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We extend the well known Bernstein inequality to the generalized derivative of a polynomial introduced by Sz-Nagy [13] with restricted zeros and obtain some new inequalities with respect to sup-norm. The obtained results include some known inequalities of Turán, Malik and Govil as special cases for the ordinary derivative. [ABSTRACT FROM AUTHOR]

Given a set of points in the complex plane, an incomplete polynomial is defined as one which has these points as zeros except one of them. Recently, the classical result known as Gauss–Lucas theorem on the location of zeros of polynomials and their derivatives was extended to the linear combinations of incomplete polynomials. In this paper, a simple proof of this result is given, and some results concerning the critical points of polynomials due to Jensen and others have extended the linear combinations of incomplete polynomials. [ABSTRACT FROM AUTHOR]