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Integral mean estimates for polynomials whose zeros are within a circle

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 ≤ μn, 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. IfP ( z ) := a n z n + j = μ n a n - j z n - j , 1 ≤ μn 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 ≤ μn, having all the zeros in |z| ≤ K ≤ 1, and thereby obtain a more general result by proving the following:

Theorem 1. IfP ( z ) := a n z n + j = μ n a n - j z n - j , 1 ≤ μn 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 =1and 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. IfP ( z ) := a n z n + j = μ n a n - j z n - j , 1 ≤ μn 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. IfP ( 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 |KK1, 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. IfP ( z ) := a n z n + j = μ n a n - j z n - j , 1 ≤ μn 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. IfP ( z ) := a n z n + j = μ n a n - j z n - j , 1 ≤ μn 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 ,1μn 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 ̄ ) ¯ and 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 ) |for|z|=1,1μn,

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 ) |=|nQ ( z ) -z Q ( z ) |for|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 || < |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 ) |,1μn.

This gives

| Q ( z ) +nm 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 ) |,1μn.
(15)

Using (14) in (15), we get

| Q ( z ) +nm 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 | |nQ ( z ) -z Q ( z ) |for|z|=1,1μn.
(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 ) ||nQ ( 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 ) ||nQ ( 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 ) +λm|=|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 ) |for|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 =1and 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.

References

  1. 1.

    Turan P: Über die Ableitung von Polynomen. Composito Math 1939, 7: 89–95.

    MathSciNet  Google Scholar 

  2. 2.

    Malik MA: On the derivative of a polynomial. J Lond Math Soc 1969,1(2):57–60.

    Article  Google Scholar 

  3. 3.

    Malik MA: An integral mean estimate for polynomials. Proc Am Math Soc 1984,91(2):281–284. 10.1090/S0002-9939-1984-0740186-3

    Article  Google Scholar 

  4. 4.

    Aziz A, Shah WM: An integral mean estimate for polynomials. Indian J Pure Appl Math 1997,28(10):1413–1419.

    MathSciNet  Google Scholar 

  5. 5.

    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.

    MathSciNet  Google Scholar 

  6. 6.

    Aziz A, Dawood QM: Inequalities for a polynomial and its derivative. J Approx Theory 1988, 54: 306–313. 10.1016/0021-9045(88)90006-8

    MathSciNet  Article  Google Scholar 

  7. 7.

    Qazi MA: On the maximum modulus of polynomials. Proc Am Math Soc 1990, 115: 337–343.

    MathSciNet  Article  Google Scholar 

  8. 8.

    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.

    MathSciNet  Article  Google Scholar 

  9. 9.

    Rahman QI, Schmeisser G: Analytic Theory of Polynomials. Oxford University Press, New York; 2002.

    Google Scholar 

  10. 10.

    Milovanovic GV, Mitrinovic DS, Rassias ThM: Topics in Polynomials, Extremal Problems, Inequalities, Zeros. World Scientific, Singapore; 1994.

    Google Scholar 

  11. 11.

    Hille E: Analytic Function Theory, Vol. II, Introduction to Higher Mathematics. Ginn and Company, New York, Toronto; 1962.

    Google Scholar 

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The authors are grateful to the referee for useful comments.

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Correspondence to Gulshan Singh.

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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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Keywords

  • Derivative of a polynomial
  • Integral mean estimates
  • Inequalities in complex domain