Zygmund's type inequality to the polar derivative of a polynomial

In this paper we improve a result recently proved by Irshad et al. [On the Inequalities Concerning to the Polar Derivative of a Polynomial with Restricted Zeroes, Thai Journal of Mathematics, 2014 (Article in Press)] and also extend Zygmund's inequality to the polar derivative of a polynomial.


Introduction and Statement of Results
Let P (z) ba a polynomial of degree n, then inequlaity (1.1) is a well known result of S. Bernstein [1]. Equality holds in (1.1) if and only if P (z) has all its zeros at the origin.
Inequality (1.1) was extended to L p -norm p ≥ 1 by Zygmund [2], who proved that if P (z) is a polynomial of degree n, then 1 2π Let α be a complex number. If P (z) is a polynomial of degree n, then the polar derivative of P (z) with respect to the point α, denoted by D α P (z), is defined by D α P (z) = nP (z) + (α − z)P ′ (z) clearly D α P (z) is a polynomial of degree at most n − 1 and it generalizes the ordinary derivative in the sense that As an extension of (1.1) to the polar derivative, Aziz and Shah [3], have shown that if P (z) is a polynomial of degree n, then for every real or complex number α with |α| > 1 and for |z| = 1, As a generalization of (1.2) to the polar derivaative Aziz et al. [4], proved the following result.
Theorem A. If P (z) is a polynomial of degree n, then for every complex number α with |α| ≥ 1 and p ≥ 1 For the Class of polynomials having no zeros in |z| < 1, inequality (1.2) was improved by D-Bruijin [5] that if P (z) = 0 in |z| < 1, then for p ≥ 1 where As an extension to the polar derivative. A. Aziz and N. Rather [6], proved the following generalization of (1.5). In fact they proved.
Theorem B. If P (z) is a polynomial of degree n which does not vanish in |z| < 1, then for every complex number α with |α| ≥ 1 and p ≥ 1 where C p is defined by (1.7).
Theorem C. If P (z) is a polynomial of degre n which does not vanish in |z| < K ≤ 1, then for every α, β ∈ C with |α| ≥ K, |β| ≤ 1 and p ≥ 1 where C p is defined by (1.7).
In this paper we prove the following more general result which also generalize Theorem B and yields a number of known polynomial inequalities.
Theorem 1. If P (z) = a n z n + n j=µ a n−j z n−j , 1 ≤ µ ≤ n be a polynomial of degree n which does not vanish in |z| < K ≤ 1, then for every α, β ∈ C with where C p is defined by (1.7), or equivalently Remark 1. If we choose µ = 1 in (1.10), we get Theorem C and if we choose β = 0 and K = 1 in (1.10), we get Theorem B.

Lemmas
For the proof of this theorem, we need the following lemmas. The first lemma is due to Gulshan Singh et al. [8].
Lemma 1. Let P (z) = a n z n + n j=µ a n−j z n−j , 1 ≤ µ ≤ n be a polynomial of degree having all its zeros in the disk |z| ≤ K, K ≤ 1, then for every real or complex number α with |α| ≥ K, K ≤ 1 and for |z| = 1 Let Q(z) be a polynomial of degree n having all its zeros in |z| < K, K ≤ 1 and P (z) is a polynomial of degree not exceeding that of Q(z).
Now consider a similar argument that for any value of β with |β| < 1, we have where |z| = 1, resulting in where |z| = 1.
That is We also conclude that for |z| = 1.
If (2.5) is not true, then there is a point z = z 0 with |z 0 | = 1, such that then |λ| < 1 with this choice, we have from (2. This completes the proof.
The next lemma is due to Aziz and Rather [4].