The Number of Zeros in a Disk of a Complex Polynomial with Coefficients Satisfying Various Monotonicity Conditions

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The Number of Zeros in a Disk of a Complex Polynomial with Coefficients Satisfying Various Monotonicity Conditions

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Submitted:

30 August 2023

Posted:

01 September 2023

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Abstract

Motivated by results on the location of zeros of a complex polynomial with monotonicity conditions on the coefficients (such as the classical Eneström-Kakeya Theorem, and its recent generalizations), we impose similar conditions and give bounds on the number of zeros in certain regions. We do so by introducing a reversal in monotonicity conditions on the real and imaginary parts of the coefficients, and also on their moduli. The results presented naturally apply to certain classes of lacunary polynomials.

Keywords:

Subject: Computer Science and Mathematics - Analysis

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 [4] and Sōichi Kakeya in 1912 [10].

Theorem 1.

Eneström-Kakeya Theorem. If

P (z) = \sum_{ℓ = 0}^{n} a_{ℓ} z^{ℓ}

is a polynomial of degree n (where z is a complex variable) with real coefficients satisfying

0 \leq a_{0} \leq a_{1} \leq \dots \leq a_{n}

, then all the zeros of P lie in

| z | \leq 1

A large body of literature on results related to the Eneström-Kakeya Theorem now exists. For a survey of results up through 2014, see [7]. Inspired by results of Aziz and Zargar [1] and Shah et al. [15], the present authors gave an Eneström-Kakeya type result [5] for polynomials

P (z) = \sum_{ℓ = 0}^{n} a_{ℓ} z^{ℓ}

such that

α_{ℓ} = R e (a_{ℓ})

and

β_{ℓ} = I m (a_{ℓ})

for

0 \leq ℓ \leq n

where, for some positive numbers

ρ_{r}

and

ρ_{i}

each at most 1, and

k_{r}

k_{i}

each at least 1, and p and q with

0 \leq q \leq p \leq n

, the coefficients satisfy

ρ_{r} α_{q} \leq α_{q + 1} \leq α_{q + 2} \leq \dots \leq α_{p - 1} \leq k_{r} α_{p}

and

ρ_{i} β_{q} \leq β_{q + 1} \leq β_{q + 2} \leq \dots \leq β_{p - 1} \leq k_{i} β_{p} .

The present authors recently generalized this result [6] by adding parameter j with

q < j < p

(which allows a reversal in the monotonicity condition) and using a total of six positive parameters

ρ_{r_{1}}

ρ_{r_{2}}

ρ_{i_{1}}

, and

ρ_{i_{2}}

each at most 1, and

k_{r}

k_{i}

each at least 1, to consider polynomials with complex coefficients satisfying

ρ_{r_{1}} α_{q} \leq α_{q + 1} \leq α_{q + 2} \leq \dots α_{j - 1} \leq k_{r} α_{j} \geq α_{j + 1} \geq \dots \geq α_{p - 1} \geq ρ_{r_{2}} α_{p}

(1)

and

ρ_{i_{1}} β_{q} \leq β_{q + 1} \leq β_{q + 2} \leq \dots β_{j - 1} \leq k_{i} β_{j} \leq β_{j + 1} \geq \dots \geq β_{p - 1} \geq ρ_{i_{2}} β_{p} .

(2)

Notice that with

ρ_{r_{1}} = k_{r} = ρ_{r_{2}} = 1

q = 0

j = p = n

0 \leq a_{0}

and each

β_{ℓ} = 0

, the above condition implies the hypotheses of the Eneström-Kakeya Theorem.

The first result concerning the number of zeros in a disk relevant to the current work, is due to Mohammad in 1965. It considers polynomials with real coefficients which satisfy the monotonicity condition of the Eneström-Kakeya Theorem (with the added condition that the constant term is nonzero) and is as follows [11].

Theorem 2.

Let

P (z) = \sum_{ℓ = 0}^{n} a_{ℓ} z^{ℓ}

be a polynomial of degree n with real coefficients such that

0 < a_{0} \leq a_{1} \leq \dots \leq a_{n}

. Then the number of zeros of

P (z)

in the disk

| z | \leq 1 / 2

does not exceed

1 + (1 / log 2) log (a_{n} / a_{0})

Another relevant result is due to Dewan [3] and concerns a monotonicity condition on the moduli of coefficients, as follows.

Theorem 3.

Let

P (z) = \sum_{ℓ = 0}^{n} a_{ℓ} z^{ℓ}

be a polynomial of degree n with complex coefficients such that for some real β,

| arg a_{ℓ} - β | \leq α \leq π / 2

for

ℓ = 0, 1, \dots, n

and

0 < | a_{0} | \leq | a_{1} | \leq \dots \leq | a_{n} |

. Then the number of zeros of

P (z)

| z | \leq 1 / 2

does not exceed

\frac{1}{log 2} log (\frac{| a_{n} | (1 + cos α + sin α) + 2 sin α \sum_{ℓ = 0}^{n - 1} | a_{ℓ} |}{| a_{0} |}) .

Though both Theorems 2 and 3 concern zeros in

| z | \leq 1 / 2

, more general results exist. For example, Pukhta [12] gave the following generalization of Theorem 3 which reduces to Theorem 3 when

δ = 1 / 2

Theorem 4.

Let

P (z) = \sum_{ℓ = 0}^{n} a_{ℓ} z^{ℓ}

be a polynomial of degree n with complex coefficients such that for some real β,

| arg a_{ℓ} - β | \leq α \leq π / 2

for

ℓ = 0, 1, \dots, n

and

0 < | a_{0} | \leq | a_{1} | \leq \dots \leq | a_{n} |

. Then, for

0 < δ < 1

, the number of zeros of

P (z)

| z | \leq δ

does not exceed

\frac{1}{log 1 / δ} log (\frac{| a_{n} | (1 + cos α + sin α) + 2 sin α \sum_{ℓ = 0}^{n - 1} | a_{ℓ} |}{| a_{0} |}) .

Recently, the number of zeros in a disk of a polynomial with coefficients satisfying a monotonicity condition, but with extra multiplicative terms on some of the coefficients, have been presented. Rather et al. [14], for example, considered polynomials with real coefficients satisfying

a_{0} \leq a_{1} \leq \dots \leq a_{n - r - 1} \leq k_{r} a_{n - r} \leq k_{r - 1} a_{n - r + 1} \leq \dots \leq k_{1} a_{n - 1} \leq k_{0} a_{n}

where

k_{ℓ} \geq 1

for

ℓ = 0, 1, \dots, r

and

0 \leq r \leq n - 1

. Rather et al. [13] (in a publication different from the previously cited one) similarly considered a monotonicity condition, but with extra additive terms on some of the coefficients. For example, they considered polynomials with real coefficients satisfying

a_{0} \leq a_{1} \leq \dots \leq a_{n - r - 1} \leq k_{r} + a_{n - r} \leq k_{r - 1} + a_{n - r + 1} \leq \dots \leq k_{1} + a_{n - 1} \leq k_{0} + a_{n}

where

k_{ℓ} \geq 0

for

ℓ = 0, 1, \dots, r

and

0 \leq r \leq n - 1

. These results of Rather et al. generalize and refine the earlier results.

The purpose of this paper is to consider complex polynomials satisfying conditions (1) and (2) (and a related condition on the moduli of the coefficients) and to give results concerning the number of zeros in a disk.

2. Results

For a polynomial of degree n with complex coefficients

a_{ℓ}

0 \leq ℓ \leq n

, where

α_{ℓ} = Re (a_{ℓ})

and

β_{ℓ} = Im (a_{ℓ})

, we impose the conditions of equations (1) and (2) to get the following.

Theorem 5.

Let

P (z) = \sum_{ℓ = 0}^{n} a_{ℓ} z^{ℓ}

be a complex polynomial of degree n with complex coefficients where

α_{ℓ} = Re (a_{ℓ})

and

β_{ℓ} = Im (a_{ℓ})

which satisfies, for some real

ρ_{r_{1}}, ρ_{r_{2}}, ρ_{i_{1}}, ρ_{i_{2}}, k_{r},

and

k_{i}

where

0 < ρ_{r_{1}} \leq 1

0 < ρ_{r_{2}} \leq 1

0 < ρ_{i_{1}} \leq 1

0 < ρ_{i_{2}} \leq 1

k_{r} \geq 1

, and

k_{i} \geq 1

, the condition

ρ_{r_{1}} α_{q} \leq α_{q + 1} \leq \dots \leq α_{j - 1} \leq k_{r} α_{j} \geq α_{j + 1} \geq \dots \geq α_{p - 1} \geq ρ_{r_{2}} α_{p}

ρ_{i_{1}} β_{q} \leq β_{q + 1} \leq \dots \leq β_{j - 1} \leq k_{i} β_{j} \geq β_{j + 1} \geq \dots \geq β_{p - 1} \geq ρ_{i_{2}} β_{p} .

Then the number of zeros of

P (z)

in the disk

| z | \leq δ

is less than

(1 / log (1 / δ)) log (M / | a_{0} |)

for

0 < δ < 1

, where

\begin{matrix} M & = & | a_{0} | + M_{q} - ρ_{r_{1}} α_{q} + | α_{q} | (1 - ρ_{r_{1}}) + 2 | α_{j} | (k_{r} - 1) + 2 k_{r} α_{j} + | α_{p} | (1 - ρ_{r_{2}}) \\ - ρ_{r_{2}} α_{p} - ρ_{i_{1}} β_{q} + | β_{q} | (1 - ρ_{i_{1}}) + 2 | β_{j} | (k_{i} - 1) + 2 k_{i} β_{j} \\ + | β_{p} | (1 - ρ_{i_{2}}) - ρ_{i_{2}} β_{p} + M_{p} + | a_{n} |, \end{matrix}

M_{q} = \sum_{ℓ = 1}^{q} | a_{ℓ} - a_{ℓ - 1} |

, and

M_{p} = \sum_{ℓ = p + 1}^{n} | a_{ℓ} - a_{ℓ - 1} |

Now we consider a condition similar to that given in equations (1) and (2), but imposed on the moduli of the complex coefficients instead of on the real and imaginary parts.

Theorem 6.

Let

P (z) = \sum_{ℓ = 0}^{n} a_{ℓ} z^{ℓ}

be a polynomial of degree n with complex coefficients satisfying

| arg a_{ℓ} - β | \leq α \leq π / 2

ℓ = q, q + 1, \dots, p

, such that for real

k, ρ_{1}, ρ_{2}

, where

k \geq 1

0 < ρ_{1} \leq 1

0 < ρ_{2} \leq 1

, we have

ρ_{1} | a_{q} | \leq | a_{q + 1} | \leq \dots \leq | a_{j - 1} | \leq k | a_{j} | \geq | a_{j + 1} | \geq \dots \geq | a_{p - 1} | \geq ρ_{2} | a_{p} | .

Then the number of zeros of

P (z)

in the disk

| z | \leq δ

is less than

(1 / log (1 / δ)) log (M / | a_{0} |)

for

0 < δ < 1

, where

\begin{matrix} M & = & | a_{0} | + M_{q} + | a_{q} | + ρ_{1} | a_{q} | (sin α - cos α - 1) + 2 \sum_{ℓ = q + 1}^{j - 1} | a_{ℓ} | sin α - 2 | a_{j} | \\ + 2 k | a_{j} | (cos α + sin α + 1) + 2 \sum_{ℓ = j + 1}^{p - 1} | a_{ℓ} | sin α + | a_{p} | \\ + ρ_{2} | a_{p} | (sin α - cos α - 1) + M_{p} + | a_{n} |, \end{matrix}

M_{q} = \sum_{ℓ = 1}^{q} | a_{ℓ} - a_{ℓ - 1} |

, and

M_{p} = \sum_{ℓ = p + 1}^{n} | a_{ℓ} - a_{ℓ - 1} |

The class of lacunary polynomials of the form

P (z) = a_{0} + \sum_{ℓ = m}^{n} a_{ℓ} z^{ℓ}

was introduced by Chan and Malik in 1983 [2] in connection with Bernstein’s Inequality [2]. For a survey of such results, see subsections 4.1.4, 6.4.2, and 6.4.3 of [8]. Theorems 5 and 6 naturally apply to such polynomials which satisfy the monotonicity condition on the remaining coefficients. For example, with coefficients

a_{1} = a_{2} = \dots = a_{q - 1} = 0

in polynomial P, we get the following corollary.

Corollary 1.

Let

P (z) = a_{0} + \sum_{ℓ = q}^{n} a_{ℓ} z^{ℓ}

be a complex polynomial of degree n with complex coefficients where

α_{ℓ} = Re (a_{ℓ})

and

β_{ℓ} = Im (a_{ℓ})

which satisfies, for some real

ρ_{r_{1}}, ρ_{r_{2}}, ρ_{i_{1}}, ρ_{i_{2}}, k_{r},

and

k_{i}

where

0 < ρ_{r_{1}} \leq 1

0 < ρ_{r_{2}} \leq 1

0 < ρ_{i_{1}} \leq 1

0 < ρ_{i_{2}} \leq 1

k_{r} \geq 1

, and

k_{i} \geq 1

, the condition

ρ_{r_{1}} α_{q} \leq α_{q + 1} \leq \dots \leq α_{j - 1} \leq k_{r} α_{j} \geq α_{j + 1} \geq \dots \geq α_{p - 1} \geq ρ_{r_{2}} α_{p}

ρ_{i_{1}} β_{q} \leq β_{q + 1} \leq \dots \leq β_{j - 1} \leq k_{i} β_{j} \geq β_{j + 1} \geq \dots \geq β_{p - 1} \geq ρ_{i_{2}} β_{p} .

Then the number of zeros of

P (z)

in the disk

| z | \leq δ

is less than

(1 / log (1 / δ)) log (M / | a_{0} |)

where M is as given in Theorem 5,

M_{q} = 0

, and

M_{p} = \sum_{ℓ = p + 1}^{n} | a_{ℓ} - a_{ℓ - 1} |

A similar corollary follows from Theorem 6. In addition, Theorems 5 and 6 naturally apply to lacunary polynomials with two gaps in their coefficients. For example, with coefficients

a_{1} = a_{2} = \dots = a_{q - 1} = 0

and

a_{p + 1} = a_{p + 2} = \dots = a_{n - 1} = 0

in polynomial P, we get the following corollary.

Corollary 2.

Let

P (z) = a_{0} + \sum_{ℓ = q}^{p} a_{ℓ} z^{ℓ} + a_{n}

be a complex polynomial of degree n with complex coefficients where

α_{ℓ} = Re (a_{ℓ})

and

β_{ℓ} = Im (a_{ℓ})

which satisfies, for some real

ρ_{r_{1}}, ρ_{r_{2}}, ρ_{i_{1}}, ρ_{i_{2}}, k_{r},

and

k_{i}

where

0 < ρ_{r_{1}} \leq 1

0 < ρ_{r_{2}} \leq 1

0 < ρ_{i_{1}} \leq 1

0 < ρ_{i_{2}} \leq 1

k_{r} \geq 1

, and

k_{i} \geq 1

, the condition

ρ_{r_{1}} α_{q} \leq α_{q + 1} \leq \dots \leq α_{j - 1} \leq k_{r} α_{j} \geq α_{j + 1} \geq \dots \geq α_{p - 1} \geq ρ_{r_{2}} α_{p}

ρ_{i_{1}} β_{q} \leq β_{q + 1} \leq \dots \leq β_{j - 1} \leq k_{i} β_{j} \geq β_{j + 1} \geq \dots \geq β_{p - 1} \geq ρ_{i_{2}} β_{p} .

Then the number of zeros of

P (z)

in the disk

| z | \leq δ

is less than

(1 / log (1 / δ)) log (M / | a_{0} |)

where M is as given in Theorem 5 and

M_{q} = M_{p} = 0

A similar corollary follows from Theorem 6.

The introduction of the reversal of the inequality at index j allows us to shift the point at which the reversal occurs. This flexibility allows us to apply Theorems 5 and 6 to a larger collection of polynomials than some of the other current results in the literature on this topic.

3. Lemmas

The number of zeros results we consider are all based on the following theorem, which appears in Titchmarsh’s The Theory of Functions [16, page 280].

Lemma 1.

Let

F (z)

be analytic in

| z | \leq R

. Let

| F (z) | \leq M

in the disk

| z | \leq R

and suppose

F (0) \neq 0

. Then for

0 < δ < 1

, the number of zeros of F in the disk

| z | \leq δ R

does not exceed

(1 / log (1 / δ)) log (M / | F (0) |)

The following lemma is due to Govil and Rahman [9].

Lemma 2.

Let

z, z^{'} \in C

with

| z | \geq | z^{'} |

. Suppose that

| arg z^{*} - β | \leq α \leq π / 2

for

z^{*} \in {z, z^{'}}

and for some real α and β. Then

| z - z^{'} | \leq (| z | - | z^{'} |) cos α + (| z | + | z^{'} |) sin α .

4. Proofs of the Results

Proof of Theorem 5.

Consider

F (z) = (1 - z) P (z) = z_{0} + \sum_{ℓ = 1}^{n} (a_{ℓ} - a_{ℓ - 1}) z^{ℓ} - a_{n} z^{n + 1} .

For

| z | = 1

we have

\begin{matrix} | F (z) | & \leq & | a_{0} | + \sum_{ℓ = 1}^{n} | a_{ℓ} - a_{ℓ - 1} {| | z |}^{ℓ} + | a_{n} {| | z |}^{n + 1} = | a_{0} | + \sum_{ℓ = 1}^{n} | a_{ℓ} - a_{ℓ - 1} | + | a_{n} | \\ = & | a_{0} | + \sum_{ℓ = 1}^{q} | a_{ℓ} - a_{ℓ - 1} | + \sum_{ℓ = q + 1}^{p} | α_{ℓ} + i β_{ℓ} - α_{ℓ - 1} - i β_{ℓ - 1} | \\ + \sum_{ℓ = p + 1}^{n} | a_{ℓ} - a_{ℓ - 1} | + | a_{n} | \\ \leq & | a_{0} | + M_{q} + \sum_{ℓ = q + 1}^{p} | α_{ℓ} - α_{ℓ - 1} | + \sum_{ℓ = q + 1}^{p} | β_{ℓ} - β_{ℓ - 1} | + M_{p} + | a_{n} |, \end{matrix}

where

M_{q} = \sum_{ℓ = 1}^{q} | a_{ℓ} - a_{ℓ - 1} |

and

M_{p} = \sum_{ℓ = p + 1}^{n} | a_{ℓ} - a_{ℓ - 1} |

. So for

| z | = 1

, we have

\begin{matrix} | F (z) | & \leq & | a_{0} | + M_{q} + | α_{q + 1} - ρ_{r_{1}} α_{q} + ρ_{r_{1}} α_{q} - α_{q} | + \sum_{ℓ = q + 2}^{j - 1} | α_{ℓ} - α_{ℓ - 1} | \\ + | α_{j} - k_{r} α_{j} + k_{r} α_{j} - α_{j - 1} | + | α_{j + 1} - k_{r} α_{j} + k_{r} α_{j} - α_{j} | + \sum_{ℓ = j + 2}^{p - 1} | α_{ℓ} - α_{ℓ - 1} | \\ + | α_{p} - ρ_{r_{2}} α_{p} + ρ_{r_{2}} α_{p} - α_{p - 1} | + | β_{q + 1} - ρ_{i_{1}} β_{q} + ρ_{i_{1}} β_{q} - β_{q} | \\ + \sum_{ℓ = q + 2}^{j - 1} | β_{ℓ} - β_{ℓ - 1} | + | β_{j} - k_{i} β_{j} + k_{i} β_{j} - β_{j - 1} | + | β_{j + 1} - k_{i} β_{j} + k_{i} β_{j} - β_{j} | \\ + \sum_{ℓ = j + 2}^{p - 1} | β_{ℓ} - β_{ℓ - 1} | + | β_{p} - ρ_{i_{2}} β_{p} + ρ_{i_{2}} β_{p} - β_{p - 1} | + M_{p} + | a_{n} | \\ \leq & | a_{0} | + M_{q} + | α_{q + 1} - ρ_{r_{1}} α_{q} | + | ρ_{r_{1}} α_{q} - α_{q} | + \sum_{ℓ = q + 2}^{j - 1} | α_{ℓ} - α_{ℓ - 1} | + | α_{j} - k_{r} α_{j} | \\ + | k_{r} α_{j} - α_{j - 1} | + | α_{j + 1} - k_{r} α_{j} | + | k_{r} α_{j} - α_{j} | + \sum_{ℓ = j + 2}^{p - 1} | α_{ℓ} - α_{ℓ - 1} | + | α_{p} - ρ_{r_{2}} α_{p} | \\ + | ρ_{r_{2}} α_{p} - α_{p - 1} | + | β_{q + 1} - ρ_{i_{1}} β_{q} | + | ρ_{i_{1}} β_{q} - β_{q} | + \sum_{ℓ = q + 2}^{j - 1} | β_{ℓ} - β_{ℓ - 1} | \\ + | β_{j} - k_{i} β_{j} | + | k_{i} β_{j} - β_{j - 1} | + | β_{j + 1} - k_{i} β_{j} | + | k_{i} β_{j} - β_{j} | + \sum_{ℓ = j + 2}^{p - 1} | β_{ℓ} - β_{ℓ - 1} | \\ + | β_{p} - ρ_{i_{2}} β_{p} | + | ρ_{i_{2}} β_{p} - β_{p - 1} | + M_{p} + | a_{n} | \\ = & | a_{0} | + M_{q} + (α_{q + 1} - ρ_{r_{1}} α_{q}) + | α_{q} | (1 - ρ_{r_{1}}) + \sum_{ℓ = q + 2}^{j - 1} (α_{ℓ} - α_{ℓ - 1}) + | α_{j} | (k_{r} - 1) \\ + (k_{r} α_{j} - α_{j - 1}) + (k_{r} α_{j} - α_{j + 1}) + | α_{j} | (k_{r} - 1) + \sum_{ℓ = j + 2}^{p - 1} (α_{ℓ - 1} - α_{ℓ}) \\ + | α_{p} | (1 - ρ_{r_{2}}) + (α_{p - 1} - ρ_{r_{2}} α_{p}) + (β_{q + 1} - ρ_{i_{1}} β_{q}) + | β_{q} | (1 - ρ_{i_{1}}) \\ + \sum_{ℓ = q + 2}^{j - 1} (β_{ℓ} - β_{ℓ - 1}) + | β_{j} | (k_{i} - 1) + (k_{i} β_{j} - β_{j - 1}) + (k_{i} β_{j} - β_{j + 1}) \\ + | β_{j} | (k_{i} - 1) + \sum_{ℓ = j + 2}^{p - 1} (β_{ℓ - 1} - β_{ℓ}) + | β_{p} | (1 - ρ_{i_{2}}) + (β_{p - 1} - ρ_{i_{2}} β_{p}) + M_{p} + | a_{n} | \\ = & | a_{0} | + M_{q} - ρ_{r_{1}} α_{q} + | α_{q} | (1 - ρ_{r_{1}}) + 2 | α_{j} | (k_{r} - 1) + 2 k_{r} α_{j} + | α_{p} | (1 - ρ_{r_{2}}) \\ - ρ_{r_{2}} α_{p} - ρ_{i_{1}} β_{q} + | β_{q} | (1 - ρ_{i_{1}}) + 2 | β_{j} | (k_{i} - 1) + 2 k_{i} β_{j} \\ + | β_{p} | (1 - ρ_{i_{2}}) - ρ_{i_{2}} β_{p} + M_{p} + | a_{n} | . \end{matrix}

(3)

Since

F (z)

is analytic in

| z | \leq 1

, by Lemma 1 and the Maximum Modulus Theorem, the number of zeros of

F (z)

(and hence of

P (z)

) in

| z | \leq δ

is less than or equal to

(1 / log (1 / δ)) log (M / | a_{0} |)

where

0 < δ < 1

, as claimed. □

Proof of Theorem 6.

Consider

F (z) = (1 - z) P (z)

. For

| z | = 1

we have,

\begin{matrix} | F (z) | & \leq & | a_{0} | + M_{q} + | a_{q + 1} - ρ_{1} a_{q} | + | ρ_{1} a_{q} - a_{q} | + \sum_{ℓ = q + 2}^{j - 1} | a_{ℓ} - a_{ℓ - 1} | \\ + | a_{j} - k a_{j} | + | k a_{j} - a_{j - 1} | + | a_{j + 1} - k a_{j} | + | k a_{j} - a_{j} | + \sum_{ℓ = j + 2}^{p - 1} | a_{ℓ} - a_{ℓ - 1} | \\ + | a_{p} - ρ_{2} a_{p} | + | ρ_{2} a_{p} - a_{p - 1} | + M_{p} + | a_{n} | a s i n () \\ \leq & | a_{0} | + M_{q} + | a_{q + 1} | cos α - ρ_{1} | a_{q} | cos α + | a_{q + 1} | sin α + ρ_{1} | a_{q} | sin α \\ + | a_{q} | (1 - ρ_{1}) + \sum_{ℓ = q + 2}^{j - 1} | a_{ℓ} | cos α - \sum_{ℓ = q + 2}^{j - 1} | a_{ℓ - 1} | cos α + \sum_{ℓ = q + 2}^{j - 1} | a_{ℓ} | sin α \\ + \sum_{ℓ = q + 2}^{j - 1} | a_{ℓ - 1} | sin α + | a_{j} | (k - 1) + k | a_{j} | cos α - | a_{j - 1} | cos α + k | a_{j} | sin α \\ + | a_{j - 1} | sin α + k | a_{j} | cos α - | a_{j + 1} | cos α + k | a_{j} | sin α + | a_{j + 1} | sin α \\ + | a_{j} | (k - 1) + \sum_{ℓ = j + 2}^{p - 1} | a_{ℓ - 1} | cos α - \sum_{ℓ = j + 2}^{p - 1} | a_{ℓ} | cos α + \sum_{ℓ = j + 2}^{p - 1} | a_{ℓ - 1} | sin α \\ + \sum_{ℓ = j + 2}^{p - 1} | a_{ℓ} | sin α + | a_{p} | (1 - ρ_{2}) + | a_{p - 1} | cos α - ρ_{2} | a_{p} | cos α + | a_{p - 1} | sin α \\ + ρ_{2} | a_{p} | sin α + M_{p} + | a_{n} | b y L e m m a . \end{matrix}

Hence

\begin{matrix} | F (z) | & \leq & | a_{0} | + M_{q} + | a_{q + 1} | cos α - ρ_{1} | a_{q} | cos α + | a_{q + 1} | sin α + ρ_{1} | a_{q} | sin α \\ + | a_{q} | (1 - ρ_{1}) + | a_{j - 1} | cos α + \sum_{ℓ = q + 2}^{j - 2} | a_{ℓ} | cos α - | a_{q + 1} | cos α \\ - \sum_{ℓ = q + 2}^{j - 2} | a_{ℓ} | cos α + | a_{j - 1} | sin α + \sum_{ℓ = q + 2}^{j - 2} | a_{ℓ} | sin α + | a_{q + 1} | sin α \\ + \sum_{ℓ = q + 2}^{j - 2} | a_{ℓ} | sin α + | a_{j} | (k - 1) + k | a_{j} | cos α - | a_{j - 1} | cos α + k | a_{j} | sin α \\ + | a_{j - 1} | sin α + k | a_{j} | cos α - | a_{j + 1} | cos α + k | a_{j} | sin α + | a_{j + 1} | sin α \\ + | a_{j} | (k - 1) + | a_{j + 1} | cos α + \sum_{ℓ = j + 2}^{p - 2} | a_{ℓ} | cos α - | a_{p - 1} | cos α \\ - \sum_{ℓ = j + 2}^{p - 2} | a_{ℓ} | cos α + | a_{j + 1} | sin α + \sum_{ℓ = j + 2}^{p - 2} | a_{ℓ} | sin α + | a_{p - 1} | sin α \\ + \sum_{ℓ = j + 2}^{p - 2} | a_{ℓ} | sin α + | a_{p} | (1 - ρ_{2}) + | a_{p - 1} | cos α - ρ_{2} | a_{p} | cos α + | a_{p - 1} | sin α \\ + ρ_{2} | a_{p} | sin α + M_{p} + | a_{n} | \\ = & | a_{0} | + M_{q} + ρ_{1} | a_{q} | (sin α - cos α - 1) + 2 | a_{q + 1} | sin α + | a_{q} | \\ + 2 \sum_{ℓ = q + 2}^{j - 2} | a_{ℓ} | sin α - 2 | a_{j} | + 2 k | a_{j} | (cos α + sin α + 1) + 2 | a_{j - 1} | sin α \\ + 2 | a_{j + 1} | sin α + 2 \sum_{ℓ = j + 2}^{p - 2} | a_{ℓ} | sin α + 2 | a_{p - 1} | sin α + | a_{p} | \\ + ρ_{2} | a_{p} | (sin α - cos α - 1) + M_{p} + | a_{n} | \\ = & | a_{0} | + M_{q} + | a_{q} | + ρ_{1} | a_{q} | (sin α - cos α - 1) + 2 \sum_{ℓ = q + 1}^{j - 1} | a_{ℓ} | sin α - 2 | a_{j} | \\ + 2 k | a_{j} | (cos α + sin α + 1) + 2 \sum_{ℓ = j + 1}^{p - 1} | a_{ℓ} | sin α + | a_{p} | \\ + ρ_{2} | a_{p} | (sin α - cos α - 1) + M_{p} + | a_{n} | . \end{matrix}

Since

F (z)

is analytic in

| z | \leq 1

, by Lemma 1 and the Maximum Modulus Theorem, the number of zeros of

F (z)

(and hence of

P (z)

) in

| z | \leq δ

is less than or equal to

(1 / log (1 / δ)) log (M / | a_{0} |)

where

0 < δ < 1

, as claimed. □

5. Discussion

As explained in the Introduction, the hypotheses applied in this paper build on similar hypotheses in the setting of results on the the location of zeros of a complex polynomial; namely, the Eneström–Kakeya Theorem and its generalizations. Future research could involve loosening or revising the monotonicity conditions of Theorems 5 and 6. For example, the mototonicity conditions of Rather et al. in [13,14], mentioned in the Introduction, could be imposed on the real and complex parts of the coefficients and on the moduli of the coefficients to produce related results. Theorems 5 and 6 concern a single reversal in the monotonicity condition, so this could be generalized to multiple reversals. In addition, combinations of the montonicity conditions presented here could be combined with others in the literature (such as those in [13,14]).

Author Contributions

Conceptualization, R.G. and M.G.; formal analysis, R.G. and M.G.; writing—original draft preparation, R.G. and M.G.; writing—review and editing, R.G. and M.G. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable

Informed Consent Statement

Not applicable

Data Availability Statement

Not applicable

Conflicts of Interest

The authors declare no conflict of interest.

References

A. Aziz and B. A. Zargar, Bounds for the zeros of a polynomial with restricted coefficients, Appl. Math. 2012, 3, 30–33. [CrossRef]
T. N. Chan and M. A. Malik, On Erdos-Lax theorem, Proc. Indian Acad. Sci. 1983, 92, 191–193.
K. K. Dewan, Extremal properties and coefficient estimates for polynomials with restricted zeros and on the location of zeros of polynomials, Ph.D. Thesis, Indian Institute of Technology, Delhi, 1980.
G. Eneström, Härledning af en allmän formel för antalet pensionärer, som vid en godtyeklig tidpunkt förefinnas inom en sluten pensionslcassa, Övfers. Vetensk.-Akad. Fórhh. 1893, 50, 405–415.
R. Gardner and M. Gladin, Generalizations of the Eneström-Kakeya Theorem Involving Weakened Hypotheses, AppliedMath 2022, 2(4), 687–699. [CrossRef]
R. Gardner and M. Gladin, The Number of Zeros in a Disk of a Complex Polynomial with Coefficients Satisfying Various Monotonicity Conditions, submitted.
R. Gardner and N. K. Govil, The Eneström-Kakeya Theorem and Some of Its Generalizations, in Current Topics in Pure and Computational Complex Analysis, ed. S. Joshi, M. Dorff, and I. Lahiri, New Delhi: Springer-Verlag, 2014; 171–200.
R. Gardner, N. K. Govil, and G. V. Milovanovi’c, Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials, Elsevier, Inc., 2022. [CrossRef]
N. K. Govil and Q. I. Rahman, On the Eneström-Kakeya theorem, Tôhoku Math. J. (2) 1968, 20, 126–136.
S. Kakeya, On the limits of the roots of an algebraic equation with positive coefficients, Tôhoku Math. J. First Ser. 1912–1913, 2, 140–142.
Q. G. Momhammad, On the zeros of the polynomials, Amer. Math. Monthly 1965 72, 631–633.
M. S. Pukhta, On the zeros of a polynomial, Appl. Math. 2011, 2, 1356–1358.
N. A. Rather, L. Ali, A. Bhat, On the number of zeros of a polynomial in a disk, Annali Del’Universita’ Di Ferrara, published online 05 May 2023. [CrossRef]
N. A. Rather, A. Bhat, and L. Ali, Number of zeros of a certain class of polynomials in a specific region, J. Class. Anal. 2021, 18(1), 29–37. arXiv:10.7153/jca-2021-18-03.
M. A. Shah, R. Swroop, H. M. Sofi, I. Nisar, A generalization of Eneström-Kakeya Theorem and a zero free region of a polynomial, J. Appl. Math. Phys. 2021, 9, 1271–1277. arXiv:10.4236/jamp.2021.96087.
E. C. Titchmarsh, The Theory of Functions, second edition, Oxford University Press, London, 1939.

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The Number of Zeros in a Disk of a Complex Polynomial with Coefficients Satisfying Various Monotonicity Conditions

Abstract

1. Introduction

2. Results

3. Lemmas

4. Proofs of the Results

5. Discussion

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

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