Note on the shape-preservation of a new class of Kantorovich-type operators via divided differences

Abstract Shape-preserving approximation is a significant approximation method that has many application areas, such as computer-based geometric design, image processing, geodesy, chemistry, and robotics. Due to this important fact, the aim of this work is to study the shape-preserving properties of a new class of Kantorovich-type operators, which are constructed via Bézier bases based on shape parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> \lambda \in {[}-1,1] . In pursuance of this goal, we commence our calculations by rewriting these operators as a summation of the classical Kantorovich-type operators and a summation of integral values of first-order divided differences of function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> </m:math> f over the interval [0, 1]. Then, adopting this representation, we establish the shape-preserving properties of these operators, such as linearity, positivity, and specifically monotonicity and convexity-preserving properties in connection with function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> </m:math> f on the interval [0, 1]. Our results reveal that while operators preserve monotonicity entirely on the interval [0, 1] for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> \lambda \in {[}-1,1] , they fail to preserve convexity for some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> \lambda on the same interval. We endorse this claim by counterexamples and give an altered result on convexity preservation for a specific class of functions when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:math> \lambda \in {[}0,1] .