The present paper deals with the modified positive linear operators that present a better degree of approximation than the original ones. This new construction of operators depend on a certain function defined on [0, 1]. Some approximation properties of these operators are given. Using the first order Ditzian-Totik modulus of smoothness, some Voronovskaja type theorems in quantitative mean are proved. The main results proved in this paper are applied for Bernstein operators, Lupas operators and genuine Bernstein-Durrmeyer operators. By numerical examples we show that depending on the choice of the function , the modified operator presents a better order of approximation than the classical ones.