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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Published: 19 August 2011

Integral mean estimates for polynomials whose zeros are within a circle

Gulshan Singh¹ &
WM Shah²

Journal of Inequalities and Applications volume 2011, Article number: 35 (2011) Cite this article

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

n ∫ 0 2 π | P (e i θ) | δ d θ 1 δ ≤ ∫ 0 2 π | 1 + K e i θ | q δ d θ 1 q δ ∫ 0 2 π | P ′ (e i θ) | p δ d θ 1 p δ .

In this paper, we extend the above inequality to the class of polynomials $P (z) : = a n z n + ∑ j = μ n a n - j z n - j$ , 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

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) | ≥ 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) | ≥ 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 ∫ 0 2 π | P (e i θ) | δ d θ 1 δ ≤ ∫ 0 2 π | 1 + e i θ | δ d θ 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 $P (z) : = a n z n + ∑ j = μ n a n - j z n - j$ , 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 ∫ 0 2 π | P (e i θ) | δ d θ 1 δ ≤ ∫ 0 2 π | 1 + K μ e i θ | δ d θ 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 $1 p + 1 q = 1$ ,

n ∫ 0 2 π | P (e i θ) | δ d θ 1 δ ≤ ∫ 0 2 π | 1 + K e i θ | q δ d θ 1 q δ ∫ 0 2 π | P ′ (e i θ) | p δ d θ 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 $P (z) : = a n z n + ∑ j = μ n a n - j z n - j$ , 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 $P (z) : = a n z n + ∑ j = μ n a n - j z n - j$ , 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 $1 p + 1 q = 1$ and for every complex number λ with |λ| < 1

n ∫ 0 2 π | P (e i θ) + λ m | δ d θ 1 δ ≤ ∫ 0 2 π | 1 + 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 1 q δ ∫ 0 2 π | P ′ (e i θ) | p δ d θ 1 p δ,

(6)

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

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

Corollary 1. If $P (z) : = a n z n + ∑ j = μ n a n - j z n - j$ , 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 $1 p + 1 q = 1$ , ^,

n ∫ 0 2 π | P (e i θ) | δ d θ 1 δ ≤ ∫ 0 2 π | 1 + 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 1 q δ ∫ 0 2 π | P ′ (e i θ) | p δ d θ 1 p δ,

(7)

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

Corollary 2. If $P (z) : = ∑ j = 0 n a j z j$ 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 $1 p + 1 q = 1$ ,

n ∫ 0 2 π | P (e i θ) + λ m | δ d θ 1 δ ≤ ∫ 0 2 π | 1 + n | a n | K 2 + | a n - 1 | n | a n | + | a n - 1 | e i θ | q δ d θ 1 q δ ∫ 0 2 π | P ′ (e i θ) | p δ d θ 1 p δ,

(8)

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

Remark 1: Since all the zeros of P(z) lie in |z| ≤ K, therefore, $1 n | a n - 1 a n | ≤ K K ≤ 1$ , it can be easily verified that

n | a n | K 2 + | a n - 1 | n | a n | + | a n - 1 | ≤ 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 $P (z) : = a n z n + ∑ j = μ n a n - j z n - j$ , 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

n ∫ 0 2 π | P (e i θ) + λ m | δ d θ 1 δ ≤ ∫ 0 2 π | 1 + 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 1 δ M a x | z | = 1 | P ′ (z) | .

(9)

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

lim δ → ∞ {1 2 π ∫ 0 2 π | P (e i θ) | δ d θ} 1 δ = M a x | z | = 1 | P (z) |,

we get from (9)

n M a x | z | = 1 | P (z) + λ m | ≤ 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

n | P (z 0) + λ m | ≤ 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 .

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) ≤ 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 $P (z) : = a n z n + ∑ j = μ n a n - j z n - j$ , 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

M a x | z | = 1 | P ′ (z) | ≥ n 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) |} .

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 $P (z) : = a 0 + ∑ j = μ n a j z j, 1 ≤ μ ≤ n$ is a polynomial of degree n having no zeros in the disk|z| < K, K ≥ 1, then

n | a 0 | K μ + 1 + μ | a μ | K 2 μ n | a 0 | + μ | a μ | K μ + 1 | P ′ (z) | ≤ | Q ′ (z) | f o r | z | = 1,

where $Q (z) = z n P (1 z ̄) ¯ a n d μ n | a μ a 0 | K μ ≤ 1$ .

Lemma 2. If $P (z) : = a n z n + ∑ j = μ n a n - j z n - j$ is a polynomial of degree n having all its zeros in the disk|z| ≤ K ≤ 1, then

| Q ′ (z) | ≤ 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 ≤ μ ≤ n,

where $Q (z) = z n P (1 z ̄) ¯$ .

Proof of Lemma 2

Since all the zeros of P(z) lie in |z| ≤ K ≤ 1, therefore all the zeros of $Q (z) = z n P (1 z ̄) ¯$ lie in $| z | ≥ 1 K ≥ 1$ . Hence, applying Lemma 1 to the polynomial $Q (z) : = ā n + ∑ j = μ n ā n - j z j$ , we get

n | a n | 1 K μ + 1 + μ | a n - μ | 1 K 2 μ n | a n | + μ | a n - μ | 1 K μ + 1 | Q ′ (z) | ≤ | P ′ (z) | .

Or, equivalently

| Q ′ (z) | ≤ 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 $Q (z) = z n P (1 z ̄) ¯$ z, we have $P (z) = z n Q (1 z ̄) ¯$ . This gives

P ′ (z) = n z n - 1 Q (1 z ̄) ¯ - z n - 2 Q ′ (1 z ̄) ¯ .

(12)

Equivalently,

z P ′ (z) = n z n Q (1 z ̄) ¯ - z n - 1 Q ′ (1 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 $H (z) = z n G (1 z ̄) ¯ = Q (z) + m λ ̄ z n$ , then by applying Lemma 2 to the polynomial G(z) = P(z) + λm, we have for |z| = 1

| H ′ (z) | ≤ 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 ≤ μ ≤ n .

This gives

| Q ′ (z) + n m λ ̄ z n - 1 | ≤ 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 ≤ μ ≤ n .

(15)

Using (14) in (15), we get

| Q ′ (z) + n m λ ̄ z n - 1 | ≤ 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 ≤ μ ≤ 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 P ′ (1 z ̄) ¯,

has all its zeros in |z| ≥ $1 K$ ≥ 1. Hence,

W (z) = n | a n | K μ - 1 + μ | a n - μ | n | a n | K 2 μ + μ | a n - μ | K μ - 1 . z (Q ′ (z) + n m 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 + 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 + 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,

∫ 0 2 π | 1 + 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 ≤ ∫ 0 2 π | 1 + 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 .

(18)

Also from (17), we have

1 + 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) + m λ ̄ z n) n Q (z) - z Q ′ (z) .

Therefore,

n | Q (z) + m λ ̄ z n | = | 1 + 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 + 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 + 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

n δ ∫ 0 2 π | P (e i θ) + λ m | δ d θ = ∫ 0 2 π | 1 + 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 .

(21)

This gives with the help of Hölder's inequality for p > 1, q > 1, with $1 p + 1 q = 1$ and for every δ > 0,

n δ ∫ 0 2 π | P (e i θ) + λ m | δ d θ ≤ ∫ 0 2 π | 1 + 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 1 q ∫ 0 2 π | P ′ (e i θ) | p δ d θ 1 p

(22)

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

n δ ∫ 0 2 π | P (e i θ) + λ m | δ d θ ≤ ∫ 0 2 π | 1 + 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 1 q ∫ 0 2 π | P ′ (e i θ) | p δ d θ 1 p

(23)

This is equivalent to

n ∫ 0 2 π | P (e i θ) + λ m | δ d θ 1 δ ≤ ∫ 0 2 π | 1 + 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 1 q δ ∫ 0 2 π | P ′ (e i θ) | p δ d θ 1 p δ

(24)

which proves the desired result.

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Acknowledgements

The authors are grateful to the referee for useful comments.

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Bharathiar University, Coimbatore, TN, 641046, India
Gulshan Singh
Department of Mathematics, Kashmir University Srinagar, 190006, India
WM Shah

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GS studied the related literature under the supervision of WMS and jointly developed the idea and drafted the manuscript. GS made the text _le and communicated the manuscript. GS also revised it as per the directions of the referee under the guidance of WMS. Both the authors read and approved the final manuscript.

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Singh, G., Shah, W. Integral mean estimates for polynomials whose zeros are within a circle. J Inequal Appl 2011, 35 (2011). https://doi.org/10.1186/1029-242X-2011-35

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Received: 26 December 2010
Accepted: 19 August 2011
Published: 19 August 2011
DOI: https://doi.org/10.1186/1029-242X-2011-35

Keywords

Derivative of a polynomial
Integral mean estimates
Inequalities in complex domain