Skew generalized von Neumann-Jordan constant in Banach Spaces

We introduce a new geometric constant CᵖNJ(ζ,η,X) in Banach spaces, which is called the skew generalized von Neumann-Jordan constant. First, the upper and lower bounds of the new constant are given for any Banach space. Then we calculate the constant values for some particular spaces. On this basis, we discuss the relation between the constant CᵖNJ(ζ,η,X) and the convexity modules δX(ε), the James constant $J(X)$. Finally, some sufficient conditions for the uniform normal structure associated with the constant CᵖNJ(ζ,η,X) are established.