Let p(z) = a 0 + n ν=μ a ν z ν , 1 μ n, be a polynomial of degree n such that p(z) = 0 in |z| < k, k > 0, then for 0 < r R k, Dewan, Yadav and Pukhta [K.K. Dewan, R.S. Yadav, M.S. Pukhta, Inequalities for a polynomial and its derivative,Equality holds for the polynomial p(z) = (z μ + k μ ) n μ 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].
Let p(z) be a polynomial of degree n having no zeros in |z|<1. In this paper, we generalize and extend a well-known result proven by Ankeny and Rivlin for the sth derivative of the polynomial. Furthermore, another well-known result proven by Rivlin is also improved, generalized and extended for the sth derivative of p(z). Our results also give a number of interesting consequences as special cases.
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