A new constant in Banach spaces based on the Zb$ \check{a} $ganu constant $ C_{Z}(B) $

Qian Li; Zhouping Yin; Yuxin Wang; Qi Liu; Hongmei Zhang; Qian Li; Zhouping Yin; Yuxin Wang; Qi Liu; Hongmei Zhang

doi:10.3934/math.2025296

AIMS Mathematics

2025, Volume 10, Issue 3: 6480-6491. doi: 10.3934/math.2025296

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Research article

A new constant in Banach spaces based on the Zb$ \check{a} $ganu constant $ C_{Z}(B) $

School of Mathematics and Physics, Anqing Normal University, Anqing 246133, China

Received: 28 December 2024 Revised: 07 March 2025 Accepted: 12 March 2025 Published: 24 March 2025
MSC : 46B20

In Banach spaces, first we present a new geometric constant, $ C_{Z}^{(q)}(t, B) $, which is closely related to the Zb$ \check{a} $ganu constant. We prove that $ \frac{(t+2)^{q}}{2^{q-1}\left(2^{q-1}+t^{q}\right)} $ and $ \frac{4}{3} $ are respectively the lower and upper bounds for $ C_{Z}^{(q)}(t, B) $. Furthermore, we also derive that $ C_{Z}^{(q)}(t, B) = C_{Z}^{(q)}(t, \widetilde{B}) $, where $ \widetilde{B} $ denotes the ultrapower spaces of $ B $. Then, we can establish certain sufficient conditions that ensure a Banach spaces possesses a normal structure, which involve different constants such as the Zb$ \check{a} $ganu constant and Domínguez–Benavides coefficient.
- geometric constants,
- Banach space,
- Zb$ \check{a} $ganu constant,
- ultrapower space,
- normal structure
Citation: Qian Li, Zhouping Yin, Yuxin Wang, Qi Liu, Hongmei Zhang. A new constant in Banach spaces based on the Zb$ \check{a} $ganu constant $ C_{Z}(B) $[J]. AIMS Mathematics, 2025, 10(3): 6480-6491. doi: 10.3934/math.2025296

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Abstract

In Banach spaces, first we present a new geometric constant, $ C_{Z}^{(q)}(t, B) $, which is closely related to the Zb$ \check{a} $ganu constant. We prove that $ \frac{(t+2)^{q}}{2^{q-1}\left(2^{q-1}+t^{q}\right)} $ and $ \frac{4}{3} $ are respectively the lower and upper bounds for $ C_{Z}^{(q)}(t, B) $. Furthermore, we also derive that $ C_{Z}^{(q)}(t, B) = C_{Z}^{(q)}(t, \widetilde{B}) $, where $ \widetilde{B} $ denotes the ultrapower spaces of $ B $. Then, we can establish certain sufficient conditions that ensure a Banach spaces possesses a normal structure, which involve different constants such as the Zb$ \check{a} $ganu constant and Domínguez–Benavides coefficient.

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