Inequalities for a polynomial and its derivative

The polar derivative of a polynomial p(z) of degree n with respect to a complex number α is a polynomial np(z)+α-zp′(z), denoted by Dαp(z). Let 1≤R≤k. For a polynomial p(z) of degree n having all its zeros in z≤k, we investigate a lower bound of modulus of Dαp(z) on z=R. Furthermore, we present an upper bound of modulus of Dαp(z) on z=R for a polynomial p(z) of degree n having no zero in z<k. In particular, our results in case R=1 generalize some well-known inequalities.

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“…Inequality (4) is best possible and equality holds for a polynomial ( ) = + . In 1999, Dewan et al [5] proved the following result.…”

Section: Introductionmentioning

confidence: 96%

“…Theorem 1 (see [5]). Let ( ) = 0 + ∑ ]= ] ] , 1 ≤ ≤ , be a polynomial of degree such that ( ) has no zero in | | < , > 0.…”

Section: Introductionmentioning

confidence: 99%

Some Bounds for the Polar Derivative of a Polynomial

Somsuwan

Nakprasit

2018

International Journal of Mathematics and Mathematical Sciences

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“…Dewan, Yadav and Pukhta [5] further improved upon as well as generalized inequality (1.8) by involving min |z|=k |p(z)|.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 93%

Inequalities for a polynomial and its derivative

Chanam¹,

Dewan²

2007

Journal of Mathematical Analysis and Applications

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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].

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“…We next prove the following result which is a generalization of the inequality (7),and result due to Gardner, Govil and Weems [8].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 96%