On the order of approximation by modified summation-integral-type operators based on two parameters

In this article, we the study generalized family of positive linear operators based on two parameters, which are a broad family of many well-known linear positive operators, e.g., Baskakov-Durrmeyer, Baskakov-Szasz, Szasz-Beta, Lupas-Beta, Lupas-Szasz, genuine Bernstein-Durrmeyer, and Paltanea. We first find direct estimates in terms of the second-order modulus of continuity, then we find an estimate of error in the Ditzian-Totik modulus of smoothness. Then we discuss the rate of approximation for functions in the Lipschitz class. Furthermore, we study the pointwise Gruss-Voronovskaja-type result and also establish the Gruss-Voronovskaja-type asymptotic formula in quantitative form.