1. Introduction
The classical Eneström–Kakeya Theorem concerns the location of the complex zeros of a real polynomial with nonnegative monotone coefficients. It was independently proved by Gustav Eneström in 1893 [
1] and Sōichi Kakeya in 1912 [
2].
Theorem 1 (Eneström–Kakeya Theorem). If is a polynomial of degree n (where z is a complex variable) with real coefficients satisfying , then all the zeros of P lie in .
A huge number of generalizations of the Eneström–Kakeya Theorem exist. Most of these involve weakening the condition on the coefficients. For a survey of such results up through 2014, see [
3]. For example, Govil and Rahman ([
4], Theorem 2) proved the following in 1968.
Theorem 2. If is a polynomial of degree n with complex coefficients such that for for some real β and , then all the zeros of P lie in With , Theorem 2 reduces to the Eneström–Kakeya Theorem.
As a corollary to a more general result, Gardner and Govil ([
5], Corollary 1) presented the following result concerning a monotonicity condition on the real and imaginary parts of the coefficients.
Theorem 3. Let be a polynomial of degree n with complex coefficients, where and for . Suppose that and . Then, all the zeros of P lie in With
for
, Theorem 3 implies a result of Joyal, Labelle, and Rahman ([
6], Theorem 3).
Aziz and Zargar [
7] gave the following result in 2012, which involves a slight generalization of the Eneström–Kakeya monotonicity condition on the real coefficients of a polynomial.
Theorem 4. Let be a polynomial of degree n with real coefficients such that, for some positive numbers k and ρ with and , the coefficients satisfy Then, all the zeros of P lie in the closed disk .
With , Theorem 4 implies the Eneström–Kakeya Theorem.
Recently, Shah et al. [
8] proved the next result, which maintains the monotonicity condition on the “central” coefficients, but imposes no condition on the “tail” coefficients.
Theorem 5. Let be a polynomial of degree n with real coefficients such that, for some positive numbers p and q with , the coefficients satisfy Then, all the zeros of P lie in the closed annulus:where and . With
and
, Theorem 5 implies a result of Joyal, Labelle, and Rahman [
6]. Additionally with
, it then implies the Eneström–Kakeya Theorem.
The purpose of this paper is to combine the hypotheses of Theorems 4 and 5 and apply them to polynomials with complex coefficients. We apply the hypotheses to the real and imaginary parts of the coefficients and to the moduli of the coefficients.
2. Results
By introducing the parameters k and of Aziz and Zargar, along with the parameters p and q of Shah et al. and imposing the hypothesis that results on the real and imaginary parts of the coefficients of a polynomial, we obtain the following.
Theorem 6. Let be a polynomial of degree n with complex coefficients. Let and for . Suppose that, for some positive numbers , , , , p, and q with , , , , and , the coefficients satisfyandThen, all the zeros of P lie in the closed annulus:where and . Notice that, when
for
and
, then Theorem 6 reduces to Theorem 5. When
for
,
,
, and
, then Theorem 6 reduces to a result of Joyal, Labelle, and Rahman [
6]. If, in addition,
, then it further reduces to the Eneström–Kakeya Theorem. With
,
, and
, then Theorem 6 reduces to Theorem 3.
By imposing a hypothesis similar to that of Theorem 6 on the moduli of the coefficients of a polynomial, we obtain the following.
Theorem 7. Let be a polynomial of degree n with complex coefficients. Suppose that, for some positive numbers k, ρ, p, and q with , , and , the coefficients satisfy for some real β and for , andThen, all the zeros of P lie in the closed annulus:where and . With
,
and
, Theorem 7 implies that
P is nonzero in
which is a slight improvement of Theorem 2. Additionally, with
, it then reduces to the Eneström–Kakeya Theorem.
In connection with the Bernstein inequalities, Chan and Malik [
9] (and, independently, Qazi [
10]) considered the class of polynomials of the form
. We can apply Theorems 6 and 7 to this class of polynomials by imposing the inequality hypotheses of those results. We obtain the following.
Corollary 1. Let be a polynomial of degree n with complex coefficients. Let and for and . Suppose that, for some positive numbers , , , , p, and q with , , , , and , the coefficients satisfyandThen, all the zeros of P lie in the closed annulus given in Theorem 6, where we replace of Theorem 6 with . Corollary 2. Let be a polynomial of degree n with complex coefficients that satisfy for and for some real β. Suppose that, for some positive numbers k, ρ, p, and q with , , and , the coefficients satisfyThen, all the zeros of P lie in the closed annulus given in Theorem 7, where we replace of Theorem 7 with . Theorems 6 and 7 also naturally apply to a polynomial of the form , where . With , we obtain corollaries similar to Corollaries 1 and 2, where is as given in the above corollaries, and .
3. A Lemma
In proving Theorem 2, Govil and Rahman used the following ([
4], Equation (6)).
Lemma 1. Let be a set of complex numbers that satisfy for and for some real β. Suppose . Then, for , we have 4. Proof of the Results
We now give a proof of Theorem 6.
Proof. Let
be a polynomial of degree
n satisfying the stated hypotheses. Define
f by the equation:
If
, then
Let
and
for
. Thus, for
,
Let
and
. Hence, for
,
Since
and
, where
, then for
,
Notice that
, where
has the same bound on
as
. Since
is analytic in
, we have
for
, by the Maximum Modulus Theorem. Thus,
for
. Replacing
z with
, we have
for
. We now have for
Therefore if
then
and, hence,
. Therefore, all the zeros of
P lie in
as claimed.
Consider the polynomial:
Let
With
and
for
, we have
Now, with
, so that
for
, we have
if
Thus, all the zeros of
whose modulus is greater than 1 lie in
Hence, all the zeros of
and of
lie in
Therefore, all the zeros of
lie in
as claimed. □
We now give a proof of Theorem 7.
Proof. Let
be a polynomial of degree
n with complex coefficients such that
for
and for some real
and
for
and
. Notice that we can assume without loss of generality that
. Consider
If
, then
where
and
. Thus, for
,
Hence, also,
for
. By the Maximum Modulus Theorem,
holds inside the unit circle
as well. If
, then
lies inside the unit circle for every real
. Thus, it follows that
for every
and
real. Hence, for every
,
if
Therefore all the zeros of
lie in
as claimed.
Consider the polynomial:
Let
This gives
Since
and
, then
Now, with
, so that
for
, we have
if
Thus, all the zeros of
whose modulus is greater than 1 lie in
Hence, all the zeros of
and of
lie in
Therefore, all the zeros of
lie in
as claimed. □
5. Discussion
As explained in the Introduction, the hypotheses applied in this paper are consistent with several existing results of the Eneström–Kakeya type. In fact, the two main theorems proven in this paper are generalizations of some of these results. Future research could involve loosening the restrictions imposed on the coefficients of a polynomial given in Theorems 6 and 7. For example, a reversal in the monotonicity could be introduced, or additional parameters related to the monotonicity of the coefficients could be added to give generalizations of these theorems. In addition, the analytic theory of functions of a quaternionic variable could be applied to quaternionic polynomials (with the same or similar conditions imposed on the coefficients) to restrict the location of the zeros of such polynomials.