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 
then
has 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 take
and 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 circle
where
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.