:OURNAL OF APPROXIMATION THEORY 69, 167-172 (1992) On a Conjecture of Z. Ditzian DING-XUAN ZHOU * Deportment of Mathematics, Zhejiang University, and Advanced Institute of Mathematics, Hangzhou, Zhejiang, People’s Republic of China 310027 Communicated bv Zeeo Ditzian Received June 5, 1990; revised April 1, 1991 A conjecture of 2. Ditzian on Bernstein polynomia!s is proved. This yields additional information on the problem of characterizing the rate of convergence for Bernstein polynomiak. ‘6Z 1992 Academic Press, Inc. The Bernstein polynomials on C[O, l] are given by B,(f, x) = i f (k/n) Pnk(-xh k=O where P&(x)=; xk(l -.y)n-k. 0 (2) The relation between the rate of the polynomials’ convergence and the smoothness of the function they approximate has been investigated in many papers. Some of these results can be stated in the following theorem, THEOREM A. For f~ C[O, 11, X=,X( 1-xi, 0 <a < 2, 0 < /3< 2, :he following statements are equivalent : (1) XPzl’IB,(f, x)-f(x)/ <MM,n-g~r; 13) (2) rx;‘If(x- t)-2f(x)+f(x+ t)l Gif,(t2/X)“? 141 Here Ml, M, are constants independent of n, x, and t. In 1972, H. Berens and G. G. Lorentz [l] proved this theorem for X= p. M. Becker [Z] gave an elementary proof of this case. The case z = 0 was * Supported by ZPSF and NSF of China. 167 ooa-9045!92 $3.00 Copyright ,ca 1992 by Academic Press. inc. All rights of rcprnduction in any form rcser~cti.