On better approximation order for the max-product Meyer-König and Zeller operator

Abstract Bede et al. (B. Bede, L. Coroianu, and S. G. Gal, Approximation and shape preserving properties of the nonlinear Meyer-König and Zeller operator of max-product kind , Numer. Funct. Anal. Optim. 31 (2010), no. 3, 232–253) defined the max-product Meyer-König and Zeller operator. They examined the approximation and shape preserving properties of this operator, and they found the order of approximation to be <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mrow> <m:msqrt> <m:mrow> <m:mi>y</m:mi> </m:mrow> </m:msqrt> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:msqrt> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msqrt> </m:mrow> </m:mfrac> </m:math> \frac{\sqrt{y}(1-y)}{\sqrt{m}} by the modulus of continuity and claimed that this order of approximation could only be improved in certain subclasses of the functions. In contrast to this claim, we demonstrate that we can obtain a better order of approximation without reducing the function class (by the classical modulus of continuity). We find the degree of approximation to be <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mstyle displaystyle="false"> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:mfrac> </m:mstyle> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mstyle displaystyle="false"> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:mfrac> </m:mstyle> </m:mrow> </m:msup> </m:mrow> </m:mfrac> </m:math> \frac{(1-y){y}^{\tfrac{1}{\alpha }}}{{m}^{1-\tfrac{1}{\alpha }}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> <m:mo form="prefix">,</m:mo> <m:mn>3</m:mn> <m:mo form="prefix">,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> </m:math> \alpha =2,3,\ldots . Since <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:mfrac> </m:math> 1-\frac{1}{\alpha } tends to 1 for enough big <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha , we improve this degree of approximation.