In recent times, Takahashi has introduced von Neumann-Jordan type constants $ C_{-\infty}(X) $. In the present manuscript, we establish a novel geometric constant $ C_{-\infty}(a, X) $ in a Banach space $ X $. Next, it is shown that $ \frac{1}{2}+\frac{2a}{4+a^2}\leqslant C_{-\infty}(a, X)\leqslant 1 $ for all $ a\geqslant0 $. Further, between the generalized James constant $ J(a, X) $ and $ C_{-\infty}(a, X) $, a relationship is investigated. For uniform normal structure, a few sufficient conditions were established. Finally, we investigate some relations between the two constants $ N(X) $ and $ C_{-\infty}(a, X) $.
Citation: Qi Liu, Anwarud Din, Yongjin Li. Some aspects of generalized von Neumann-Jordan type constant[J]. AIMS Mathematics, 2021, 6(6): 6309-6321. doi: 10.3934/math.2021370
In recent times, Takahashi has introduced von Neumann-Jordan type constants $ C_{-\infty}(X) $. In the present manuscript, we establish a novel geometric constant $ C_{-\infty}(a, X) $ in a Banach space $ X $. Next, it is shown that $ \frac{1}{2}+\frac{2a}{4+a^2}\leqslant C_{-\infty}(a, X)\leqslant 1 $ for all $ a\geqslant0 $. Further, between the generalized James constant $ J(a, X) $ and $ C_{-\infty}(a, X) $, a relationship is investigated. For uniform normal structure, a few sufficient conditions were established. Finally, we investigate some relations between the two constants $ N(X) $ and $ C_{-\infty}(a, X) $.
| [1] |
D. Amir, On Jung's constant and related constants in normed linear spaces, Pac. J. Math., 118 (1985), 1–15. doi: 10.2140/pjm.1985.118.1
|
| [2] | A. G. Aksoy, M. A. Khamsi, Nonstandard methods in fixed point theory, Spinger-Verlag, Heidelberg, 1990. |
| [3] |
J. Alonso, P. Martin, P. Papini, Wheeling around von Neumann-Jordan constant in Banach spaces, Stud. Math., 188 (2008), 135–150. doi: 10.4064/sm188-2-3
|
| [4] | M. S. Brodskii, D. P. Milman, On the center of a convex Set, Dokl. Akad. Nauk SSSR(N.S.), 59 (1948), 837–840. |
| [5] |
J. A. Clarkson, The von Neumann–Jordan constant for the Lebesgue space, Ann. of Math., 38 (1937), 114–115. doi: 10.2307/1968512
|
| [6] |
J. A. Clarkson, Uniformly convex spaces, T. Am. Math. Soc., 40 (1936), 396–414. doi: 10.1090/S0002-9947-1936-1501880-4
|
| [7] |
S. Dhompongsa, A. Kaewkhao, S. Tasena, On a generalized James constant, J. Math. Anal. Appl., 285 (2003), 419–435. doi: 10.1016/S0022-247X(03)00408-6
|
| [8] |
S. Dhompongsa, P. Piraisangjun, S. Saejung, Generalized Jordan-von Neumann constants and uniform normal structure, B. Aust. Math. Soc., 67 (2003), 225–240. doi: 10.1017/S0004972700033694
|
| [9] |
M. Dinarvand, On a generalized geometric constant and sufficient conditions for normal structure in Banach spaces, Acta Math. Sci., 37 (2017), 1209–1220. doi: 10.1016/S0252-9602(17)30068-1
|
| [10] | J. Gao, On some geometric parameters in Banach spaces, J. Math. Anal. Appl., 1 (2007), 114–122. |
| [11] |
J. Gao, K. S. Lau, On two classes of Banach spaces with uniform normal structure, Stud. Math., 99 (1991), 41–56. doi: 10.4064/sm-99-1-41-56
|
| [12] | J. Gao, Research on normal structure in a Banach space via some parameters in its dual space, Commun. Korean Math. Soc., 34 (2019), 465–475. |
| [13] | A. Jim$\acute{e}$nez-Melado, E. Llorens-Fuster, S. Saejung, The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces, P. Am. Math. Soc., 134 (2006), 355–364. |
| [14] |
M. A. Khamsi, Uniform smoothness implies super-normal structure property, Nonlinear Anal., 19 (1992), 1063–1069. doi: 10.1016/0362-546X(92)90124-W
|
| [15] |
M. Kato, Y. Takahashi, On the von Neumann–Jordan constant for Banach spaces, P. Am. Math. Soc., 125 (1997), 1055–1062. doi: 10.1090/S0002-9939-97-03740-4
|
| [16] |
M. Kato, L. Maligranda, Y. Takahashi, On James Jordan–von Neumann constants and the normal structure coefficient of Banach spaces, Stud. Math., 144 (2001), 275–295. doi: 10.4064/sm144-3-5
|
| [17] | M. Kato, L. Maligranda, Y. Takahashi, Von Neumann–Jordan constant and some geometrical constants of Banach spaces (Nonlinear Analysis and Convex Analysis, Research Institute for Mathematical Sciences, 1031 (1998), 68–74. |
| [18] |
N. Komuro, K. Saito, R. Tanaka, On the class of Banach spaces with James constant $\sqrt2$, Math. Nachr., 289 (2016), 1005–1020. doi: 10.1002/mana.201500238
|
| [19] | S. Prus, M. Szczepanik, New coefficients related to uniform normal structure, J. Nonlinear Convex A., 2 (2001), 203–216. |
| [20] | S. Prus, Geometrical background of metric fixed point theory, in: W.A.Kirk, B. Sims(Eds.), Handbook of Metric Fixed Point Theory, (2001), 93–132. Springer, Dordrecht.). |
| [21] | Y. Takahashi, Some geometric constants of Banach spaces—a unified approach, Banach and Function Spaces II, eds. M. Kato and L. Maligranda, Yokohama Publishers, Yokohama, (2008), 191–220. |
| [22] | C. Yang, F. Wang, On estimates of the generalized Jordan-von Neumann constant of Banach spaces, Journal of Inequalities in Pure and Applied Mathematics, 7 (2006), 1–5. |
| [23] | Z. Zuo, C. Tang, On James and Jordan von-Neumann type constants and the normal structure in Banach spaces, Topol. Method. Nonl. An., 49 (2017), 615–623. |
Qi Liu, Anwarud Din, Yongjin Li. Some aspects of generalized von Neumann-Jordan type constant[J]. AIMS Mathematics, 2021, 6(6): 6309-6321. doi: 10.3934/math.2021370
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