Weighted Bleimann-butzer And Hahn-durrmeyer, Count The Approximation And Conformal

In this paper, we introduce and discuss the operators of Bleinmann-Butzer and Hahn-Durrmeyer with double parametersα,β, namely weighted BBH-D operator. Wenot only obtain the complete asymptotic expansion for the operators and their derivatives but also discuss the shape preserving properties of the operator. The weighted BBH-D operator is defined bywhere v^(α,β)(x) = x^α(1+x)^-β.Basing on the relation of the weighted BBH-D operator and the Bernstein-Durrmeyer operator with the Jacobi' s right, we obtain the complete asymptotic expansion for the operators and their derivatives as followsTheorem 1 Let q∈N , x∈(0,+∞), and bounded f is 2q times differentiable at x, r∈N , Then we haveTheorem 2 Let q∈N, x∈(0,+∞), and bounded f is 2q + r times differentiable at x, r∈N , Then we haveFor q=1, We get the Asymptotic bounded expansion of Voronovskaja-type for the operators.Corollary 3 Let bounded f is differentiable at x , x∈(0,+∞), ThenForα=0,β= 2, We get the Corollary 2.7.Corollary 4 Let q∈N , x∈(0,+∞), and bounded f is 2q times differentiable at x , ThenLast we discuss the shape preserving properties of the operator:Theorem 5 Let bounded f is differentiable at x , x∈(0,+∞), and here exista constant M such as f(x) = O(e^-Mx)(xâ†'+∞), Then BBH-D operator preserves the monotonicity.Theorem 6 Let bounded f is two times differentiable at x , x∈(0,+∞), andhere exist a constant M such as f'(x) = O(e^-Mx) (xâ†'+∞), Then BBH-D operatorpreserves the converxity.By K-Functionals we obtain the following direct theorem.Theorem 7 Letf is bounded continuous on[0,+∞), then...