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

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.