Integral mean estimates for polynomials whose zeros are within a circle

Gulshan Singh; WM Shah

doi:10.1186/1029-242X-2011-35

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Integral mean estimates for polynomials whose zeros are within a circle

Gulshan Singh¹Email author and
WM Shah²

Journal of Inequalities and Applications20112011:35

https://doi.org/10.1186/1029-242X-2011-35

Received: 26 December 2010

Accepted: 19 August 2011

Published: 19 August 2011

Abstract

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 $\frac{1}{p} + \frac{1}{q} = 1$ , Aziz and Ahmad (Glas Mat Ser III 31:229-237, 1996) proved that

n {\{\int_{0}^{2 π} | P (e^{i θ}) |^{δ} d θ\}}^{\frac{1}{δ}} \leq {\{\int_{0}^{2 π} | 1 + K e^{i θ} |^{q δ} d θ\}}^{\frac{1}{q δ}} {\{\int_{0}^{2 π} | P^{'} (e^{i θ}) |^{p δ} d θ\}}^{\frac{1}{p δ}} .

In this paper, we extend the above inequality to the class of polynomials $P (z) : = a_{n} z^{n} + \sum_{j = μ}^{n} a_{n - j} z^{n - j}$ , 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

Keywords

Derivative of a polynomial Integral mean estimates Inequalities in complex domain

1 Introduction and statement of results

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

M a x_{| z | = 1} | P^{'} (z) | \geq \frac{n}{2} M a x_{| z | = 1} | P (z) | .

(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

M a x_{| z | = 1} | P^{'} (z) | \geq \frac{n}{1 + K} M a x_{| z | = 1} | P (z) | .

(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

n {\{\int_{0}^{2 π} | P (e^{i θ}) |^{δ} d θ\}}^{\frac{1}{δ}} \leq {\{\int_{0}^{2 π} | 1 + e^{i θ} |^{δ} d θ\}}^{\frac{1}{δ}} M a x_{| z | = 1} | P^{'} (z) | .

(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

P (z) : = a_{n} z^{n} + \sum_{j = μ}^{n} a_{n - j} z^{n - j}

, 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,

n {\{\int_{0}^{2 π} | P (e^{i θ}) |^{δ} d θ\}}^{\frac{1}{δ}} \leq {\{\int_{0}^{2 π} | 1 + K^{μ} e^{i θ} |^{δ} d θ\}}^{\frac{1}{δ}} M a x_{| z | = 1} | P^{'} (z) | .

(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

\frac{1}{p} + \frac{1}{q} = 1

,

n {\{\int_{0}^{2 π} | P (e^{i θ}) |^{δ} d θ\}}^{\frac{1}{δ}} \leq {\{\int_{0}^{2 π} | 1 + K e^{i θ} |^{q δ} d θ\}}^{\frac{1}{q δ}} {\{\int_{0}^{2 π} | P^{'} (e^{i θ}) |^{p δ} d θ\}}^{\frac{1}{p δ}} .

(5)

If we let p → ∞ (so that q → 1) in (5), we get (3).

In this paper, we consider a class of polynomials $P (z) : = a_{n} z^{n} + \sum_{j = μ}^{n} a_{n - j} z^{n - j}$ , 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

P (z) : = a_{n} z^{n} + \sum_{j = μ}^{n} a_{n - j} z^{n - j}

, 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

\frac{1}{p} + \frac{1}{q} = 1

and for every complex number λ with |λ| < 1

\begin{gathered} n {\{\int_{0}^{2 π} | P (e^{i θ}) + λ m |^{δ} d θ\}}^{\frac{1}{δ}} \\ \leq {\{\int_{0}^{2 π} | 1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] e^{i θ} |^{q δ} d θ\}}^{\frac{1}{q δ}} {\{\int_{0}^{2 π} | P^{'} (e^{i θ}) |^{p δ} d θ\}}^{\frac{1}{p δ}}, \end{gathered}

(6)

where m = Min_|z|=K|P (z)|.

If we take λ = 0 in Theorem 1, we get the following:

Corollary 1. If

P (z) : = a_{n} z^{n} + \sum_{j = μ}^{n} a_{n - j} z^{n - j}

, 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

\frac{1}{p} + \frac{1}{q} = 1

, ^,

\begin{gathered} n {\{\int_{0}^{2 π} | P (e^{i θ}) |^{δ} d θ\}}^{\frac{1}{δ}} \\ \leq {\{\int_{0}^{2 π} | 1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] e^{i θ} |^{q δ} d θ\}}^{\frac{1}{q δ}} {\{\int_{0}^{2 π} | P^{'} (e^{i θ}) |^{p δ} d θ\}}^{\frac{1}{p δ}}, \end{gathered}

(7)

For μ = 1 in Theorem 1, we have the following:

Corollary 2. If

P (z) : = \sum_{j = 0}^{n} a_{j} z^{j}

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

\frac{1}{p} + \frac{1}{q} = 1

,

\begin{gathered} n {\{\int_{0}^{2 π} | P (e^{i θ}) + λ m |^{δ} d θ\}}^{\frac{1}{δ}} \\ \leq {\{\int_{0}^{2 π} | 1 + [\frac{n | a_{n} | K^{2} + | a_{n - 1} |}{n | a_{n} | + | a_{n - 1} |}] e^{i θ} |^{q δ} d θ\}}^{\frac{1}{q δ}} {\{\int_{0}^{2 π} | P^{'} (e^{i θ}) |^{p δ} d θ\}}^{\frac{1}{p δ}}, \end{gathered}

(8)

where m = Min_|z|=K|P (z)|.

Remark 1: Since all the zeros of P(z) lie in |z| ≤ K, therefore,

\frac{1}{n} | \frac{a_{n - 1}}{a_{n}} | \leq K K \leq 1

, it can be easily verified that

\frac{n | a_{n} | K^{2} + | a_{n - 1} |}{n | a_{n} | + | a_{n - 1} |} \leq K .

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

P (z) : = a_{n} z^{n} + \sum_{j = μ}^{n} a_{n - j} z^{n - j}

, 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

\begin{gathered} n {\{\int_{0}^{2 π} | P (e^{i θ}) + λ m |^{δ} d θ\}}^{\frac{1}{δ}} \\ \leq {\{\int_{0}^{2 π} | 1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] e^{i θ} |^{δ} d θ\}}^{\frac{1}{δ}} M a x_{| z | = 1} | P^{'} (z) | . \end{gathered}

(9)

Also if we let δ → ∞ in the Corollary 3 and note that

lim_{δ \to \infty} {\frac{1}{2 π} \int_{0}^{2 π} | P (e^{i θ}) |^{δ} d θ}^{\frac{1}{δ}} = M a x_{| z | = 1} | P (z) |,

we get from (9)

n M a x_{| z | = 1} | P (z) + λ m | \leq \frac{n | a_{n} | (K^{2 μ} + K^{μ - 1}) + μ | a_{n - μ} | (1 + K^{μ - 1})}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |} M a x_{| z | = 1} | P^{'} (z) | f o r | z | = 1 .

(10)

If z₀ be such that Max_|z|=1|P (z)| = |P (z₀)|, then from (10), we have

\begin{gathered} n | P (z_{0}) + λ m | \leq \\ \frac{n | a_{n} | (K^{2 μ} + K^{μ - 1}) + μ | a_{n - μ} | (1 + K^{μ - 1})}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |} M a x_{| z | = 1} | P^{'} (z) | f o r | z | = 1 . \end{gathered}

Choosing an argument of λ such that

| P (z_{0}) + λ m | = | P (z_{0}) | + | λ | m,

we get

n (| P (z_{0}) | + | λ | m) \leq \frac{n | a_{n} | (K^{2 μ} + K^{μ - 1}) + μ | a_{n - μ} | (1 + K^{μ - 1})}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |} M a x_{| z | = 1} | P^{'} (z) | .

(11)

From inequality (11), we conclude the following:

Corollary 4. If

P (z) : = a_{n} z^{n} + \sum_{j = μ}^{n} a_{n - j} z^{n - j}

, 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

\begin{gathered} M a x_{| z | = 1} | P^{'} (z) | \geq \\ n \frac{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}{n | a_{n} | (K^{2 μ} + K^{μ - 1}) + μ | a_{n - μ} | (1 + K^{μ - 1})} {M a z_{| z | = 1} | P (z) | + t M i n_{| z | = K} | P (z) |} . \end{gathered}

Further, if we take K = t = μ = 1 in the Corollary 4, we get a result of Aziz and Dawood [6].

2. Lemmas

For the proof of this theorem, we need the following lemmas.

The first lemma is due to Qazi [7].

Lemma 1. If

P (z) : = a_{0} + \sum_{j = μ}^{n} a_{j} z^{j}, 1 \leq μ \leq n

is a polynomial of degree n having no zeros in the disk|z| < K, K ≥ 1, then

[\frac{n | a_{0} | K^{μ + 1} + μ | a_{μ} | K^{2 μ}}{n | a_{0} | + μ | a_{μ} | K^{μ + 1}}] | P^{'} (z) | \leq | Q^{'} (z) | f o r | z | = 1,

where $Q (z) = z^{n} \bar{P (\frac{1}{\bar{z}})} a n d \frac{μ}{n} | \frac{a_{μ}}{a_{0}} | K^{μ} \leq 1$ .

Lemma 2. If

P (z) : = a_{n} z^{n} + \sum_{j = μ}^{n} a_{n - j} z^{n - j}

is a polynomial of degree n having all its zeros in the disk|z| ≤ K ≤ 1, then

| Q^{'} (z) | \leq [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] | P^{'} (z) | f o r | z | = 1, 1 \leq μ \leq n,

where $Q (z) = z^{n} \bar{P (\frac{1}{\bar{z}})}$ .

Proof of Lemma 2

Since all the zeros of P(z) lie in |z| ≤ K ≤ 1, therefore all the zeros of

Q (z) = z^{n} \bar{P (\frac{1}{\bar{z}})}

lie in

| z | \geq \frac{1}{K} \geq 1

. Hence, applying Lemma 1 to the polynomial

Q (z) : = ā_{n} + \sum_{j = μ}^{n} ā_{n - j} z^{j}

, we get

[\frac{n | a_{n} | \frac{1}{K^{μ + 1}} + μ | a_{n - μ} | \frac{1}{K^{2 μ}}}{n | a_{n} | + μ | a_{n - μ} | \frac{1}{K^{μ + 1}}}] | Q^{'} (z) | \leq | P^{'} (z) | .

Or, equivalently

| Q^{'} (z) | \leq [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] | P^{'} (z) |, | z | = 1 .

This proves Lemma 2.

Remark 1: Lemma 3 of Govil and Mc Tume [8] is a special case of this lemma when μ = 1.

Proof of Theorem 1

Let

Q (z) = z^{n} \bar{P (\frac{1}{\bar{z}})}

z, we have

P (z) = z^{n} \bar{Q (\frac{1}{\bar{z}})}

. This gives

P^{'} (z) = n z^{n - 1} \bar{Q (\frac{1}{\bar{z}})} - z^{n - 2} \bar{Q^{'} (\frac{1}{\bar{z}})} .

(12)

Equivalently,

z P^{'} (z) = n z^{n} \bar{Q (\frac{1}{\bar{z}})} - z^{n - 1} \bar{Q^{'} (\frac{1}{\bar{z}})} .

(13)

This implies

| P^{'} (z) | = | n Q (z) - z Q^{'} (z) | f o r | z | = 1 .

(14)

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.

If

H (z) = z^{n} \bar{G (\frac{1}{\bar{z}})} = Q (z) + m \bar{λ} z^{n}

, then by applying Lemma 2 to the polynomial G(z) = P(z) + λm, we have for |z| = 1

| H^{'} (z) | \leq [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] | G^{'} (z) |, 1 \leq μ \leq n .

This gives

| Q^{'} (z) + n m \bar{λ} z^{n - 1} | \leq [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] | P^{'} (z) |, 1 \leq μ \leq n .

(15)

Using (14) in (15), we get

| Q^{'} (z) + n m \bar{λ} z^{n - 1} | \leq [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] | n Q (z) - z Q^{'} (z) | f o r | z | = 1, 1 \leq μ \leq n .

(16)

Since P(z) has all its zeros in |z| ≤ K ≤ 1, by Gauss{Lucas theorem so does P^′(z). It follows that nQ(z) - zQ^′(z), which is simply (see (12))

z^{n - 1} \bar{P^{'} (\frac{1}{\bar{z}})},

has all its zeros in |z| ≥

\frac{1}{K}

≥ 1. Hence,

W (z) = [\frac{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}] . \frac{z (Q^{'} (z) + n m \bar{λ} z^{n - 1})}{(n Q (z) - z Q^{'} (z))}

(17)

is analytic for |z| < 1, |W(z)| ≤ 1 for |z| = 1 and W(0)=0. Thus, the function

1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] . W (z)

is subordinate to the function

1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] z

for |z| < 1. Hence, by a property of subordination (for reference see [[9], p. 36, Theorem 1.6.17] or [[10], p. 454] or [11]), we have for each δ > 0 and 0 ≤ θ < 2π,

\begin{gathered} \int_{0}^{2 π} | 1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] W (e^{i θ}) |^{δ} d θ \\ \leq \int_{0}^{2 π} | 1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] e^{i θ} |^{δ} d θ . \end{gathered}

(18)

Also from (17), we have

1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] W (z) = \frac{n (Q (z) + m \bar{λ} z^{n})}{n Q (z) - z Q^{'} (z)} .

Therefore,

n | Q (z) + m \bar{λ} z^{n} | = | 1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] W (z) | | n Q (z) - z Q^{'} (z) |,

(19)

which implies

n | H (z) | = | 1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] W (z) | | n Q (z) - z Q^{'} (z) | .

(20)

Using (14) and the fact that |H(z)| = |G(z)| = |P(z) + λm| for |z| = 1, we get from (20)

n | P (z) + λ m | = | 1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] W (z) | | P^{'} (z) | f o r | z | = 1 .

Hence, for each δ > 0 and 0 ≤ θ < 2π, we have

\begin{gathered} n^{δ} \int_{0}^{2 π} | P (e^{i θ}) + λ m |^{δ} d θ \\ = \int_{0}^{2 π} | 1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - v} |}] W (e^{i θ}) |^{δ} | P^{'} (e^{i θ}) |^{δ} d θ . \end{gathered}

(21)

This gives with the help of Hölder's inequality for p > 1, q > 1, with

\frac{1}{p} + \frac{1}{q} = 1

and for every δ > 0,

\begin{gathered} n^{δ} \int_{0}^{2 π} | P (e^{i θ}) + λ m |^{δ} d θ \\ \leq {\{\int_{0}^{2 π} | 1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] W (e^{i θ}) |^{q δ} d θ\}}^{\frac{1}{q}} {\{\int_{0}^{2 π} | P^{'} (e^{i θ}) |^{p δ} d θ\}}^{\frac{1}{p}} \end{gathered}

(22)

Combining (18) and (22), we get for δ > 0 and 0 ≤ θ < 2π,

\begin{gathered} n^{δ} \int_{0}^{2 π} | P (e^{i θ}) + λ m |^{δ} d θ \\ \leq {\{\int_{0}^{2 π} | 1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] e^{i θ} |^{q δ} d θ\}}^{\frac{1}{q}} {\{\int_{0}^{2 π} | P^{'} (e^{i θ}) |^{p δ} d θ\}}^{\frac{1}{p}} \end{gathered}

(23)

This is equivalent to

\begin{gathered} n {\{\int_{0}^{2 π} | P (e^{i θ}) + λ m |^{δ} d θ\}}^{\frac{1}{δ}} \\ \leq {\{\int_{0}^{2 π} | 1 + [\frac{n | a_{n} | K^{2 μ} + μ | a_{n - μ} | K^{μ - 1}}{n | a_{n} | K^{μ - 1} + μ | a_{n - μ} |}] e^{i θ} |^{q δ} d θ\}}^{\frac{1}{q δ}} {\{\int_{0}^{2 π} | P^{'} (e^{i θ}) |^{p δ} d θ\}}^{\frac{1}{p δ}} \end{gathered}

(24)

which proves the desired result.

Declarations

Acknowledgements

The authors are grateful to the referee for useful comments.

Authors’ Affiliations

(1)

Bharathiar University

(2)

Department of Mathematics, Kashmir University Srinagar

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