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On Location of Zeros of Polynomials

M S PUKHTA

Division of Agri. Statistics, Sher-e-Kashmir, University of Agricultural Sciences and Technology of Kashmir, India

Abstract: If be a polynomial of degree n such that aj=aj+ibj where aj and bj, j = 0, 1, ….., n are real numbers. In this paper we obtain a generalization of well known result of Eneström -Kakeya concerning the bounds for the moduli of the zeros of polynomials with complex coefficients which improve upon some results due to A. Aziz and Q.G Mohammad and others.

Keywords: complex polynomials, zeros, Eneström – Kakeya Theorem

Received April 30, 2014; Revised May 19, 2014; Accepted May 25, 2014

Citation: M S PUKHTA, On Location of Zeros of Polynomials. American Journal of Mathematical Analysis, 2 (2). 23-27. January 2014. doi:10.12691/ajma-2-2-2

1. Introduction and Statement of Results

The following result known as Eneström-Kakeya theorem, is well known in the theory of distribution of zeros of polynomials was firstly proved by Eneström [2] and Kakeya [5] and later independently by Hurwitz [3].

Theorem 1.1. If is a polynomial of degree n such that

then all the zeros of lie in .

Joyal, Labelle and Rahman [4] extended Theorem A to the polynomial whose coefficients are monotonic but not necessarily non-negative by proving the following:

Theorem 1.2. Let be a polynomial of degree n such that

then all the zeros of lie in

Aziz and Zargar [6] relaxed the hypothesis of Eneström-Kakeya theorem and proved the following extension of Theorem 1.2.

Theorem 1.3. Letbe a polynomial of degree n such that for some,

then all the zeros of lie in

By using Schwartz lemma, Aziz and Mohammad [1] generalized Eneström -Kakeya theorem in a different way and proved:

Theorem 1.4. If be a polynomial of degree n with real positive coefficients. If can be found such that

then all the zeros of lie in .

In this paper, we prove some generalizations and extensions of Theorem1.1, Theorem 1.2, Theorem 1.3 and Theorem 1.4. In this direction we first present the following interesting result which is generalization of Theorem 1.4.

Theorem 1.5. If be a polynomial of degree n such that where and , j = 0, 1, ….., n are real numbers and for certain non-negative real numbers with and

then all the zeros of lie in

where

Remark 1.1. If in Theorem 1.5, we assume that all the coefficients are real and positive, then M=0 and it reduces to Theorem 1.4 due to Aziz and Mohammad [1].

In particular, if we choose t₂=0 and in Theorem 1.5, then we have the following result.

Corollary 1.1. If be a polynomial of degree n such that where and , j = 0, 1, 2 ….., n are real numbers. If be such that

then all the zeros of lie in

where

Remark 1.2. If we consider all the coefficients in Corollary 1.1 to be real and choose t = 1, then we get Theorem 1.2due to Joyal, Labelle and Rahman [4]. In addition, if we choose all coefficients to be positive, then we obtain the well-known Eneström-Kakeya Theorem (Theorem 1.1).

Next we prove the following result which is also a generalization of Theorem

Theorem.1.6. Let be a polynomial of degree n such that where and , j = 0, 1, ….., n are real numbers and for certain non-negative real numbers with and

and for some

and

then all the zeros of lie in

where

Remark 1.3. If we choose and in Theorem 1.6, we get the following result.

Corollary 1.2. Let be a polynomial of degree n such that where and , j = 0, 1, 2,….., n are real numbers and for certain non-negative real number t

and for some ,

Then all the zeros of lie in

where

Remark 1.4. If we consider all the coefficients in Corollary 1.2 to be real and positive, then we obtain Theorem 1.3 due to Aziz and Zargar [1].

Remark 1.5. Again, if we choose all the coefficients in Corollary 1 to be real and consider k=1 and t=1, we get Theorem B due to Joyal, Labelle and Rahman [5].

2. Proof of the Theorem

Proof of Theorem1.5. Consider the polynomial

Further, let

[picture] | (3.1)

where

Now

This gives after using hypothesis, for |z| = 1/t₁

Clearly

Therefore, by Schwarz’s Lemma

Hence from (3.1), if , then we get

If .

That is

Thus in .

Consequently, all the zeros of G(z) lie in.

As we conclude that all the zeros of F(z) and hence all the zeros of p(z) lie in

This completes the proof of the theorem 1.5.

Proof of Theorem 1.6. Consider the polynomial

This gives

Therefore for , we have

if

Hence all the zeros of whose modulus is greater than lie in the circle

Since all the zeros whose modulus is less than already lies in this circle, we conclude that all the zeros of and hence lie in

This completes the proof of the Theorem1.6.

References

[1]   | A. Aziz and Q. G. Mohammad, Zero free regions for polynomials and some generalizations of Enström-Kakeya theorem, Cand. Math. Bull., 27(1984), 265-272.
[2]   | G. Eneström, Remarqueesur un The’ore’me’relatif aux racines de I’ equation anxn + an-1xn-1 + ……+ a₀ = 0 outous les coefficients sont reels et possitifs, Tôhoku Math. J., 18 (1920), 34-36.
[3]   | A. Hurwitz, Űberdie Nullslenllen der Bessel Schen Function, Math. Ann., 33(1989)246-266.
[4]   | A.Joyal, G.Labelle and Q.I. Rahman, On the location of zeros of polynomials, Canad. Math. Bull., 10 (1967), 53-63.
[5]   | S.Kakeya, On the limits of the roots of an algebraic equation with positive coefficients, Tôhoku Math. J., 2 (1912-13), 140-142.
[6]   | A. Aziz and B. A. Zargar, Some extensions of Enström-Kakeya Theorem, Glasnik Mate., 31(1996), 239-244.