Integral mean estimates for polynomials whose zeros are within a circle
© Singh and Shah; licensee Springer. 2011
Received: 26 December 2010
Accepted: 19 August 2011
Published: 19 August 2011
Let P(z) be a polynomial of degree n having all its zeros in |z| ≤ K ≤ 1, then for each δ > 0, p > 1, q > 1 with , Aziz and Ahmad (Glas Mat Ser III 31:229-237, 1996) proved that
In this paper, we extend the above inequality to the class of polynomials , 1 ≤ μ ≤ n, having all its zeros in |z| ≤ K ≤ 1, and obtain a generalization as well as refinement of the above result.
Mathematics Subject Classification (2000)
30A10, 30C10, 30C15
KeywordsDerivative of a polynomial Integral mean estimates Inequalities in complex domain
1 Introduction and statement of results
Malik  also obtained a generalization of (1) in the sense that the right-hand side of (1) is replaced by a factor involving the integral mean of |P(z)| on |z| = 1. In fact, he proved the following:
The result is sharp, and equality in (3) holds for P(z) = (z + 1) n . If we let δ → ∞ in (3), we get (1).
As a generalization of Theorem A, Aziz and Shah  proved the following:
Aziz and Ahmad  generalized (3) in the sense that Max|z|=1|P′(z)| on |z| = 1 on the right-hand side of (3) is replaced by a factor involving the integral mean of |P′(z)| on |z| = 1 and proved the following:
If we let p → ∞ (so that q → 1) in (5), we get (3).
In this paper, we consider a class of polynomials , 1 ≤ μ ≤ n, having all the zeros in |z| ≤ K ≤ 1, and thereby obtain a more general result by proving the following:
where m = Min|z|=K|P (z)|.
If we take λ = 0 in Theorem 1, we get the following:
For μ = 1 in Theorem 1, we have the following:
where m = Min|z|=K|P (z)|.
It shows that for λ = 0, Corollary 2 provides a refinement of the result of Aziz and Ahmad .
The next result immediately follows from Theorem 1, if we let p → ∞ (so that q → 1)
From inequality (11), we conclude the following:
Further, if we take K = t = μ = 1 in the Corollary 4, we get a result of Aziz and Dawood .
For the proof of this theorem, we need the following lemmas.
The first lemma is due to Qazi .
Proof of Lemma 2
This proves Lemma 2.
Remark 1: Lemma 3 of Govil and Mc Tume  is a special case of this lemma when μ = 1.
Proof of Theorem 1
Let m = Min|z|=K|P (z)|, so that m ≤ |P (z)| for |z| = K. Therefore, for every complex number λ with |λ| < 1, we have |mλ| < |P(z)| on |z| = K. Since P(z) has all its zeros in |z| ≤ K ≤ 1, by Rouche's theorem, it follows that all the zeros of the polynomial G(z) = P(z) + λm lie in |z| ≤ K ≤ 1.
which proves the desired result.
The authors are grateful to the referee for useful comments.
- Turan P: Über die Ableitung von Polynomen. Composito Math 1939, 7: 89–95.MathSciNetGoogle Scholar
- Malik MA: On the derivative of a polynomial. J Lond Math Soc 1969,1(2):57–60.View ArticleGoogle Scholar
- Malik MA: An integral mean estimate for polynomials. Proc Am Math Soc 1984,91(2):281–284. 10.1090/S0002-9939-1984-0740186-3View ArticleGoogle Scholar
- Aziz A, Shah WM: An integral mean estimate for polynomials. Indian J Pure Appl Math 1997,28(10):1413–1419.MathSciNetGoogle Scholar
- Aziz A, Ahmad N: Integral mean estimates for polynomials whose zeros are within a circle. Glas Mat Ser III 1996,31((51)2):229–237.MathSciNetGoogle Scholar
- Aziz A, Dawood QM: Inequalities for a polynomial and its derivative. J Approx Theory 1988, 54: 306–313. 10.1016/0021-9045(88)90006-8MathSciNetView ArticleGoogle Scholar
- Qazi MA: On the maximum modulus of polynomials. Proc Am Math Soc 1990, 115: 337–343.MathSciNetView ArticleGoogle Scholar
- Govil NK, Mc Tume GN: Some generalizations involving the polar derivative for an inequality of Paul Turan. Acta Math Thunder 2004,104(1–2):115–126.MathSciNetView ArticleGoogle Scholar
- Rahman QI, Schmeisser G: Analytic Theory of Polynomials. Oxford University Press, New York; 2002.Google Scholar
- Milovanovic GV, Mitrinovic DS, Rassias ThM: Topics in Polynomials, Extremal Problems, Inequalities, Zeros. World Scientific, Singapore; 1994.Google Scholar
- Hille E: Analytic Function Theory, Vol. II, Introduction to Higher Mathematics. Ginn and Company, New York, Toronto; 1962.Google Scholar
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