On generalized Voronovskaja theorem for Bernstein polynomials

We establish a new quantitative variant of Floater’s theorem dealing with the generalization of Voronovskaja’s theorem for Bernstein polynomials. Our estimate improves the recent quantitative version of Floater’s theorem proved in [Gonska, H. and Ras¸a, I., Asymptotic behaviour of differentiated Bernstein polynomials, Mat. Vesnik 61 (2009), 53-60]. Moreover, as a particular case, we recover the main result of [Tachev, G. T., Voronovskaja’s theorem revisited, J. Math. Anal. Appl. 343 (2008), 399-404]