In this paper, we shall introduce a skew generalized Zb \(\check{a}\) ganu constant \(C^{(p)}_{Z}(\varsigma ,\upsilon ,X)\) . First, we give the upper and lower bounds of this constant for any Banach spaces, as well as the exact values of the constant for some specific Banach spaces. The relationships between this constant and a few other constants are then shown, including the J(X), \(C_\textrm{NJ}(X)\) , and \(C^p_\textrm{NJ}(\varsigma , \upsilon , X)\) constants. Furthermore, a characterization of uniformly non-square is provided, indicating that X possesses the fixed point property. A sufficient condition that implies normal structure is also established by the \(C^{(p)}_{Z}(\varsigma ,\upsilon , X)\) constant. Finally, based on the \(C^{(p)}_{Z}(\varsigma ,\upsilon , X)\) constant, another new constant \(\widetilde{C}^{(p)}_{Z}(\varsigma ,\upsilon , X)\) is also introduced, its range of values for any Banach spaces and the exact values for some specific Banach space are studied.
