Inequalities for the Sth Derivative of Polynomials Not Vanishing inside A C...

GULSHAN SINGH

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Inequalities for the Sth Derivative of Polynomials Not Vanishing inside A Circle

GULSHAN SINGH

Govt. Department of Education Jammu and Kashmir, India

Abstract

Let P(z) be a polynomial of degree n having all its zeros in , then for , Bidkham and Dewan [J. Math. Anal. Appl. 166(1992), 191-193] proved max In this paper, we prove an interesting generalization as well as an improvement of this result by considering the sth derivative of lacunary type of polynomials P(z) of degree n > 3.

Cite this article:

  • SINGH, GULSHAN. "Inequalities for the Sth Derivative of Polynomials Not Vanishing inside A Circle." American Journal of Mathematical Analysis 3.1 (2015): 1-4.
  • SINGH, G. (2015). Inequalities for the Sth Derivative of Polynomials Not Vanishing inside A Circle. American Journal of Mathematical Analysis, 3(1), 1-4.
  • SINGH, GULSHAN. "Inequalities for the Sth Derivative of Polynomials Not Vanishing inside A Circle." American Journal of Mathematical Analysis 3, no. 1 (2015): 1-4.

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1. Introduction and Statement of Results

Let be a polynomial of degree n and P’(z) its derivative, then it is known that

(1)

The above result, which is an immediate consequence of Bernstein's inequality on the derivative of a trigonometric polynomial is best possible with equality holding for the polynomial , where is a complex number.

If we restrict ourselves to the class of polynomials having all their zeros in , inequality (1) can be sharpened. In fact, Erdös conjectured and later Lax [6] proved that if P(z)≠ 0 in , then

(2)

On the other hand, if the polynomial P(z) of degree n has all its zeros in , then it was proved by Turán [10], that max

(3)

Inequalities (2) and (3) are best possible and become equality for polynomials which have all zeros on .

Inequality (2) was refined by Aziz and Dawood [1] by showing that under the same hypothesis that

(4)

Equality in (4) holds for .

For the class of polynomials P(z) of degree n having all their zeros in , Malik [7] proved:

(5)

Inequality (5) was further improved by Govil [5] who under the same hypothesis proved:

(6)

Chan and Malik [3] obtained a generalization of (5) by considering the lacunary type of polynomials and obtained the following:

Theorem A: Let , be a polynomial of degree n having all its zeros in , then

(7)

The result is best possible and extremal polynomial is ; where n is a multiple of .

The next result was proved by Pukhta [8], who infact proved:

Theorem B: Let , be a polynomial of degree n having all its zeros in , then

(8)

The result is best possible and extremal polynomial is ; where n is a multiple of .

Bidkham and Dewan [2] obtained a generalization of (5) by proving the following result:

Theorem C: Let P(z) be a polynomial of degree n having all its zeros in , then for

(9)

The result is best possible and equality holds for P(z) = (z + k)n.

In this paper, we prove the following generalization as well as an improvement of Theorem C by considering the sth derivative of P(z).

Theorem 1: If , n>3, is a polynomial of degree n having all its zeros in , then for , and

2. Lemmas

For the proof of above theorem, we need the following lemmas. The first result is due to Qazi [9, Lemma 1].

Lemma 1: If , is a polynomial of degree n having all its zeros in , then

The next lemma is due to Dewan, Kour and Mir [4].

Lemma 2: Let be a polynomial of degree n, then for R 1;

(10)

and

(11)

Lemma 3: If , n > 3, is a polynomial of degree n having all its zeros in , then for , we have

Proof of Lemma 3: Since P(z) is a polynomial of degree n > 3, the polynomial P′(z) is of degree n 3, hence on applying inequality (10) of Lemma 2 to the polynomial P′(z), we obtain

This proves Lemma 3.

Lemma 4: If , n > 3, is a polynomial of degree n having all its zeros in , then for R 1,

Proof of Lemma 4: For each and for we have

Hence

(12)

which when combined with Lemma 3, gives

which gives

Hence the proof.

Lemma 5: If P(z) is a polynomial of degree n having all its zeros in , then for .

This Lemma is due to Govil [5].

Proof of Theorem 1: Since P(z) has all its zeros in and if , then G(z) = P(Rz) has all its zeros in , therefore by applying Lemma 5 to G(z), we obtain

which implies

which is equivalent to

(13)

Inequality (13) in conjunction with Lemma 4 yields

The proof of Theorem 1 is completed.

References

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[2]  M. Bidkham and K. K. Dewan, Inequalities for a polynomial and its derivative, J. Math. Anal. Appl., Vol. 166 (1992), 319-324.
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[3]  T. N. Chan and M. A. Malik, On Erdos-Lax Theorem, Proc. Indian Acad. Sci. (Math. Sci.), 92 (3) (1983), 191-193.
In article      
 
[4]  K. K. Dewan, Jagjeet Kaur and Abdullah Mir, Inequalities for the derivative of a polynomial, J. Math. Anal. Appl., 269 (2002), 489-499.
In article      CrossRef
 
[5]  N. K. Govil, Some inequalities for derivatives of polynomials, J. Approx. Theory, 66 (1) (1991), 29-35.
In article      
 
[6]  P. D. Lax, Proof of a conjecture of P. Erdos on the derivative of a polynomial, Amer. Math. Soc., Bulletin, 50 (1944), 509-513.
In article      
 
[7]  M. A. Malik, On the derivative of a polynomial, J. London Math. Soc., 2 (1) (1969), 57-60.
In article      
 
[8]  M. S. Pukhta Extremal Problems for Polynomials and on Location of Zeros of Polynomials, Ph. D Thesis, Jamia Millia Islamia, New Delhi (1995).
In article      
 
[9]  M. A. Qazi, On the maximum modulus of polynomials, Proc. Amer. Math. Soc., 115(1992), 337-343.
In article      CrossRef
 
[10]  P. Turán, Uber die Ableitung von Polynomen, Compositio Mathematica, 7 (1939), 89-95 (German).
In article      
 
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