Stud. Univ. Babe¸ s-Bolyai Math. 57(2012), No. 1, 83–88 On the Conjecture of Cao, Gonska and Kacs´o Gancho T. Tachev Abstract. We consider the question if lower estimates in terms of the second or- der Ditzian-Totik modulus are possible, when we measure the pointwise approx- imation of continuous function by Bernstein operator. In this case we confirm the conjecture made by Cao, Gonska and Kacs´o. To prove this we first estab- lish sharp upper and lower bounds for pointwise approximation of the function g(x)= x ln(x) + (1 - x) ln(1 - x),x ∈ [0, 1] by Bernstein operator. Mathematics Subject Classification (2010): 41A10, 41A15, 41A25, 41A36. Keywords: Bernstein operator, lower bounds, Ditzian-Totik moduli of smooth- ness. 1. Introduction In [6] Cao, Gonska and Kacs´ o formulated the following Conjecture 1.1. Let T n : C[a, b] → C[a, b] be a sequence of linear operators and ε n > 0, lim n→∞ ε n =0,ϕ(x)= ϕ(x) [a,b] =  (x − a)(b − x), and 0 ≤ β<λ ≤ 1 fixed. If for every f ∈ C[a, b] one has |T n (f,x) − f (x)|≤ C(f )ω ϕ λ 2 ( f ; ε n ϕ 1−λ (x) ) , (1.1) then lower pointwise estimates c(f )ω ϕ β 2 ( f ; ε n ϕ 1−λ (x) ) ≤|T n (f,x) − f (x)|,f ∈ C[a, b], (1.2) do not hold in general. The case β = 0 was already solved by the same authors in Theorem 3.1 in [5]. The aim of this note is to confirm conjecture above for the case when T n is replaced by the Bernstein operator B n . Instead of T n we consider further only the classical Bernstein operator B n applied to a continuous on [0, 1] function f (x) and defined by B n (f ; x)= n  k=0 f  k n  ·  n k  x k (1 − x) n−k ,x ∈ [0, 1]. (1.3)