KYUNGPOOK Math. J. 46(2006), 185-188 Durrmyer Type Summation Integral Operators Niraj Kumar School of Applied Sciences, Netaji Subhas Institute of Technology Sector 3, Dwarka, New Delhi 110045, India e-mail : neeraj@nsit.ac.in Abstract. In the present paper, we give the applications of the optimum bound for Bernstein basis functions. It is noted that using the optimum bound the main results of Aniol and Taberska [Ann. Soc. Math. Pol. Seri, Commentat. Math., 30(1990), 9-17], [Approx. Theory and its Appl. 11:2(1995), 94-105] and V. Gupta [Soochow J. Math., 23(1)(1997) 115-118] can be improved which were not pointed out earlier. 1. Introduction Durrmeyer [4] introduced the integral modification of the Bernstein polynomials to approximate Lebesgue integrable functions on the interval [0, 1]. The operators introduced by Durrmeyer are defined as (1.1) B n (f,x)=(n + 1) n  k=0 p n,k (x)  1 0 p n,k (t)f (t)dt, x ∈ [0, 1], where p n,k (x)= ( n k ) x k (1 - x) n-k . Guo [5] estimated the rate of convergence for the operator (1.1) for functions of bounded variation, using some results of probability theory. After this Aniol and Taberska (see e.g. [1] and [2]) generalized and extended the results of Guo [5]. Recently Gupta [6] introduced a slight but interesting integral modification of the Bernstein polynomials and studied the rate of convergence for functions of bounded variation. The operators introduced by Gupta [6] are defined by (1.2) P n (f,x)= n  k=0 p n,k (x)  1 0 b n,k (t)f (t)dt, x ∈ [0, 1], where p n,k =(-1) k x k k! Φ (k) n (x), b n,k (t)=(-1) k+1 t k k! Φ (k+1) n (t),b n,n (t) = 0 and Φ n (x) = (1 - x) n . Received May 31, 2004, and, in revised form, August 23, 2004. 2000 Mathematics Subject Classification: 41A16, 41A25. Key words and phrases: Durrmeyer operators, Bernstein polynomials, rate of conver- gence, Bernstein basis function. 185