Abstract

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

1. Introduction and Statement of Results

Let be a polynomial of degree n such that

then according to a classical result usually known as Eneström-Kakeya theorem [11], does not vanish in. Applying this result to the polynomial, the following more general result is immediate.

Theorem A. If is a polynomial of degree n such that for some

thenhas all the zeros in.

In the literature, [1-15], 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:

Theorem B. If is a polynomial of degree n such that for some and,

then 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.

Theorem C. Let be a polynomial of degree n with complex coefficients such that for some

and

then has all its zeros in the ring-shaped region given by

Here

where

and

By using Schwarz’s Lemma, Aziz and Mohammad [1] generalized Eneström-Kakeya theorem in a different way and proved:

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

where then all the zeros of 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. Infact we prove Theorem 1. Let be a polynomial of degree n such that

where and, are real numbers and for certain non negative real numbers with and

then all the zeros of lie in

Here

where

Assuming that all the coefficients are real, the following result is immediate:

Corollary 1. Let be a polynomial of degree n with real coefficients such that for certain non negative real numbers, with and

then all the zeros of lie in

Here

where

If in Corollary 1, we assume that all the coefficients are positive and then we have the following:

Corollary 2. Let be a polynomial of degree n such that for some real number

then all the zeros of lie in

In particular, if, Corollary 2 gives the following improvement of Eneström-Kakeya theorem.

Corollary 3. Let be a polynomial of degree n such that

then all the zeros of lie in

We next prove the following more general result which include many known results as special cases.

Theorem 2. Let be a polynomial of degree n such that where and, are real numbers. If can be found such that for a certain integer,

then all the zeros of lie in

(1)

where

Remark 1. Theorem B is a special case of Theorem 2, if we take and assume that all the coefficients, are real.

The following result follows immediately from Theorem 2 by taking and assuming, to be a real.

Corollary 4. Let be a polynomial of degree n with real coefficients. If can be found such that

then all the zeros of lie in

Remark 2. For and, Corollary 4 reduces to a result of Joyal, Labelle and Rahman [9].

We also prove the following result which is of independent interest.

Theorem 3. Let be a polynomial of degree n such that where and,

are real numbers. If can be found such that for a certain integer,

and

then all the zeros of lie in

(2)

where

.

Remark 3. Theorem 4 of [4] immediately follows from Theorem 3 when, and the coefficients, are real.

On combining Theorem 2 and Theorem 3 the following more interesting result is immediate.

Corollary 5. Let be a polynomial of degree n such that where and, are real. If can be found such that for a certain integer,

         

then all the zeros of lie in the intersection of the two circles given by (1) and (2).

If we takeand the coefficients are real in Theorem 3, we get the following result.

Corollary 6. Let be a polynomial of degree n with real coefficients. If can be found such that

then all the zeros of lie in

The following result also follows from Theorem 3, when, the coefficients, are real and.

Corollary 7. Let be a polynomial of degree n with real coefficients. If for some,

then has all the zeros in

2. Lemmas

For proving the above theorems, we require the following lemmas. The first Lemma which we need is due to Rahman and Schmeisser [11].

Lemma 1. If is analytic in, , where, , on, then for,

From Lemma 1, one can easily deduce the following :

Lemma 2. If is analytic in, , and for, then

The next Lemma is due to Aziz and Mohammad [2].

Lemma 3. Let, be a polynomial of degree n with complex coefficients.

Then for every positive real number r, all the zeros of lie in the disk

(3)

3. Proofs of the Theorems

Proof of Theorem 1. Consider the polynomial

    (4)

Further, let

(5)

where

Now

This gives after using hypothesis, for

Clearly, and

for

Thus, it follows by Lemma 2 that

From (5), we get

if

This gives if

Consequently, all the zeros of lie in

Since, it follows that all the zeros of and hence all the zeros of lie in

(6)

Again from (4)

(7)

where

Therefore, for, we have by using the hypothesis

Therefore, it follows again by Lemma 2 that

Using this result in (7), we get

if

Thus if

This shows that all the zeros of and hence of the polynomial lie in

(8)

Combining (6) and (8), we get the desired result.

Proof of Theorem 2. Consider the polynomial

Since is a polynomial of degree n + 2, it follows by applying Lemma 3 to with and, that all the zeros of lie in

(9)

Now

Using the hypothesis, we get

Hence by (9) all the zeros of lie in the circlewhere

Since every zero of is also a zero of, the theorem is proved completely.

This gives

Let , we get by using the hypothesis

if

Thus if

T his shows that those zeros of whose modulus is greater than, lie in the circle

It can be easily verified that those zeros of whose modulus is less than, lie in the circle as well. Therefore, we conclude that all zeros of and hence lie in

This completes the proof of the theorem.

4. References