On the Approximation Properties of Modified Bernstein–Kantorovich Operators and Their Subfamilies

ABSTRACT Motivated by the construction of Bernstein–Kantorovich operators preserving affine functions, we introduce a Bernstein–Kantorovich type operators together with its subfamilies , and . We investigate and compare their approximation properties with the other Bernstein–Kantorovich operators. Our analysis demonstrates that, proposed operators, especially , have significant approximation properties and preserve affine functions. Furthermore, it is shown that, for our Bernstein Kantorovich type operators , , as for . In addition, some Voronoskaja‐type theorems are established for the proposed operators. We also observe that, the operators possesses shape‐preserving properties. In the final section, the obtained results are illustrated and supported by graphical representations.