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 $C_Z^{(q)}(t,B)=C_Z^{(q)}(t,\tilde{B})$, where $\tilde{B}$ 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.