Inequalities for the polar derivative of a polynomial with restricted zeros

Abstract

Abstract. For a polynomial p(z) of degree n, we consider an operator In this paper we extend above inequality for the polynomials having no zeros in |z| < 1, except s-fold zeros at the origin. Our result generalize certain well-known polynomial inequalities.

L^q inequalities for polynomials with restricted zeros concerning the growth

“…Concerning the inequalities for |D α P (z)| in terms of |P (z)| an interested reader may go through [9,11], [14]- [17]. Now, the following result is regarding the growth of polar derivative of a polynomial.…”

“…There exist number of results in the literature as refinements and generalizations of inequalities of (1.6) and (1.7) in various ways for references see [9,11,17,18]. Aziz and Rather [3] considered a more general case for investigating the dependence of P (Rz) − βP (rz) q on P q for every β ∈ C with |β| ≤ 1, R > r ≥ 1, q > 0 and extended inequality (1.8) for 0 < q < 1 as following.…”

Generalizations of some Bernstein-type Inequalities for the Polar Derivative of a Polynomial

“…Many of these generalizations involve the comparison of polar derivative D β P (z) with various choices of P (z), β and other parameters. The latest research and development on this topic can be found in the papers ( [5,8,10,11,[13][14][15][16]20]).…”

Note on an inequality of M.A. Malik

“…In the literature, we see that inequalities have been extended from ordinary derivative to polar derivative of polynomials, and for some of the recent research and developments in this direction, one can consult the papers ( [6], [11]- [16], [19], [20]). As a polar derivative generalization of (3), Aziz and Rather [1] proved that if P (z) is a polynomial of degree n having all its zeros in |z| ≤ k, k ≤ 1, then for every complex number α with |α| ≥ k,…”