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In this paper, we establish some upper bound estimates for the maximum modulus of the higher order derivative of a polynomial with restricted zeros on a circle of |z| = R, R ≥ 1, in terms of the maximum modulus of the same polynomial on the unit circle |z| = 1. These results generalize and sharpen some known results in this direction. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n with zero of multiplicity s at the origin and the remaining zeros be 0. In this paper, we investigate the relative growth of a polynomial p(z) with respect to two circles z=r and z = R and obtain inequalities about the dependence of p(rz) on p(Rz), where z = 1, for 0 while taking into account the placement of the zeros of the underlying polynomial. Our results improve as well as generalize certain well-known polynomial inequalities. Some numerical examples are also given in order to illustrate and compare graphically the obtained inequalities with some recent results. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n having no zero in /z

Abstract: Let , , be a polynomial of degree n such that in , , then for , Dewan, Yadav and Pukhta [K.K. Dewan, R.S. Yadav, M.S. Pukhta, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 2 (2) (1999) 203–205] proved Equality holds for the polynomial where n is a multiple of μ. In this paper, we obtain an improvement of the above inequality by involving some of the coefficients. As an application of our result, we further improve upon a result recently proved by Aziz and Shah [A. Aziz, W.M. Shah, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 7 (3) (2004) 379–391]. [Copyright &y& Elsevier]