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In this paper we establish some Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} inequalities for polynomials having no zeros in |z|<1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|z|<1,$$\end{document} where k≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 1$$\end{document} except for t-fold zeros at origin. Our results not only generalize some known polynomial inequalities, but also a variety of interesting results can be deduced from these by a fairly uniform procedure.


Introduction and statement of results
Let be a polynomial of degree at most n and P � (z) its derivative, then The inequality (1.1) is a classical result of Bernstein [17] whereas the inequality (1.2) is due to Zygmund [18]. Inequality (1.3) is a simple consequence of a result due to Hardy [13]. Arestove [2] proved that (1.2) remains true for 0 < q < 1 as well.
Inequalities (1.2) and (1.3) can be sharpened if we restrict ourselves to the class of polynomials having no zeros in |z| < k where k ≥ 1 . In case k = 1, (1.2) can be replaced [8,16] by where For k ≥ 1 , Govil and Rahman [12] have shown that, if P(z) does not vanish in |z| < k , then for every q ≥ 1, where The validity of (1.5) for 0 < q < 1 is verified in [5,13]. On the other hand, the extension of (1.5) was proved by Aziz and Rather [5].
Aziz and Shah [6] investigated the dependence of max |z|=1 |P(Rz) − P(z)| on max |z|=1 |P(z)| and proved that if inequalities for polynomials If we divide the two sides of (1.6) by R − 1 , make R ⟶ 1 , we get a result due to Qazi [15].
Under the same hypothesis, Gardner, Govil and Weems [10] proved that where m = min |z|=k |P(z)| and As a generalization of (1.8) to the L r norm of P(z) for every r > 0 , Abdullah Mir et.el. [1]proved: , be a polynomial of degree n which does not vanish in |z| < k, k ≥ 1, then for every complex number with | | ≤ 1 and for each r > 0 where S 0 is defined by (1.9).
They also generalized inequality (1.6) by proving that: , be a polynomial of degree n which does not vanish in |z| < k, k ≥ 1 and m = min |z|=k |P(z)|, then for every complex number with | | ≤ 1, r > 0, R > 1 and real In this paper, we consider a class of polynomials P(z) = z t (a t + ∑ n j= a j z j−t ), t + 1 ≤ ≤ n and prove more general results which not only generalize Theorems A and B but among other things provide generalizations for some well known polynomial inequalities in L q spaces.

Main results
Theorem 2.1 If P(z) = z t (a t + ∑ n j= a j z j−t ), t + 1 ≤ ≤ n, is a polynomial of degree n, which does not vanish in |z| < k, k ≥ 1, except for t-fold zeros at origin. Then for every complex number with | | ≤ 1 and for every q > 0,

for polynomials
Taking t = 0 in (2.1), we get Theorem A. If we take min |z|=k| |P(z)| = 0 we obtain the following result which is a special case of Theorem 2.1.

Corollary 2.2 If
is a polynomial of degree n, which does not vanish in |z| < k, k ≥ 1, except for t-fold zeros at origin then for each q > 0, where If we let q → ∞ and t → 0, in (2.3), we get a result due to Qazi [15]. Several other interesting results can be obtained from Corollary 2.2. To mention a few using the fact that S 2 ≥ k in (2.3), we immediately get the following: is a polynomial of degree n, which does not vanish in |z| < k, k ≥ 1, except for t-fold zeros at origin then for each q > 0, Instead of proving Theorem 2.1, we prove a more general result which includes not only Theorem 2.1 and inequality (1.7) as a special case, but also leads to a standard development of interesting generalizations of some well known results.

Theorem 2.4
If P(z) = z t (a t + ∑ n j= a j z j−t ), t + 1 ≤ ≤ n , is a polynomial of degree n, which does not vanish in |z| < k, k ≥ 1, except for t-fold zeros at origin and m = min |z|=k| |P(z)| , then for every q > 0, R > 1 and for every complex number with | | ≤ 1, where If we let q → ∞ in (2.5) and choose the argument of with | | = 1 suitably, we get Corollary 2.5 If P(z) = z t (a t + ∑ n j= a j z j−t ), t + 1 ≤ ≤ n , is a polynomial of degree n, which does not vanish in |z| < k, k ≥ 1, except for t-fold zeros at origin then Dividing the two sides of (2.7) by R-1 and letting R → 1 , we immediately get Corollary 2.6 If P(z) = z t (a t + ∑ n j= a j z j−t ), t + 1 ≤ ≤ n , is a polynomial of degree n, which does not vanish in |z| < k, k ≥ 1, except for t-fold zeros at origin then inequalities for polynomials where Taking t = 0 , we get a result due to (Gardner, Govil and Weems) [10].
If we divide the two sides of (2.5) by R − 1 and letting R → 1 and note that (R) → S 1 as R → 1 , we get (2.2) of Theorem 2.1.

Lemmas
For the proofs of these theorems we need the following lemmas: , is a polynomial of degree n having no zeros in |z| < k, k ≥ 1, then for R > 1 and |z| = 1, where The above lemma is due to Aziz and Shah [6].
The following lemma is due to Aziz and Rather [5].

Lemma 3.2 If P(z) is a polynomial of degree n having all its zeros in
is a non-increasing function of x.

Proof of Lemma 3.3
Taking the first derivative test for S(x), the proof will follow. ◻ Lemma 3.4 If P(z) = ∑ n j=0 a j z j , is a polynomial of degree n, P(z) ≠ 0 in |z| < k, then |P(z)| > m for |z| < k, and in particular |a 0 | > m, where m = min |z|=k |P(z)|.

Lemma 3.5
If P(z) = z t (a 0 + ∑ n j= a j z j−t ), 1 + t ≤ ≤ n , is a polynomial of degree n having no zeros in |z| < k, k ≥ 1, except for t-fold zeros at origin and m = min |z|=k |P(z)| , then Proof of Lemma 3.5 First we show that the inequality holds for all R ≥ 1 and 1 + t ≤ ≤ n . For R = 1 , it is trivial and for R ≥ 1 , It follows when = n − t . So to establish (3.5), it suffices to consider the case when t + 1 ≤ ≤ n − 1 and R > 1.

3 inequalities for polynomials
This implies Which is (3.5).