A Class Of Integral Type Operator Conformal Approximation Problem

In this paper, we mainly discuss the shape preserving properties of Baskakov-Kantorovich operator. Baskakov operator Vn is defined bywhere.In [4] given the following representation for Vn:, where (iV,)t>0 is a standard poisson and is aGamma process independent of (Nt)t>0. E denotes mathematical expectation.On the basis of this probabilistic representation ,the authors have obtained the preserving monotonicity, convexity and so on in [2]-[5], The Kantorovich deformation of Baskakov operator is defined byIn this paper, we obtain the shape preserving properties of Baskakov桲antorovich operator (namely B-K operator):1. B-K operator preserves the monotonicity and convexity. (Theorem 2. 1 and Theorem 2. 3);2. If continuous nonnegative function f satisfies: x-1f(x) is nonincreasing, then B-K operator of f have the same property. (Theorem3.1);3. If continuous function f is defined on [0,) , and for which f(0) = 0 , then , when f is subadditive and increasing on [0, ) meansVn(f;x)is subadditive function on [0, ) for (Theorem 3.3); 4. i) If Ff is convex, then F_x has the same property;ii)If f is starshaped on the average when x-1Ff(x) is noincreasing , then F (x) is noincreasing ;iii)If f is subadditive on the average and increasing, then F (x) issubadditive function on [0, ) for .Where the average function Ff of f is the function defined for allx>0 by Ff(x)=dt (Theorem 4. 1-Theorem 4. 3); 5. we obtain the best constants both in preservation inequalities concerning the first modulus and in preservation of Lipschitz classes offirst order (Theorem 5.2).In 66, we discuss the properties of preserving monotonicity and convexity for some type of positive operators for Durrmeyer deformation .These operators include Bernstein-Durrmeyer operator Baskakov-Durrmeyer operator and Szdsz-Mirakjan-Durrmeyer operator .At last, we obtain the representation of Jacobi polynomial kernel for Meyer-Konig and Zeller-Durrmeyer operator.