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Abstract In the present paper, we study a new type of Bernstein operators depending on the parameter λ∈[−1,1] $\lambda\in[-1,1]$. The Kantorovich modification of these sequences of linear positive operators will be considered. A quantitative Voronovskaja type theorem by means of Ditzian–Totik modulus of smoothness is proved. Also, a Grüss–Voronovskaja type theorem for λ-Kantorovich operators is provided. Some numerical examples which show the relevance of the results are given.