Let P(z) be a polynomial of degree n and P′(z) be its derivative. If P(z) has all its zeros in |z| ≤ 1, then it was shown by Turan [1] that
(1)
Inequality (1) is best possible with equality for P(z) = αzn + β, where |α| = |β|. As an extension of (1), Malik [2] proved that if P(z) has all its zeros in |z| ≤ K, where K ≤ 1, then
(2)
Malik [3] 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:
Theorem A. If P(z) has all its zeros in |z| ≤ 1, then for each δ > 0
(3)
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 [4] proved the following:
Theorem B. If, 1 ≤ μ ≤ n is a polynomial of degree n having all its zeros in the disk |z| ≤ K, K ≤ 1, then for each δ > 0,
(4)
Aziz and Ahmad [5] 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:
Theorem C. If P(z) is a polynomial of degree n having all its zeros in |z| ≤ K ≤ 1, then for δ > 0, p > 1, q > 1 with,
(5)
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:
Theorem 1. If, 1 ≤ μ ≤ n is a polynomial of degree n having all its zeros in the disk |z| ≤ K, K ≤ 1, then for each δ > 0, q > 1, p > 1 withand for every complex number λ with |λ| < 1
(6)
where m = Min|z|=K|P (z)|.
If we take λ = 0 in Theorem 1, we get the following:
Corollary 1. If, 1 ≤ μ ≤ n is a polynomial of degree n having all its zeros in the disk |z| ≤ K, K ≤ 1, then for each δ > 0, q > 1, p > 1 with, ,
(7)
For μ = 1 in Theorem 1, we have the following:
Corollary 2. Ifis a polynomial of degree n having all its zeros in the disk |z| ≤ K, K ≤ 1, then for each δ > 0, q > 1, p > 1 with,
(8)
where m = Min|z|=K|P (z)|.
Remark 1: Since all the zeros of P(z) lie in |z| ≤ K, therefore, , it can be easily verified that
It shows that for λ = 0, Corollary 2 provides a refinement of the result of Aziz and Ahmad [5].
The next result immediately follows from Theorem 1, if we let p → ∞ (so that q → 1)
Corollary 3. If, 1 ≤ μ ≤ n is a polynomial of degree n having all its zeros in the disk |z| ≤ K, K ≤ 1, then for each δ > 0 and for every complex number λ with |λ| < 1
(9)
Also if we let δ → ∞ in the Corollary 3 and note that
we get from (9)
(10)
If z0 be such that Max|z|=1|P (z)| = |P (z0)|, then from (10), we have
Choosing an argument of λ such that
we get
(11)
From inequality (11), we conclude the following:
Corollary 4. If, 1 ≤ μ ≤ n is a polynomial of degree n having all its zeros in the disk |z| ≤ K, K ≤ 1, then for 0 ≤ t ≤ 1, we have
Further, if we take K = t = μ = 1 in the Corollary 4, we get a result of Aziz and Dawood [6].