Convexity preservation of five-point binary subdivision scheme with a parameter

A five-point binary subdivision scheme with a parameter u   is presented. The generating polynomial method and the Hölder exponent are used to investigate the uniform convergence and CkCk continuity of this subdivision scheme, where k depends on the choice of the parameter u. Moreover, the conditions on the initial points are discussed for the given limit curve to be convexity preserving and an example is given to illustrate our conclusion.