Location Of The Zeros Of Polynomials

In this paper, we prove some extensions and generalizations of the classical Enestrom-Kakeya theorem.


Introduction and Statement of Results
Let   Eneström-Kakeya theorem [11],   P z does not vanish in > 1 z .Applying this result to the polynomial   P tz , the following more general result is immediate.
 is a polynomial of degree n such that for some > 0 t then   P z has all the zeros in z t  .In the literature, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], there exist extensions and generalizations of Eneström-Kakeya theorem.Joyal, Labelle and Rahman [9] extended this theorem to polynomials whose coefficients are monotonic but not necessarily non negative and the result was further generalized by Dewan and Bidkham [6] to read as:  is a polynomial of degree n such that for some > 0 t and 0 < n   , then   P z has all the zeros in the circle Govil and Rahman [8] extended Theorem A to the polynomials with complex coefficients.As a refinement of the result of Govil and Rahman, Govil and Jain [7] proved the following.then   P z has all its zeros in the ring-shaped region given by where   > 0 t t  can be found such that then all the zeros of   P z lie in In this paper, we also make use of a generalized form of Schwarz's Lemma and prove some more general results which include not only the above theorems as special cases, but also lead to a standard development of interesting generalizations of some well known results.
then all the zeros of   Assuming that all the coefficients , 0,1, , j j n    are real, the following result is immediate:  be a polynomial of degree n with real coefficients such that for certain non negative real numbers 1 t , 2 t with 1 2 t t  and then all the zeros of   where Remark 2. For 1 = 1 t and 2 = 0 t , Corollary 4 reduces to a result of Joyal, Labelle and Rahman [9].
We also prove the following result which is of independent interest.
where j a and j b , = 0,1,..., then all the zeros of   where are real.On combining Theorem 2 and Theorem 3 the following more interesting result is immediate.
where j a and j b , = 0,1,..., j n are real.If 1 2 > 0 t t  can be found such that for a certain integer  , then all the zeros of   P z lie in the intersection of the two circles given by (1) and (2).
If we take = 1 n   and the coefficients , are real in Theorem 3, we get the following result.
The following result also follows from Theorem

Lemmas
For proving the above theorems, we require the following lemmas.The first Lemma which we need is due to Rahman and Schmeisser [11].
From Lemma 1, one can easily deduce the following : The next Lemma is due to Aziz and Mohammad [2].
be a polynomial of degree n with complex coefficients.
Then for every positive real number r, all the zeros of

Proofs of the Theorems
Proof of Theorem 1.Consider the polynomial This gives after using hypothesis, for From (5), we get , it follows that all the zeros of   F z and hence all the zeros of   Again from ( 4) where Therefore, for 1 = z t , we have by using the hypothesis Therefore, it follows again by Lemma 2 that Using this result in (7), we get This shows that all the zeros of   F z and hence of the polynomial   Combining ( 6) and ( 8), we get the desired result.Proof of Theorem 2. Consider the polynomial follows by applying Lemma 3 to   f z with = 1 p n and 1 = r t , that all the zeros of   Hence by (9) all the zeros of   where Since every zero of   P z is also a zero of   f z , the theorem is proved completely.
z t , we get by using the hypothesis T his shows that those zeros of   f z whose modulus is greater than 1 t , lie in the circle It can be easily verified that those zeros of   f z whose modulus is less than 1 t , lie in the circle as well.
Therefore, we conclude that all zeros of   a classical result usually known as of degree n with complex coefficients such that for some 


By using Schwarz's Lemma, Aziz and Mohammad[1] generalized Eneström-Kakeya theorem in a different way and proved: of degree n with real positive coefficients.If 1 2