufficient conditions for uniform normal structure of Banach spaces and their duals, J. Math. Anal. Appl. 330 (1), 597-604, 2007.
- [15] B. Sims, Ultra-Techniques in Banach Space Theory, in: Pure and Applied Mathematics, Queen’s University 60, Kingston, Canada, 1982.
- [16] X. Wang, Y. Cui and C. Zhang, The generalized von Neumann-Jordan constant and normal structure in Banach spaces, Ann. Funct. Anal. 6 (4), 206-214, 2015.
- [17] C. Yang and F. Wang, An extension of a simply inequality between Von Neumann- Jordan and James constants in Banach spaces, Acta Math. Sinica Engl. Ser. 9, 1287- 1296, 2017.
- [18] Z. Zuo and Y. Cui, A coefficient related to some geometrical properties of Banach space, J. Inequal. Appl. Article ID 319804, doi:10.1155/2009/319804, 2009.
- [19] J.A. Clarkson, The von Neumann-Jordan constant for the Lebesgue space, Ann. of Math. 38, 114-115, 1937.
Year 2025,
Volume: 54 Issue: 2, 436 - 444, 28.04.2025
Abstract
References
- [1] A.G. Aksoy and M.A. Khamsi, Nonstandard Methods, Fixed Point Theory, Springer, New York, 1990.
- [2] J. Alonso, H. Martini and S. Wu, Orthogonality types in normed linear spaces, in: Papadopoulos, A. (ed.) Chapter 4 of Surveys in Geometry, 97-170, Springer, Cham, 2022.
- [3] M. Baronti, E. Casini and P.L. Papini, Revisiting the rectangular constant in Banach spaces, Bull. Aust. Math. Soc. 105 (1), 124-133, 2022.
- [4] M. Baronti and P.L. Papini, Parameters in Banach spaces and orthogonality, Constructive Mathematical Analysis 5 (1), 37-45, 2022.
- [5] Y. Cui, W. Huang, H. Hudzik and R. Kaczmarek, Generalized von Neumann-Jordan constant and its relationship to the fixed point property, Fixed Point Theory Appl 40, 1-11, 2015.
- [6] J. Gao and K.S. Lau, On the geometry of spheres in normed linear spaces, J. Austral. Math. Soc. Ser. A 48 (1), 101-112, 1990.
- [7] R.C. James, Uniformly non-square Banach spaces, Ann. of Math. 80, 542-550, 1964.
- [8] J. Lindenstrauss, On the modulus of smoothness and divergent series in Banach spaces, Michigan Math. J. 10, 241-252, 1963.
- [9] Q. Liu and Y. Li, On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces, Mathematics 9, 116, 2021.
- [10] Q. Liu, C. Zhou, M. Sarfraz and Y. Li, On new moduli related to the generalization of the Parallelogram law, Bull. Malays. Math. Sci. Soc. 45, 307-321, 2022.
- [11] E.M. Mazcuñán-Navarro, Banach space properties sufficient for normal structure, J. Math. Anal. Appl. 337 (1), 197-218, 2008.
- [12] H. Mizuguchi, A lower bound for the constant A1(X) in normed linear spaces, Beitr. Algebra Geom. 64, 535-543, 2023.
- [13] H. Mizuguchi, The von Neumann-Jordan and another constants in Radon planes, Monatsh. Math. 195, 307-322, 2021.
- [14] S. Saejung, Sufficient conditions for uniform normal structure of Banach spaces and their duals, J. Math. Anal. Appl. 330 (1), 597-604, 2007.
- [15] B. Sims, Ultra-Techniques in Banach Space Theory, in: Pure and Applied Mathematics, Queen’s University 60, Kingston, Canada, 1982.
- [16] X. Wang, Y. Cui and C. Zhang, The generalized von Neumann-Jordan constant and normal structure in Banach spaces, Ann. Funct. Anal. 6 (4), 206-214, 2015.
- [17] C. Yang and F. Wang, An extension of a simply inequality between Von Neumann- Jordan and James constants in Banach spaces, Acta Math. Sinica Engl. Ser. 9, 1287- 1296, 2017.
- [18] Z. Zuo and Y. Cui, A coefficient related to some geometrical properties of Banach space, J. Inequal. Appl. Article ID 319804, doi:10.1155/2009/319804, 2009.
- [19] J.A. Clarkson, The von Neumann-Jordan constant for the Lebesgue space, Ann. of Math. 38, 114-115, 1937.
There are 19 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Operator Algebras and Functional Analysis |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | August 27, 2024 |
| Publication Date | April 28, 2025 |
| Submission Date | March 11, 2024 |
| Acceptance Date | May 7, 2024 |
| Published in Issue | Year 2025 Volume: 54 Issue: 2 |