Zeros of Quaternionic Polynomials with Incomplete Monotonicity Conditions on the Coefficients

Abstract

The classical Eneström–Kakeya Theorem restricts the location of the complex zeros of polynomials with real, positive, monotone increasing coefficients. That is, for ${\displaystyle p\left(z\right)=\sum _{v=0}^n{a}_v{z}^v}$, where $0\le a_0\le a_1\le \cdots \le a_n$, the zeros of p lie in the unit disk $|z|\le 1$ in the complex plane. Following the introduction of an analytic theory of functions of a quaternionic variable, this result was extended to polynomials of a quaternionic variable. Numerous generalizations of both the complex and quaternionic versions of the Eneström–Kakeya Theorem have appeared which modify the monotonicity condition and extend results to complex and quaternionic coefficients. We give a related theorem which generalizes several of the known results and includes them as corollaries. We impose a type-of-monotonicity condition on some of the real and imaginary parts of the coefficients of the polynomial.

1. Introduction

Finding the zeros of polynomials has a rich history. Algebraic formulae giving the zeros of polynomials with real coefficients of degree up to four were known as early as the mid-1540s [1]. With Abel’s proof of the insolvability of the quintic, the limitations of algebraic solutions became apparent. Restrictions of the location of the zeros became a necessary and popular area of study. The main result of interest to us was proved independently by Gustav Eneström in 1893 and Sōichi Kakeya in 1912 [2,3]. As a result, it is called the Eneström–Kakeya Theorem. It involves polynomials with real coefficients, but holds for both real and complex zeros of the polynomial.

Theorem 1 (The Eneström–Kakeya Theorem).

A polynomial with real coefficients, ${\displaystyle P\left(z\right)=\sum _{v=0}^n{a}_v{z}^v}$ where $0\le a_0\le a_1\le \cdots \le a_n$, has all of its zeros in $|z|\le 1$.

A large number of generalizations of this theorem exists involving real and complex polynomials. These often result by varying the monotonicity condition on the coefficients in different ways. Several such results concerning complex polynomials are given in Section 2.

With the recent introduction of an analytic theory of functions of a quaternionic variable [4], the Eneström–Kakeya Theorem has been extended to polynomials of a quaternionic variable [5] as follows.

Theorem 2 (The Quaternionic Eneström–Kakeya Theorem).

A polynomial of a quaternionic variable with real coefficients, ${\displaystyle P\left(qq\right)=\sum _{v=0}^n{q}^v{a}_v}$ where $0\le a_0\le a_1\le \cdots \le a_n$, has all of its zeros in $|q|\le 1$.

Several generalizations of this result also exist, including consideration of polynomials with quaternionic coefficients. Some of these are given in Section 4.

2. Previous Results for Complex Polynomials

Several generalizations of the Eneström–Kakeya Theorem in the setting of polynomials with complex coefficients exist. In this section, we mention several such results which are relevant to this paper.

In 1980 Aziz and Mohammad [6] introduced a “t condition with a reversal” on the real and imaginary parts of the coefficients of an analytic function and used this to get a bound on the zeros of the function as follows.

Theorem 3.

Let $f\left(z\right)={\sum }_{v=0}^{\infty }{a}_v{z}^v$ be analytic in $|z|\le t$. If $Re\left(a_v\right)={\alpha }_v$ and $Im\left(a_v\right)={\beta }_v$ for all $v=0,1,2,\dots$ and for some k and r,

$$0<{\alpha }_0\le t{\alpha }_1\le t^2{\alpha }_2\le \cdots \le t^k{\alpha }_k\ge t^{k+1}{\alpha }_{k+1}\ge \cdots$$

$$and\beta a_0\le t{\beta }_1\le t^2{\beta }_2\le \cdots \le t^r{\beta }_r\ge t^{r+1}{\beta }_{r+1}\ge \cdots$$

then $f\left(z\right)$ has all its zeros in ${\displaystyle |z|\ge \frac{t|a_0|}{2({\alpha }_k{t}^k+{\beta }_r{t}^r)-({\alpha }_1+{\beta }_0)}.}$

A related result for polynomials was given in [7]:

Theorem 4.

Let $P\left(z\right)={\sum }_{v=0}^n{a}_v{z}^v$ be a polynomial of degree n. If $Re\left(a_v\right)={\alpha }_v$ and $Im\left(a_v\right)={\beta }_v$ for $v=0,1,2,\dots ,n$, $a_n\ne 0$, and for some μ and θ and some $t\ge 0$,

$${\alpha }_0\le t{\alpha }_1\le t^2{\alpha }_2\le \cdots \le t^{\mu }{\alpha }_{\mu }\ge t^{\mu +1}{\alpha }_{\mu +1}\ge \cdots \ge t^n{\alpha }_n,$$

$${\beta }_0\le t{\beta }_1\le t^2{\beta }_2\le \cdots \le t^{\theta }{\beta }_{\theta }\ge t^{\theta +1}{\beta }_{\theta +1}\ge \cdots \ge t^n{\beta }_n,$$

then $P\left(z\right)$ has all its zeros in $R_1\le |z|\le R_2$ where

$$R_1=min\left\{\left(t|a_0|/(2(t^{\mu }{\alpha }_{\mu }+t^{\theta }{\beta }_{\theta })-({\alpha }_0+{\beta }_0)-t^n({\alpha }_n+{\beta }_n-|a_n|)\right),t\right\}and$$

$$R_2=max\left\{\right(|a_0|t^{n+1}-t^{n-1}({\alpha }_0+{\beta }_0)-t({\alpha }_n+{\beta }_n)+(t_2+1)(t^{n-\mu -1}{\alpha }_{\mu }+t^{n-\theta -1}{\beta }_{\theta })$$

$$+(t^2-1)\left(\sum _{v=1}^{\mu -1}{t}^{n-v-1}{\alpha }_v+\sum _{v=1}^{\theta -1}{t}^{n-v-1}{\beta }_v\right)$$

$$+(1-t^2)\left(\sum _{v=\mu +1}^{n-1}{t}^{n-v-1}{\alpha }_v+\sum _{v=\theta +1}^{n-1}{t}^{n-v-1}{\beta }_v\right)/|a_n|,{\displaystyle \frac{1}{t}}\}.$$

The advantages of the type of hypotheses used in Theorem 4 are illustrated by observing that a number of corollaries easily follow. For example, with $t=1$, $k=n$, and ${\beta }_v=0$ for $v=0,1,2,\dots ,n$, Theorem 4 implies the Eneström–Kakeya Theorem. By manipulating k and r, and with $t=1$ we can apply the result to any polynomial with coefficients which have monotone real and imaginary parts (whether monotone increasing, or monotone decreasing). For example, with $k=0$, $r=n$, and $t=1$ we have the hypotheses ${\alpha }_0\ge {\alpha }_1\ge \cdots \ge {\alpha }_n$ and ${\beta }_0\le {\beta }_1\le \cdots \le {\beta }_n$.

A related monotonicity condition on the real coefficients of a polynomial was given in [8]. In this case, an incomplete monotonicity condition is used. Monotonicity is imposed on some of the coefficients and excludes the condition on the first q coefficients and the last $n-p$ coefficients. As with the Eneström–Kakeya Theorem the coefficients are real but zeros are allowed to be complex.

Theorem 5.

Let $P\left(z\right)={\sum }_{v=0}^n{a}_v{z}^v$ be a polynomial of degree n with real coefficients such that for some positive numbers s and r with $0\le r\le s\le n$, the coefficients satisfy $a_r\le a_{r+1}\le \cdots \le a_{s-1}\le a_s$. Then all the zeros of P lie in the closed annulus

$$min\left\{1,{\displaystyle \frac{|a_0|}{M_r-a_r+a_s+M_s+|a_n|}}\right\}\le |z|\le {\displaystyle \frac{|a_0|+M_r-a_r+a_s+M_s}{|a_n|}},$$

where ${\displaystyle M_r=\sum _{v=1}^r|a_v-a_{v-1}|}$ and ${\displaystyle M_s=\sum _{v=s+1}^n|a_v-a_{v-1}|.}$

Notice that with $r=0$ and $s=n$ Theorem 5 implies the Eneström–Kakeya Theorem (and also gives a corresponding inner radius of the zero-containing region).

The monotonicity condition of Theorem 5 was generalized to include multiplicative parameters applied to the first and last coefficients in the monotonicity condition and extended to the real and imaginary parts of the coefficients in ([9], Theorem 6).

Theorem 6.

Let $P\left(z\right)={\sum }_{v=0}^n{a}_v{z}^v$ be a polynomial of degree n with complex coefficients. Let ${\alpha }_v=Re\left(a_v\right)$ and ${\beta }_v=Im\left(a_v\right)$ for $0\le v\le n$. Suppose that, for some positive numbers ${\rho }_{R_1}$, ${\rho }_{R_2}$, ${\rho }_{I_1}$, ${\rho }_{I_2}$, r, and s with ${\rho }_{R_2}\ge 1$, ${\rho }_{I_2}\ge 1$, $0<{\rho }_{R_1}\le 1$, $0<{\rho }_{I_1}\le 1$, and $0\le r\le s\le n$, the coefficients satisfy

$${\rho }_{R_1}{\alpha }_r\le {\alpha }_{r+1}\le {\alpha }_{r+2}\le \cdots \le {\alpha }_{s-1}\le {\rho }_{R_2}{\alpha }_{s}and$$

$${\rho }_{I_1}{\beta }_r\le {\beta }_{r+1}\le {\beta }_{r+2}\le \cdots \le {\beta }_{s-1}\le {\rho }_{I_2}{\beta }_s.$$

Then the zeros of P lie in

$$min\{1,|a_0|/(M_r+(1-{\rho }_{R_1})|{\alpha }_r|-{\rho }_{R_1}{\alpha }_r+(1-{\rho }_{I_1})|{\beta }_r|-{\rho }_{I_1}{\beta }_r+{\rho }_{R_2}{\alpha }_s+({\rho }_{R_2}-1)|{\alpha }_s|$$

$$+{\rho }_{I_2}{\beta }_s+({\rho }_{I_2}-1)|{\beta }_s|+M_s+|a_n|)\}<|z|\le (|a_0|+M_r+(1-{\rho }_{R_1})|{\alpha }_r|-{\rho }_{R_1}{\alpha }_r$$

$$+(1-{\rho }_{I_1})|{\beta }_r|-{\rho }_{I_1}{\beta }_r+({\rho }_{R_2}-1)|{\alpha }_s|+{\rho }_{R_2}{\alpha }_s+({\rho }_{I_2}-1)|{\beta }_s|+{\rho }_{I_2}{\beta }_s+M_s)/|a_n|,$$

where ${\displaystyle M_r=\sum _{v=1}^r|a_v-a_{v-1}|}$ and ${\displaystyle M_s=\sum _{v=s+1}^n|a_v-a_{v-1}|}$.

We get Theorem 5 as a corollary to Theorem 6 by setting ${\beta }_v=0$ for $v=0,1,\dots ,n$ and setting ${\rho }_{R_1}={\rho }_{R_2}=1$. If we also set $r=0$ and $s=n$ then we get ([10], Theorem 3).

3. Some Properties of Quaternionic Polynomials

The quaternions $\mathbb{H}$ form a noncommutative division ring (in fact, they are the standard example of such a structure). The absence of commutativity means that we lose the factor theorem and that the zero-containing set is much more complicated than it is, for example, in the complex setting. As a simple example, the set of zeros of the polynomial equation $q^2+1=0$ is $\{x_{1}i+x_{2}j+x_{3}k\mid x_1,x_2,x_3\in \mathbb{R},x_1^2+x_2^2+x_3^2=1\}$. So not only does the second-degree polynomial equation $q^2+1=0$ have more than two solutions, it has uncountably infinitely many solutions! However with an appropriate definition of multiplicity of a zero of a quaternionic polynomial, the sum of the multiplicities of the zeros equals the degree of the polynomial. See ([11], Definition 2.6) where such a definition is given and it is shown that, when applied to the complex setting, it reduces to the usual definition of multiplicity of a polynomial.

Another complication presented by noncommutativity is ambiguity as to how to multiply polynomials (or series). Writing a polynomial with the indeterminate on the left and the coefficients on the right, we define the regular product of $P_1\left(q\right)={\sum }_{v=0}^n{q}^v{a}_v$ and $P_2\left(q\right)={\sum }_{w=0}^m{q}^w{b}_w$ as

$$(P_1\ast P_2)\left(q\right)=\sum _{v=0,1,\dots ,n;w=0,1,\dots ,m}{q}^{v+w}{a}_v{b}_w,$$

where the symbol ∗ represents the regular product. As shown in [12], the regular product of polynomials f and g is zero, $(f\ast g)\left(q_0\right)=0$, if and only if

$$f\left(q_0\right)=0,\mathrm{or}f\left(q_0\right)\ne 0\mathrm{implies}g\left(f{\left(q_0\right)}^{-1}{q}_{0}f\left(q_0\right)\right)=0.$$

(1)

Since we are seeking results applicable to quaternionic polynomials instead of complex polynomials, we cannot simply apply well-known results from the complex setting. In addition to the unusual properties of quaternionic polynomials mentioned above with which we must deal, we also need quaternionic versions of the Maximum Modulus Theorem and Schwarz’s Lemma.

The Maximum Modulus Theorem in the quaternionic setting is as follows ([4], Theorem 3.4).

Theorem 7.

Let $B=B(0,r)$ be an open ball in $\mathbb{H}$ with center 0 and radius $r>0$, and let $f:B\to \mathbb{H}$ be a regular function. If $|f|$ has a relative maximum at a point $a\in B$, then f is constant on B.

The complex version of Schwarz’s Lemma (see [13], 5.5, p. 168); a quaternionic version of Schwarz’s Lemma appears in ([14], Lemma 1), as follows.

Theorem 8.

Let $f\left(q\right)={\sum }_{v=0}^{\infty }{q}^v{a}_v$ for $|q|\le R$, where the coefficients $a_v$ for $0\le v<\infty$, and variable q are quaternions. Suppose $f\left(0\right)=a_0=0$. Then

$$|f\left(q\right)|\le {\displaystyle \frac{M|q|}{R}}forany|q|\le R,$$

where $M={max}_{|q|=R}|f\left(q\right)|.$

4. Previous Eneström–Kakeya-Type Results for Quaternionic Polynomials

Following the introduction of the Eneström–Kakeya Theorem in 2020 (Theorem 2), a large number of generalizations followed. In this section we state several which are relevant to the present work.

The first relevant generalization appeared in the same paper which presented Theorem 2 ([5], Theorem 9). It involved a monotone increasing condition on the real part and the imaginary parts of the coefficients, as follows.

Theorem 9.

If ${\displaystyle P\left(q\right)=\sum _{v=0}^n{q}^v{a}_v}$ is a quaternionic polynomial of degree n, where $a_v={\alpha }_v+{\beta }_{v}i+{\gamma }_{v}j+{\delta }_{v}k$ for $v=0,1,\dots ,n$, and satisfying ${\alpha }_0\le {\alpha }_1\le \cdots \le {\alpha }_n$, ${\beta }_0\le {\beta }_1\le \cdots \le {\beta }_n$, ${\gamma }_0\le {\gamma }_1\le \cdots \le {\gamma }_n$, ${\delta }_0\le {\delta }_1\le \cdots \le {\delta }_n$, then all the zeros of p lie in

$$|q|\le {\displaystyle \frac{(|{\alpha }_0|-{\alpha }_0+{\alpha }_n)+(|{\beta }_0|-{\beta }_0+{\beta }_n)+(|{\gamma }_0|-{\gamma }_0+{\gamma }_n)+(|{\delta }_0|-{\delta }_0+{\delta }_n)}{|a_n|}}.$$

If ${\gamma }_v={\delta }_v=0$ for $v=0,1,\dots ,n$ in Theorem 9, then it implies Theorem 2.

A result generalizing Theorem 9 in two ways is in ([15], Theorem 3.1). First, it does not impose any condition on the coefficients $a_0,a_1,\dots ,a_{r-1}$, and second it introduces new parameters $\lambda >0$ and $\kappa \ge 1$ which are multiplied by the parts of $a_r$ and $a_n$, respectively, in a monotonicity condition.

Theorem 10.

If $P\left(q\right)={\sum }_{v=0}^n{q}^v{a}_v$ is a polynomial of degree n with quaternionic coefficients and quaternionic variable where $a_v={\alpha }_v+i{\beta }_v+j{\gamma }_v+k{\delta }_v$ for $v=0,1,\dots ,n$ such that for some $\kappa \ge 1$ and for some $\lambda >0$, we have:

$$\lambda {\alpha }_r\le {\alpha }_{r+1}\le \cdots \le {\alpha }_{n-1}\le \kappa {\alpha }_n,\lambda {\beta }_r\le {\beta }_{r+1}\le \cdots \le {\beta }_{n-1}\le \kappa {\beta }_n,$$

$$\lambda {\gamma }_r\le {\gamma }_{r+1}\le \cdots \le {\gamma }_{n-1}\le \kappa {\gamma }_n,\lambda {\delta }_r\le {\delta }_{r+1}\le \cdots \le {\delta }_{n-1}\le \kappa {\delta }_n,$$

where $0\le r\le n-1$, then all the zeros of $P\left(q\right)$ lie in

$$\begin{array}{ccc}\hfill |q|& \le & {\displaystyle \frac{1}{|a_n|}}(|{\alpha }_0|+M_r+(2\kappa -1){\alpha }_n+|\lambda -1||{\alpha }_r|-\lambda {\alpha }_r\hfill \\ & & +|{\beta }_0|+(2\kappa -1){\beta }_n+|\lambda -1||{\beta }_r|-\lambda {\beta }_r\hfill \\ & & +|{\gamma }_0|+(2\kappa -1){\gamma }_n+|\lambda -1||{\gamma }_r|-\lambda {\alpha }_r\hfill \\ & & +|{\delta }_0|+(2\kappa -1){\delta }_n+|\lambda -1||{\delta }_r|-\lambda {\delta }_r),\hfill \end{array}$$

where

$$M_r=\sum _{v=1}^r(|{\alpha }_v-{\alpha }_{v-1}|+|{\beta }_v-{\beta }_{v-1}|+|{\gamma }_v-{\gamma }_{v-1}|+|{\gamma }_v-{\gamma }_{v-1}|).$$

With $r=0$ and $\lambda =\kappa =1$, Theorem 10 reduces to Theorem 9. With ${\gamma }_v={\delta }_v=0$ for $v=0,1,\dots ,n$, the hypotheses of Theorem 10 are the same as the hypotheses of Theorem 6, so that Theorem 10 implies the outer radius of the zero-containing region in the complex plane of Theorem 6.

A generalization of Theorem 9 is given in ([16], Theorem 3.2) which imposes a monotonicity condition with a reversal on each of the parts of the coefficients.

Theorem 11.

If $P\left(q\right)={\sum }_{v=0}^n{q}^v{a}_v$ is a polynomial of degree n with quaternionic coefficients and quaternionic variable where $a_v={\alpha }_v+i{\beta }_v+j{\gamma }_v+k{\delta }_v$ for $v=0,1,\dots ,n$ such that, we have:

$${\alpha }_0\le {\alpha }_1\le \cdots \le {\alpha }_{\mu }\ge {\alpha }_{\mu +1}\cdots \ge {\alpha }_n,{\beta }_0\le {\beta }_1\le \cdots \le {\beta }_{\mu }\ge {\beta }_{\mu +1}\cdots \ge {\beta }_n$$

$${\gamma }_0\le {\gamma }_1\le \cdots \le {\gamma }_{\mu }\ge {\gamma }_{\mu +1}\cdots \ge {\gamma }_n,{\delta }_0\le {\delta }_1\le \cdots \le {\delta }_{\mu }\ge {\delta }_{\mu +1}\cdots \ge {\delta }_n,$$

then all the zeros of $P\left(q\right)$ lie in

$$\begin{array}{ccc}\hfill |q|& \le & {\displaystyle \frac{1}{|a_n|}}\left(\right(2{\alpha }_{\mu }-{\alpha }_n+|{\alpha }_0|-{\alpha }_0)+(2{\beta }_{\mu }-{\beta }_n+|{\beta }_0|-{\beta }_0)\hfill \\ & & +(2{\gamma }_{\mu }-{\gamma }_n+|{\gamma }_0|-{\gamma }_0)+(2{\delta }_{\mu }-{\delta }_n+|{\delta }_0|-{\delta }_0).\hfill \end{array}$$

With $\mu =n$, Theorem 11 reduces to Theorem 9.

A result generalizing both Theorem 9 and Theorem 11 is based on the t condition of Theorems 3 and 4 applied to the parts of the quaternionic coefficients. It is given in [14].

Theorem 12.

Let $P\left(q\right)={\sum }_{v=0}^n{q}^v{a}_v$ be a polynomial of degree n with quaternionic coefficients where $a_v={\alpha }_v+{\beta }_{v}i+{\gamma }_{v}j+{\delta }_{v}k$ for $v=0,1,2,\dots ,n$. If $a_n\ne 0$ and for some μ, θ, λ, and ω and for some $t\ge 0,$

$${\alpha }_0\le t{\alpha }_1\le t^2{\alpha }_2\le \cdots \le t^{\mu }{\alpha }_{\mu }\ge t^{\mu +1}{\alpha }_{\mu +1}\ge \cdots \ge t^n{\alpha }_n,$$

$${\beta }_0\le t{\beta }_1\le t^2{\beta }_2\le \cdots \le t^{\theta }{\beta }_{\theta }\ge t^{\theta +1}{\beta }_{\theta +1}\ge \cdots \ge t^n{\beta }_n,$$

$${\gamma }_0\le t{\gamma }_1\le t^2{\gamma }_2\le \cdots \le t^{\lambda }{\gamma }_{\lambda }\ge t^{\lambda +1}{\gamma }_{\lambda +1}\ge \cdots \ge t^n{\gamma }_n,and$$

$${\delta }_0\le t{\delta }_1\le t^2{\delta }_2\le \cdots \le t^{\omega }{\delta }_{\omega }\ge t^{\omega +1}{\delta }_{\omega +1}\ge \cdots \ge t^n{\delta }_n,$$

then $P\left(q\right)$ has all its zeroes in $R_1\le |q|\le R_2,$ where

$$\begin{array}{ccc}\hfill R_1& =& min\left\{\right(t|a_0|/(2(t^{\mu }{\alpha }_{\mu }+t^{\theta }{\beta }_{\theta }+t^{\lambda }{\gamma }_{\lambda }+t^{\omega }{\delta }_{\omega })-({\alpha }_0+{\beta }_0+{\gamma }_0+{\delta }_0)\hfill \\ & & -t^n({\alpha }_n+{\beta }_n+{\gamma }_n+{\delta }_n-|a_n|)),t\}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill R_2& =& max\left\{\right(|a_0|t^{n+1}-t^{n-1}({\alpha }_0+{\beta }_0+{\gamma }_0+{\delta }_0)-t({\alpha }_n+{\beta }_n+{\gamma }_n+{\delta }_n)\hfill \\ & & +(t^2+1)(t^{n-\mu -1}{\alpha }_{\mu }+t^{n-\theta -1}{\beta }_{\theta }+t^{n-\lambda -1}{\gamma }_{\lambda }+t^{n-\omega -1}{\delta }_{\omega })\hfill \\ & & +(t^2-1)\left(\sum _{v=1}^{\mu -1}{t}^{n-v-1}{\alpha }_v+\sum _{v=1}^{\theta -1}{t}^{n-v-1}{\beta }_v+\sum _{v=1}^{\lambda -1}{t}^{n-v-1}{\gamma }_v+\sum _{v=1}^{\omega -1}{t}^{n-v-1}{\delta }_v\right)\hfill \\ & & +(1-t^2)(\sum _{v=\mu +1}^{n-1}{t}^{n-v-1}{\alpha }_v+\sum _{v=\theta +1}^{n-1}{t}^{n-v-1}{\beta }_v+\sum _{v=\lambda +1}^{n-1}{t}^{n-v-1}{\gamma }_v\hfill \\ & & +\sum _{v=\omega +1}^{n-1}{t}^{n-v-1}{\delta }_v\left)\right)/|a_n|,1/t\}.\hfill \end{array}$$

With $t=1$ and $\theta =\lambda =\omega =\mu$, Theorem 12 reduces to Theorem 11. It then further reduces to Theorem 9, and ultimately to the Eneström–Kakeya Theorem (in both the quaternionic and complex settings).

5. Results

We now give a result which, under certain conditions, includes the hypotheses of each of the theorems of Section 2 and Section 4. Some new conditions are also imposed on the parts of the coefficients. A multiplicative parameter on the parts of the rth and the sth coefficients is imposed (similar to Theorem 6). We require the t condition, a reversal, and a multiplicative parameter on the parts of the coefficients which appear at the reversal. We also leave the option of imposing no condition on the parts of the first r coefficients nor on the parts of the last $n-s$ coefficients. In this way, we get each of the results of Section 4 as corollaries.

Theorem 13.

Let $P\left(q\right)={\sum }_{v=0}^n{q}^v{a}_v$ be a quaternionic polynomial of degree n with coefficients $a_v={\alpha }_v+{\beta }_{v}i+{\gamma }_{v}j+{\delta }_{v}k.$ Let ${\rho }_{R_1}$, ${\rho }_{R_2}$, ${\rho }_{I_1}$, ${\rho }_{I_2}$, ${\rho }_{J_1}$, ${\rho }_{J_2}$, ${\rho }_{K_1}$, ${\rho }_{K_2}$, ${\rho }_R$, ${\rho }_I$, ${\rho }_J$, and ${\rho }_K$ be positive real numbers, and let $t\ge 0$. Let r, μ, and s be integers between 0 and n. Suppose

$$\begin{array}{cc}\hfill t^r{\rho }_{R_1}{\alpha }_r& \le t^{r+1}{\alpha }_{r+1}\le \cdots \le t^{\mu -1}{\alpha }_{\mu -1}\le t^{\mu }{\rho }_R{\alpha }_{\mu }\ge t^{\mu +1}{\alpha }_{\mu +1}\ge \cdots \ge t^{s-1}{\alpha }_{s-1}\ge t^s{\rho }_{R_2}{\alpha }_s,\hfill \\ \hfill t^r{\rho }_{I_1}{\beta }_r& \le t^{r+1}{\beta }_{r+1}\le \cdots \le t^{\mu -1}{\beta }_{\mu -1}\le t^{\mu }{\rho }_I{\beta }_{\mu }\ge t^{\mu +1}{\beta }_{\mu +1}\ge \cdots \ge t^{s-1}{\beta }_{s-1}\ge t^s{\rho }_{I_2}{\beta }_s,\hfill \\ \hfill t^r{\rho }_{J_1}{\gamma }_r& \le t^{r+1}{\gamma }_{r+1}\le \cdots \le t^{\mu -1}{\gamma }_{\mu -1}\le t^{\mu }{\rho }_J{\gamma }_{\mu }\ge t^{\mu +1}{\gamma }_{\mu +1}\ge \cdots \ge t^{s-1}{\gamma }_{s-1}\ge t^s{\rho }_{J_2}{\gamma }_s,\hfill \\ \hfill t^r{\rho }_{K_1}{\delta }_r& \le t^{r+1}{\delta }_{r+1}\le \cdots \le t^{\mu -1}{\delta }_{\mu -1}\le t^{\mu }{\rho }_K{\delta }_{\mu }\ge t^{\mu +1}{\delta }_{\mu +1}\ge \cdots \ge t^{s-1}{\delta }_{s-1}\ge t^s{\rho }_{K_2}{\delta }_s.\hfill \end{array}$$

Then the zero of $p\left(q\right)$ satisfy

$$min\left\{{\displaystyle \frac{|a_0|t^2}{M_1}},t\right\}\le |q|\le max\left\{{\displaystyle \frac{M_2}{|a_n|}},{\displaystyle \frac{1}{t}}\right\}$$

where

$$\begin{array}{ccc}\hfill M_1& =& M_r+M_s+\left(\right(|{\rho }_{R_1}-1||{\alpha }_r|-{\rho }_{R_1}{\alpha }_r)+(|{\rho }_{I_1}-1||{\beta }_r|-{\rho }_{I_1}{\beta }_r)\hfill \\ & & +(|{\rho }_{J_1}-1||{\gamma }_r|-{\rho }_{J_1}{\gamma }_r)+(|{\rho }_{K_1}-1||{\delta }_r|-{\rho }_{K_1}{\delta }_r))t^{r+1}\hfill \\ & & +2\left(\right({\rho }_R{\alpha }_{\mu }+|{\rho }_R-1||{\alpha }_{\mu }|)+({\rho }_I{\beta }_{\mu }+|{\rho }_I-1||{\beta }_{\mu }|)\hfill \\ & & +({\rho }_J{\gamma }_{\mu }+|{\rho }_J-1||{\gamma }_{\mu }|)+({\rho }_K{\delta }_{\mu }+|{\rho }_K-1||{\delta }_{\mu }|))t^{\mu +1}\hfill \\ & & +\left(\right(|{\rho }_{R_2}-1||{\alpha }_s|-{\rho }_{R_2}{\alpha }_{s}t)+(|{\rho }_{I_2}-1||{\beta }_s|-{\rho }_{I_2}{\beta }_{s}t)\hfill \\ & & +(|{\rho }_{J_2}-1||{\gamma }_s|-{\rho }_{J_2}{\gamma }_{s}t)+(|{\rho }_{K_2}-1||{\delta }_s|-{\rho }_{K_2}{\delta }_{s}t\left)\right)t^s+|a_n|t^{n+1}and\hfill \\ \hfill M_2& =& |a_0|t^{n+1}+M_r+M_s+\left(\right(|{\rho }_{R_1}-1||{\alpha }_r|-{\rho }_{R_1}{\alpha }_r)+(|{\rho }_{I_1}-1||{\beta }_r|-{\rho }_{I_1}{\beta }_r)\hfill \\ & & +(|{\rho }_{J_1}-1||{\gamma }_r|-{\rho }_{J_1}{\gamma }_r)+(|{\rho }_{K_1}-1||{\delta }_r|-{\rho }_{K_1}{\delta }_r))t^{n-r-1}\hfill \\ & & +(t^2-1)\sum _{v=r+1}^{\mu -1}({\alpha }_v+{\beta }_v+{\gamma }_v+{\delta }_v)t^{n-v-1}+(t^2+1)\left(\right(|{\rho }_R-1||{\alpha }_{\mu }|+{\rho }_R{\alpha }_{\mu })\hfill \\ & & +(|{\rho }_I-1||{\beta }_{\mu }|+{\rho }_I{\beta }_{\mu })+(|{\rho }_J-1||{\gamma }_{\mu }|+{\rho }_J{\gamma }_{\mu })+(|{\rho }_K-1||{\delta }_{\mu }|+{\rho }_K{\delta }_{\mu }))t^{n-\mu -1}\hfill \\ & & +(1-t^2)\sum _{v=\mu +1}^{s-1}({\alpha }_v+{\beta }_v+{\gamma }_v+{\delta }_v)t^{n-v-1}+\left(\right(|{\rho }_{R_2}-1||{\alpha }_s|-{\rho }_{R_2}{\alpha }_s)\hfill \\ & & +(|{\rho }_{I_2}-1||{\beta }_s|-{\rho }_{I_2}{\beta }_s)+(|{\rho }_{J_2}-1||{\gamma }_s|-{\rho }_{J_2}{\gamma }_s)+(|{\rho }_{K_2}-1||{\delta }_s|-{\rho }_{K_2}{\delta }_s))t^{n-s+1},\hfill \end{array}$$

where

$$M_r=\sum _{v=1}^r{t}^v|a_{v}t-a_{v-1}|and{M}_s=\sum _{v=s+1}^n{t}^v|a_{v}t-a_{v-1}|.$$

To get the previously mentioned results as corollaries, we assign specific values to the parameters of Theorem 13. With $r=0$, $s=n$, ${\rho }_{R_1}={\rho }_{I_1}={\rho }_{J_1}={\rho }_{K_1}={\rho }_{R_2}={\rho }_{I_2}={\rho }_{J_2}={\rho }_{K_2}={\rho }_R={\rho }_I={\rho }_K={\rho }_K=1$, Theorem 13 gives Theorem 12 in the case where $\theta =\lambda =\omega =\mu$. If, in addition, we take $t=1$ in Theorem 13 then get Theorem 11 as a corollary. With these values of the parameters, along with setting $\mu =n$ yields Theorem 9 as a corollary.

Theorem 13 also implies results of Section 2. With $r=0$, $s=n$, ${\rho }_{R_1}={\rho }_{I_1}={\rho }_{R_2}={\rho }_{I_2}={\rho }_R={\rho }_I=1$, and ${\gamma }_n={\delta }_v=0$ for $v=0,1,\dots ,n$, Theorem 4 results in the case $\theta =\mu$. With ${\rho }_{R_1}={\rho }_{R_2}={\rho }_R=1$, $\mu =s$, and ${\beta }_v={\gamma }_v={\delta }_v=0$ for $v=0,1,\dots ,n$, we get Theorem 5. Theorem 13 also implies the Eneström–Kakeya Theorem in both the complex setting (Theorem 1) and the quaternionic setting (Theorem 2).

Of course other corollaries follow as well be adjusting the many parameters of Theorem 13. As one example, we get the following generalization of Theorem 11 by setting $r=0$, $s=n$, and $t=1$.

Corollary 1.

Let $P\left(q\right)={\sum }_{v=0}^n{q}^v{a}_v$ is a polynomial of degree n with quaternionic coefficients and quaternionic variable where $a_v={\alpha }_v+i{\beta }_v+j{\gamma }_v+k{\delta }_v$ for $v=0,1,\dots ,n$. Let ${\rho }_{R_1}$, ${\rho }_{R_2}$, ${\rho }_{I_1}$, ${\rho }_{I_2}$, ${\rho }_{J_1}$, ${\rho }_{J_2}$, ${\rho }_{K_1}$, ${\rho }_{K_2}$, ${\rho }_R$, ${\rho }_I$, ${\rho }_J$, and ${\rho }_K$ be positive real numbers. Let μ be an integer between 0 and n. Suppose

$${\rho }_{R_1}{\alpha }_0\le {\alpha }_1\le \cdots \le {\rho }_R{\alpha }_{\mu }\ge {\alpha }_{\mu +1}\cdots \ge {\rho }_{R_2}{\alpha }_n,{\rho }_{I_1}{\beta }_0\le {\beta }_1\le \cdots \le {\rho }_I{\beta }_{\mu }\ge {\beta }_{\mu +1}\cdots \ge {\rho }_{I_2}{\beta }_n$$

$${\rho }_{J_1}{\gamma }_0\le {\gamma }_1\le \cdots \le {\rho }_J{\gamma }_{\mu }\ge {\gamma }_{\mu +1}\cdots \ge {\rho }_{J_2}{\gamma }_n,{\rho }_{K_1}{\delta }_0\le {\delta }_1\le \cdots \le {\rho }_K{\delta }_{\mu }\ge {\delta }_{\mu +1}\cdots \ge {\rho }_{K_2}{\delta }_n,$$

then all the zeros of $P\left(q\right)$ lie in

$$min\left\{{\displaystyle \frac{|a_0|t^2}{M_1}},1\right\}\le |q|\le {\displaystyle \frac{M_2}{|a_n|}}$$

where

$$\begin{array}{ccc}\hfill M_1& =& \left(\right(|{\rho }_{R_1}-1||{\alpha }_0|-{\rho }_{R_1}{\alpha }_0)+(|{\rho }_{I_1}-1||{\beta }_0|-{\rho }_{I_1}{\beta }_0)\hfill \\ & & +(|{\rho }_{J_1}-1||{\gamma }_0|-{\rho }_{J_1}{\gamma }_0)+(|{\rho }_{K_1}-1||{\delta }_0|-{\rho }_{K_1}{\delta }_0))\hfill \\ & & +2\left(\right({\rho }_R{\alpha }_{\mu }+|{\rho }_R-1||{\alpha }_{\mu }|)+({\rho }_I{\beta }_{\mu }+|{\rho }_I-1||{\beta }_{\mu }|)\hfill \\ & & +({\rho }_J{\gamma }_{\mu }+|{\rho }_J-1||{\gamma }_{\mu }|)+({\rho }_K{\delta }_{\mu }+|{\rho }_K-1||{\delta }_{\mu }|))\hfill \\ & & +\left(\right(|{\rho }_{R_2}-1||{\alpha }_n|-{\rho }_{R_2}{\alpha }_n)+(|{\rho }_{I_2}-1||{\beta }_n|-{\rho }_{I_2}{\beta }_{s}n)\hfill \\ & & +(|{\rho }_{J_2}-1||{\gamma }_n|-{\rho }_{J_2}{\gamma }_n)+(|{\rho }_{K_2}-1||{\delta }_n|-{\rho }_{K_2}{\delta }_{s}n\left)\right)+|a_n|and\hfill \\ \hfill M_2& =& |a_0|+((|{\rho }_{R_1}-1||{\alpha }_0|-{\rho }_{R_1}{\alpha }_0)+(|{\rho }_{I_1}-1||{\beta }_0|-{\rho }_{I_1}{\beta }_0)\hfill \\ & & +(|{\rho }_{J_1}-1||{\gamma }_0|-{\rho }_{J_1}{\gamma }_0)+(|{\rho }_{K_1}-1||{\delta }_0|-{\rho }_{K_1}{\delta }_0))\hfill \\ & & +2\left(\right(|{\rho }_R-1||{\alpha }_{\mu }|+{\rho }_R{\alpha }_{\mu })+(|{\rho }_I-1||{\beta }_{\mu }|+{\rho }_I{\beta }_{\mu })\hfill \\ & & +(|{\rho }_J-1||{\gamma }_{\mu }|+{\rho }_J{\gamma }_{\mu })+(|{\rho }_K-1||{\delta }_{\mu }|+{\rho }_K{\delta }_{\mu }))\hfill \\ & & +\left(\right(|{\rho }_{R_2}-1||{\alpha }_n|-{\rho }_{R_2}{\alpha }_n)+(|{\rho }_{I_2}-1||{\beta }_n|-{\rho }_{I_2}{\beta }_n)\hfill \\ & & +(|{\rho }_{J_2}-1||{\gamma }_n|-{\rho }_{J_2}{\gamma }_n)+(|{\rho }_{K_2}-1||{\delta }_n|-{\rho }_{K_2}{\delta }_n)).\hfill \end{array}$$

With ${\rho }_{R_1}={\rho }_{R_2}={\rho }_{I_1}={\rho }_{I_2}={\rho }_{J_1}={\rho }_{J_2}={\rho }_{K_1}={\rho }_{K_2}={\rho }_R={\rho }_I={\rho }_J={\rho }_K=1$, Corollary 1 yields Theorem 11. Another corollary can be similarly obtained. If we do not restrict $r=0$ and $s=n$, then we have a result with hypotheses similar to both those of Theorem 12 and (when $t=1$) similar to those of Theorem 11, but the monotonicity condition imposed on the coefficients is incomplete in that it is imposed only on the parts of coefficients $a_r$ through $a_s$. This further emphasizes the flexibility of the incomplete monotonicity condition and shows how it results in the admissibility of polynomials not covered by the other results.

We now see that Theorem 13 is a generalization of several of the results given in Section 4 concerning the location of the zeros of a quaternionic polynomial. Therefore, Theorem 13 gives the same bounds on the zero-containing regions as those other results when both apply. The advantage of Theorem 13 and Corollary 1 is that they are applicable to a more diverse collection of quaternionic polynomials than the existing such results. As a simple example to illustrate this, consider the polynomial $P\left(q\right)=1+2q+q^2+2q^3+q^4$. Notice that none of Theorems 9, 10, 11, 12 apply to this polynomial. However, with ${\alpha }_0=1$, ${\alpha }_1=2$, ${\alpha }_2=1$, ${\alpha }_3=2$, ${\alpha }_4=1$, ${\rho }_{R_1}=2$, ${\rho }_R=2$, ${\rho }_2=2$, $\mu =2$, $n=4$, and the remaining parameters 0, we have that Theorem 13 applies. In fact, it gives that the quaternionic zeros of this polynomial lie in the region $1/5\le |q|\le 5$.

6. Proof of the Result

Proof. 

First, notice that if $p\left(q\right)\ast (t-q)=0$ for $q=q_0$ if and only if $p\left(q_0\right)=0$, or $p\left(q_0\right)\ne 0$ implies $t-p{\left(q_0\right)}^{-1}{q}_{0}f\left(q_0\right)=0$ (that is, $q_0=t$), by Equation (1). So the zeros of $p\left(q\right)\ast (t-q)$ are just the zeros of $p\left(q\right)$ and $q=t$. Define $f\left(q\right)$ by

$$p\left(q\right)\ast (t-q)=a_{0}t+\sum _{v=1}^n{q}^v(a_{v}t-a_{v-1})-q^{n+1}{a}_n=a_{0}t+f\left(q\right).$$

For $|q|=t$ we have

$$\begin{array}{ccc}\hfill |f(q)|& =& \left|\sum _{v=1}^n{q}^v(a_{v}t-a_{v-1})-q^{n+1}{a}_n\right|\hfill \\ & \le & \sum _{v=1}^n|a_{v}t-a_{v-1}|t^v+|a_n|t^{n+1}\hfill \\ & =& \sum _{v=1}^r|a_{v}t-a_{v-1}|t^v+\sum _{v=r+1}^{\mu }|a_{v}t-a_{v-1}|t^v+\sum _{v=\mu +1}^s|a_{v}t-a_{v-1}|t^v\hfill \\ & & +\sum _{v=s+1}^n|a_{v}t-a_{v-1}|t^v+|a_n|t^{n+1}\hfill \\ & =& M_r+M_s+\sum _{v=r+1}^{\mu }|a_{v}t-a_{v-1}|t^v+\sum _{v=\mu +1}^s|a_{v}t-a_{v-1}|t^v+|a_n|t^{n+1}\hfill \\ & & where{M}_r=\sum _{v=1}^r|a_{v}t-a_{v-1}|t^{v}and{M}_s=\sum _{v=s+1}^n|a_{v}t-a_{v-1}|t^v\hfill \\ & \le & M_r+M_2+\sum _{v=r+1}^{\mu }|{\alpha }_{v}t-{\alpha }_{v-1}|t^v+\sum _{v=\mu +1}^s|{\alpha }_{v}t-{\alpha }_{v-1}|t^v+\sum _{v=r+1}^{\mu }|{\beta }_{v}t-{\beta }_{v-1}|t^v\hfill \\ & & +\sum _{v=\mu +1}^s|{\beta }_{v}t-{\beta }_{v-1}|t^v+\sum _{v=r+1}^{\mu }|{\gamma }_{v}t-{\gamma }_{v-1}|t^v+\sum _{v=\mu +1}^s|{\gamma }_{v}t-{\gamma }_{v-1}|t^v\hfill \\ & & +\sum _{v=r+1}^{\mu }|{\delta }_{v}t-{\delta }_{v-1}|t^v+\sum _{v=\mu +1}^s|{\delta }_{v}t-{\delta }_{v-1}|t^v+|a_n|t^{n+1}\hfill \\ & =& M_r+M_s+|{\alpha }_{r+1}t-{\rho }_{R_1}{\alpha }_r+{\rho }_{R_1}{\alpha }_r-{\alpha }_r|t^{r+1}+|{\alpha }_{r+2}t-{\alpha }_{r+1}|t^{r+2}\hfill \\ & & +|{\alpha }_{r+3}t-{\alpha }_{r+2}|t^{r+3}+\cdots +|{\alpha }_{\mu -1}t-{\alpha }_{\mu -2}|t^{\mu -1}\hfill \\ & & +|{\alpha }_{\mu }t-k_R{\alpha }_{\mu }t+k_R{\alpha }_{\mu }t-{\alpha }_{\mu -1}|t^{\mu }+|{\alpha }_{\mu +1}t-k_R{\alpha }_{\mu }+k_R{\alpha }_{\mu }-{\alpha }_{\mu }|t^{\mu +1}\hfill \\ & & +|{\alpha }_{\mu +2}t-{\alpha }_{\mu +1}|t^{\mu +2}+|{\alpha }_{\mu +3}t-{\alpha }_{\mu +2}|t^{\mu +3}+\cdots \hfill \\ & & +|{\alpha }_{s-1}t-{\alpha }_{s-2}|t^{s-1}+|{\alpha }_{s}t-{\rho }_{R_2}{\alpha }_{s}t+{\rho }_{R_2}{\alpha }_{s}t-{\alpha }_{s-1}|t^s\hfill \\ & & +|{\beta }_{r+1}t-{\rho }_{I_1}{\beta }_r+{\rho }_{I_1}{\beta }_r-{\beta }_r|t^{r+1}+|{\beta }_{r+2}t-{\beta }_{r+1}|t^{r+2}\hfill \\ & & +|{\beta }_{r+3}t-{\beta }_{r+2}|t^{r+3}+\cdots +|{\beta }_{\mu -1}t-{\beta }_{\mu -2}|t^{\mu -1}\hfill \\ & & +|{\beta }_{\mu }t-k_I{\beta }_{\mu }t+k_I{\beta }_{\mu }t-{\beta }_{\mu -1}|t^{\mu }+|{\beta }_{\mu +1}t-k_I{\beta }_{\mu }+k_I{\beta }_{\mu }-{\beta }_{\mu }|t^{\mu +1}\hfill \\ & & +|{\beta }_{\mu +2}t-{\beta }_{\mu +1}|t^{\mu +2}+|{\beta }_{\mu +3}t-{\beta }_{\mu +2}|t^{\mu +3}+\cdots \hfill \\ & & +|{\beta }_{s-1}t-{\beta }_{s-2}|t^{s-1}+|{\beta }_{s}t-{\rho }_{I_2}{\beta }_{s}t+{\rho }_{I_2}{\beta }_{s}t-{\beta }_{s-1}|t^s\hfill \\ & & +|{\gamma }_{r+1}t-{\rho }_{J_1}{\gamma }_r+{\rho }_{J_1}{\gamma }_r-{\gamma }_r|t^{r+1}+|{\gamma }_{r+2}t-{\gamma }_{r+1}|t^{r+2}\hfill \\ & & +|{\gamma }_{r+3}t-{\gamma }_{r+2}|t^{r+3}+\cdots +|{\gamma }_{\mu -1}t-{\gamma }_{\mu -2}|t^{\mu -1}\hfill \\ & & +|{\gamma }_{\mu }t-k_J{\gamma }_{\mu }t+k_J{\gamma }_{\mu }t-{\gamma }_{\mu -1}|t^{\mu }+|{\gamma }_{\mu +1}t-k_J{\gamma }_{\mu }+k_J{\gamma }_{\mu }-{\gamma }_{\mu }|t^{\mu +1}\hfill \\ & & +|{\gamma }_{\mu +2}t-{\gamma }_{\mu +1}|t^{\mu +2}+|{\gamma }_{\mu +3}t-{\gamma }_{\mu +2}|t^{\mu +3}+\cdots \hfill \\ & & +|{\gamma }_{s-1}t-{\gamma }_{s-2}|t^{s-1}+|{\gamma }_{s}t-{\rho }_{J_2}{\gamma }_{s}t+{\rho }_{J_2}{\gamma }_{s}t-{\gamma }_{s-1}|t^s\hfill \\ & & +|{\delta }_{r+1}t-{\rho }_{K_1}{\delta }_r+{\rho }_{K_1}{\delta }_r-{\delta }_r|t^{r+1}+|{\delta }_{r+2}t-{\delta }_{r+1}|t^{r+2}\hfill \\ & & +|{\delta }_{r+3}t-{\delta }_{r+2}|t^{r+3}+\cdots +|{\delta }_{\mu -1}t-{\delta }_{\mu -2}|t^{\mu -1}\hfill \\ & & +|{\delta }_{\mu }t-k_K{\delta }_{\mu }t+k_K{\delta }_{\mu }t-{\delta }_{\mu -1}|t^{\mu }+|{\delta }_{\mu +1}t-k_K{\delta }_{\mu }+k_K{\delta }_{\mu }-{\delta }_{\mu }|t^{\mu +1}\hfill \\ & & +|{\delta }_{\mu +2}t-{\delta }_{\mu +1}|t^{\mu +2}+|{\delta }_{\mu +3}t-{\delta }_{\mu +2}|t^{\mu +3}+\cdots \hfill \\ & & +|{\delta }_{s-1}t-{\delta }_{s-2}|t^{s-1}+|{\delta }_{s}t-{\rho }_{R_2}{\delta }_{s}t+{\rho }_{R_2}{\delta }_{s}t-{\delta }_{s-1}|t^s+|a_n|t^{n+1}\hfill \\ & \le & M_r+M_s+(|{\rho }_{R_1}-1||{\alpha }_r|-{\rho }_{R_1}{\alpha }_r)t^{r+1}\hfill \\ & & +2(k_R{\alpha }_{\mu }+|k_R-1||{\alpha }_{\mu }|)t^{\mu +1}+(|{\rho }_{R_2}-1||{\alpha }_s|-{\rho }_{R_2}{\alpha }_{s}t)t^s\hfill \\ & & +(|{\rho }_{I_1}-1||{\beta }_r|-{\rho }_{I_1}{\beta }_r)t^{r+1}+2(k_I{\beta }_{\mu }+|k_I-1||{\beta }_{\mu }|)t^{\mu +1}\hfill \\ & & +(|{\rho }_{I_2}-1||{\beta }_s|-{\rho }_{I_2}{\beta }_{s}t)t^s+(|{\rho }_{J_1}-1||{\gamma }_r|-{\rho }_{J_1}{\gamma }_r)t^{r+1}\hfill \\ & & +2(k_J{\gamma }_{\mu }+|k_J-1||{\gamma }_{\mu }|)t^{\mu +1}+(|{\rho }_{J_2}-1||{\gamma }_s|-{\rho }_{J_2}{\gamma }_{s}t)t^s\hfill \\ & & +(|{\rho }_{K_1}-1||{\delta }_r|-{\rho }_{K_1}{\delta }_r)t^{r+1}+2(k_K{\delta }_{\mu }+|k_K-1||{\delta }_{\mu }|)t^{\mu +1}\hfill \\ & & +(|{\rho }_{K_2}-1||{\delta }_s|-{\rho }_{K_2}{\delta }_{s}t)t^s+|a_n|t^{n+1}\hfill \\ & =& M_r+M_s+\left(\right(|{\rho }_{R_1}-1||{\alpha }_r|-{\rho }_{R_1}{\alpha }_r)+(|{\rho }_{I_1}-1||{\beta }_r|-{\rho }_{I_1}{\beta }_r)\hfill \\ & & +(|{\rho }_{J_1}-1||{\gamma }_r|-{\rho }_{J_1}{\gamma }_r)+(|{\rho }_{K_1}-1||{\delta }_r|-{\rho }_{K_1}{\delta }_r))t^{r+1}\hfill \\ & & +2\left(\right(k_R{\alpha }_{\mu }+|k_R-1||{\alpha }_{\mu }|)+(k_I{\beta }_{\mu }+|k_I-1||{\beta }_{\mu }|)\hfill \\ & & +(k_J{\gamma }_{\mu }+|k_J-1||{\gamma }_{\mu }|)+(k_K{\delta }_{\mu }+|k_K-1||{\delta }_{\mu }|))t^{\mu +1}\hfill \\ & & +\left(\right(|{\rho }_{R_2}-1||{\alpha }_s|-{\rho }_{R_2}{\alpha }_{s}t)+(|{\rho }_{I_2}-1||{\beta }_s|-{\rho }_{I_2}{\beta }_{s}t)\hfill \\ & & +(|{\rho }_{J_2}-1||{\gamma }_s|-{\rho }_{J_2}{\gamma }_{s}t)+(|{\rho }_{K_2}-1||{\delta }_s|-{\rho }_{K_2}{\delta }_{s}t\left)\right)t^s+|a_n|t^{n+1}\hfill \\ & =& M_1.\hfill \end{array}$$

Applying Schwarz’s Lemma (Theorem 8) to $f\left(q\right)$ gives $|f\left(q\right)|\le M_1|q|/t$ for $|q|\le t$. We now have for $|q|\le t$

$$|p\left(q\right)\ast (t-q)|=|t{a}_0+f\left(q\right)|\ge |a_0|t-|f\left(q\right)|\ge |a_0|t-{\displaystyle \frac{M_1|q|}{t}}.$$

Therefore, if ${\displaystyle |q|\le min\left\{\frac{|a_0|t^2}{M_1},t\right\}}$ then $p\left(q\right)\ne 0$, as claimed.

Define $g\left(q\right)$ by

$$p\left(q\right)\ast (t-q)=a_{0}t+\sum _{v=1}^n{q}^v(a_{v}t-a_{v-1})-q^{n+1}{a}_n=-q^{n+1}{a}_n+g\left(q\right).$$

Then

$$\left|q^n\ast g\left({\displaystyle \frac{1}{q}}\right)\right|=\left|q^n{a}_{0}t+\sum _{v=1}^n{q}^{n-v}(a_{v}t-a_{v-1})\right|.$$

For $|q|=t$ we have

$$\begin{array}{ccc}\hfill \left|q^n\ast g\left({\displaystyle \frac{1}{q}}\right)\right|& \le & |a_0|t^{n+1}+\sum _{v=1}^n|a_{v}t-a_{v-1}|t^{n-v}\hfill \\ & =& |a_0|t^{n+1}+\sum _{v=1}^r|a_{v}t-a_{v-1}|t^{n-v}+\sum _{v=r+1}^{\mu }|a_{v}t-a_{v-1}|t^{n-v}\hfill \\ & & \sum _{v=\mu +1}^s|a_{v}t-a_{v-1}|t^{n-v}+\sum _{v=s+1}^n|a_{v}t-a_{v-1}|t^{n-v}\hfill \\ & =& |a_n|t^{n+1}+M_r+M_s+\sum _{v=r+1}^{\mu }|a_{v}t-a_{v-1}|t^{n-v}+\sum _{v=\mu +1}^s|a_{v}t-a_{v-1}|t^{n-v}\hfill \\ & \le & |a_n|t^{n+1}+M_r+M_s+\sum _{v=r+1}^{\mu }|{\alpha }_{v}t-{\alpha }_{v-1}|t^{n-v}+\sum _{v=\mu +1}^s|{\alpha }_{v}t-{\alpha }_{v-1}|t^{n-v}\hfill \\ & & +\sum _{v=r+1}^{\mu }|{\beta }_{v}t-{\beta }_{v-1}|t^{n-v}+\sum _{v=\mu +1}^s|{\beta }_{v}t-{\beta }_{v-1}|t^{n-v}\hfill \\ & & +\sum _{v=r+1}^{\mu }|{\gamma }_{v}t-{\gamma }_{v-1}|t^{n-v}+\sum _{v=\mu +1}^s|{\gamma }_{v}t-{\gamma }_{v-1}|t^{n-v}\hfill \\ & & +\sum _{v=r+1}^{\mu }|{\delta }_{v}t-{\delta }_{v-1}|t^{n-v}+\sum _{v=\mu +1}^s|{\delta }_{v}t-{\delta }_{v-1}|t^{n-v}\hfill \\ & =& |a_0|t^{n+1}+M_r+M_s+|{\alpha }_{r+1}t-{\rho }_{R_1}{\alpha }_r+{\rho }_{R_1}{\alpha }_r-{\alpha }_r|t^{n-r-1}\hfill \\ & & +|{\alpha }_{r+2}t-{\alpha }_{r+1}|t^{n-r-2}+|{\alpha }_{r+3}t-{\alpha }_{r+2}|t^{n-r-3}+|{\alpha }_{r+4}t-{\alpha }_{r+3}|t^{n-r-4}\hfill \\ & & +\cdots +|{\alpha }_{\mu -1}t-{\alpha }_{\mu -2}|t^{n-\mu +1}+|{\alpha }_{\mu }t-k_R{\alpha }_{\mu }t+k_R{\alpha }_{n}t-{\alpha }_{\mu -1}|t^{n-\mu }\hfill \\ & & +|{\alpha }_{\mu +1}t-k_R{\alpha }_{\mu }+k_R{\alpha }_n-{\alpha }_{\mu }|t^{n-\mu -1}+|{\alpha }_{\mu +2}t-{\alpha }_{\mu +1}|t^{n-\mu -2}\hfill \\ & & +|{\alpha }_{\mu +3}t-{\alpha }_{\mu +2}|t^{n-\mu -3}+\cdots +|{\alpha }_{s-1}t-{\alpha }_{s-2}|t^{n-s+1}\hfill \\ & & +|{\alpha }_{s}t-{\rho }_{R_2}{\alpha }_{s}t+{\rho }_{R_2}{\alpha }_{s}t-{\alpha }_{s-1}|t^{n-s}\hfill \\ & & +|{\beta }_{r+1}t-{\rho }_{I_1}{\beta }_r+{\rho }_{I_1}{\beta }_r-{\beta }_r|t^{n-r-1}\hfill \\ & & +|{\beta }_{r+2}t-{\beta }_{r+1}|t^{n-r-2}+|{\beta }_{r+3}t-{\beta }_{r+2}|t^{n-r-3}+|{\beta }_{r+4}t-{\beta }_{r+3}|t^{n-r-4}\hfill \\ & & +\cdots +|{\beta }_{\mu -1}t-{\beta }_{\mu -2}|t^{n-\mu +1}+|{\beta }_{\mu }t-k_I{\beta }_{\mu }t+k_I{\beta }_{n}t-{\beta }_{\mu -1}|t^{n-\mu }\hfill \\ & & +|{\beta }_{\mu +1}t-k_I{\beta }_{\mu }+k_I{\beta }_n-{\beta }_{\mu }|t^{n-\mu -1}+|{\beta }_{\mu +2}t-{\beta }_{\mu +1}|t^{n-\mu -2}\hfill \\ & & +|{\beta }_{\mu +3}t-{\beta }_{\mu +2}|t^{n-\mu -3}+\cdots +|{\beta }_{s-1}t-{\beta }_{s-2}|t^{n-s+1}\hfill \\ & & +|{\beta }_{s}t-{\rho }_{I_2}{\beta }_{s}t+{\rho }_{I_2}{\beta }_{s}t-{\beta }_{s-1}|t^{n-s}\hfill \\ & & +|{\gamma }_{r+1}t-{\rho }_{J_1}{\gamma }_r+{\rho }_{J_1}{\gamma }_r-{\gamma }_r|t^{n-r-1}\hfill \\ & & +|{\gamma }_{r+2}t-{\gamma }_{r+1}|t^{n-r-2}+|{\gamma }_{r+3}t-{\gamma }_{r+2}|t^{n-r-3}+|{\gamma }_{r+4}t-{\gamma }_{r+3}|t^{n-r-4}\hfill \\ & & +\cdots +|{\gamma }_{\mu -1}t-{\gamma }_{\mu -2}|t^{n-\mu +1}+|{\gamma }_{\mu }t-k_J{\gamma }_{\mu }t+k_J{\gamma }_{n}t-{\gamma }_{\mu -1}|t^{n-\mu }\hfill \\ & & +|{\gamma }_{\mu +1}t-k_J{\gamma }_{\mu }+k_J{\gamma }_n-{\gamma }_{\mu }|t^{n-\mu -1}+|{\gamma }_{\mu +2}t-{\gamma }_{\mu +1}|t^{n-\mu -2}\hfill \\ & & +|{\gamma }_{\mu +3}t-{\gamma }_{\mu +2}|t^{n-\mu -3}+\cdots +|{\gamma }_{s-1}t-{\gamma }_{s-2}|t^{n-s+1}\hfill \\ & & +|{\gamma }_{s}t-{\rho }_{J_2}{\gamma }_{s}t+{\rho }_{J_2}{\gamma }_{s}t-{\gamma }_{s-1}|t^{n-s}\hfill \\ & & +|{\delta }_{r+1}t-{\rho }_{K_1}{\delta }_r+{\rho }_{K_1}{\delta }_r-{\delta }_r|t^{n-r-1}\hfill \\ & & +|{\delta }_{r+2}t-{\delta }_{r+1}|t^{n-r-2}+|{\delta }_{r+3}t-{\delta }_{r+2}|t^{n-r-3}+|{\delta }_{r+4}t-{\delta }_{r+3}|t^{n-r-4}\hfill \\ & & +\cdots +|{\delta }_{\mu -1}t-{\delta }_{\mu -2}|t^{n-\mu +1}+|{\delta }_{\mu }t-k_K{\delta }_{\mu }t+k_K{\delta }_{n}t-{\delta }_{\mu -1}|t^{n-\mu }\hfill \\ & & +|{\delta }_{\mu +1}t-k_K{\delta }_{\mu }+k_K{\delta }_n-{\delta }_{\mu }|t^{n-\mu -1}+|{\delta }_{\mu +2}t-{\delta }_{\mu +1}|t^{n-\mu -2}\hfill \\ & & +|{\delta }_{\mu +3}t-{\delta }_{\mu +2}|t^{n-\mu -3}+\cdots +|{\delta }_{s-1}t-{\delta }_{s-2}|t^{n-s+1}\hfill \\ & & +|{\delta }_{s}t-{\rho }_{K_2}{\delta }_{s}t+{\rho }_{K_2}{\delta }_{s}t-{\delta }_{s-1}|t^{n-s}\hfill \\ & \le & |a_0|t^{n+1}+M_r+M_s+(|{\rho }_{R_1}-1||{\alpha }_r|-{\rho }_{R_1}{\alpha }_r)t^{n-r-1}\hfill \\ & & +{\alpha }_{r+1}(t^{n-r}-t^{n-r-2})+{\alpha }_{r+2}(t^{n-r-1}-t^{n-r-3})+\cdots \hfill \\ & & +{\alpha }_{\mu -2}(t^{n-\mu +3}-t^{n-\mu +1})+{\alpha }_{\mu -1}(t^{n-\mu +2}-t^{n-\mu })\hfill \\ & & +(|k_R-1||{\alpha }_n|+k_R{\alpha }_{\mu })t^{n-\mu +1}+(|k_R-1||{\alpha }_{\mu }|+k_R{\alpha }_{\mu })t^{n-\mu -1}+\hfill \\ & & {\alpha }_{\mu +1}(t^{n-\mu -2}-t^{n-\mu })+{\alpha }_{\mu +2}(t^{n-\mu -3}-t^{n-\mu -1})+\cdots \hfill \\ & & +{\alpha }_{s-2}(t^{n-s-1}-t^{n-s+1})+{\alpha }_{s-1}(t^{n-s}-t^{n-s+2})\hfill \\ & & +(|{\rho }_{R_2}-1||{\alpha }_s|-{\rho }_{R_2}{\alpha }_s)t^{n-s+1}\hfill \\ & & +(|{\rho }_{I_1}-1||{\beta }_r|-{\rho }_{I_1}{\beta }_r)t^{n-r-1}+{\beta }_{r+1}(t^{n-r}-t^{n-r-2})\hfill \\ & & +{\beta }_{r+2}(t^{n-r-1}-t^{n-r-3})+\cdots +{\beta }_{\mu -2}(t^{n-\mu +3}-t^{n-\mu +1})\hfill \\ & & +{\beta }_{\mu -1}(t^{n-\mu +2}-t^{n-\mu })+(|k_I-1||{\beta }_n|+k_I{\beta }_{\mu })t^{n-\mu +1}\hfill \\ & & +(|k_I-1||{\beta }_{\mu }|+k_I{\beta }_{\mu })t^{n-\mu -1}+{\beta }_{\mu +1}(t^{n-\mu -2}-t^{n-\mu })\hfill \\ & & +{\beta }_{\mu +2}(t^{n-\mu -3}-t^{n-\mu -1})+\cdots +{\beta }_{s-2}(t^{n-s-1}-t^{n-s+1})\hfill \\ & & +{\beta }_{s-1}(t^{n-s}-t^{n-s+2})+(|{\rho }_{R_2}-1||{\beta }_s|-{\rho }_{I_2}{\beta }_s)t^{n-s+1}\hfill \\ & & +(|{\rho }_{J_1}-1||{\gamma }_r|-{\rho }_{J_1}{\gamma }_r)t^{n-r-1}+{\gamma }_{r+1}(t^{n-r}-t^{n-r-2})\hfill \\ & & +{\gamma }_{r+2}(t^{n-r-1}-t^{n-r-3})+\cdots +{\gamma }_{\mu -2}(t^{n-\mu +3}-t^{n-\mu +1})\hfill \\ & & +{\gamma }_{\mu -1}(t^{n-\mu +2}-t^{n-\mu })+(|k_J-1||{\gamma }_n|+k_J{\gamma }_{\mu })t^{n-\mu +1}\hfill \\ & & +(|k_J-1||{\gamma }_{\mu }|+k_J{\gamma }_{\mu })t^{n-\mu -1}+{\gamma }_{\mu +1}(t^{n-\mu -2}-t^{n-\mu })\hfill \\ & & +{\gamma }_{\mu +2}(t^{n-\mu -3}-t^{n-\mu -1})+\cdots +{\gamma }_{s-2}(t^{n-s-1}-t^{n-s+1})\hfill \\ & & +{\gamma }_{s-1}(t^{n-s}-t^{n-s+2})+(|{\rho }_{J_2}-1||{\gamma }_s|-{\rho }_{J_2}{\gamma }_s)t^{n-s+1}\hfill \\ & & +(|{\rho }_{K_1}-1||{\delta }_r|-{\rho }_{K_1}{\delta }_r)t^{n-r-1}+{\delta }_{r+1}(t^{n-r}-t^{n-r-2})\hfill \\ & & +{\delta }_{r+2}(t^{n-r-1}-t^{n-r-3})+\cdots +{\delta }_{\mu -2}(t^{n-\mu +3}-t^{n-\mu +1})\hfill \\ & & +{\delta }_{\mu -1}(t^{n-\mu +2}-t^{n-\mu })+(|k_K-1||{\delta }_n|+k_K{\delta }_{\mu })t^{n-\mu +1}\hfill \\ & & +(|k_K-1||{\delta }_{\mu }|+k_K{\delta }_{\mu })t^{n-\mu -1}+{\delta }_{\mu +1}(t^{n-\mu -2}-t^{n-\mu })\hfill \\ & & +{\delta }_{\mu +2}(t^{n-\mu -3}-t^{n-\mu -1})+\cdots +{\delta }_{s-2}(t^{n-s-1}-t^{n-s+1})\hfill \\ & & +{\delta }_{s-1}(t^{n-s}-t^{n-s+2})+(|{\rho }_{K_2}-1||{\delta }_s|-{\rho }_{K_2}{\delta }_s)t^{n-s+1}\hfill \\ & =& |a_0|t^{n+1}+M_r+M_s+(|{\rho }_{R_1}-1||{\alpha }_r|-{\rho }_{R_1}{\alpha }_r)t^{n-r-1}\hfill \\ & & +(t^2-1)\sum _{v=r+1}^{\mu -1}{\alpha }_v{t}^{n-v-1}+(t^2+1)(|k_R-1||{\alpha }_{\mu }|+k_R{\alpha }_{\mu })t^{n-\mu -1}\hfill \\ & & +(1-t^2)\sum _{v=\mu +1}^{s-1}{\alpha }_v{t}^{n-v-1}+(|{\rho }_{R_2}-1||{\alpha }_s|-{\rho }_{R_2}{\alpha }_s)t^{n-s+1}\hfill \\ & & +(|{\rho }_{I_1}-1||{\beta }_r|-{\rho }_{I_1}{\beta }_r)t^{n-r-1}+(t^2-1)\sum _{v=r+1}^{\mu -1}{\beta }_v{t}^{n-v-1}\hfill \\ & & +(t^2+1)(|k_I-1||{\beta }_{\mu }|+k_I{\beta }_{\mu })t^{n-\mu -1}+(1-t^2)\sum _{v=\mu +1}^{s-1}{\beta }_v{t}^{n-v-1}\hfill \\ & & +(|{\rho }_{I_2}-1||{\beta }_s|-{\rho }_{I_2}{\beta }_s)t^{n-s+1}\hfill \\ & & +(|{\rho }_{J_1}-1||{\gamma }_r|-{\rho }_{J_1}{\gamma }_r)t^{n-r-1}+(t^2-1)\sum _{v=r+1}^{\mu -1}{\gamma }_v{t}^{n-v-1}\hfill \\ & & +(t^2+1)(|k_J-1||{\gamma }_{\mu }|+k_J{\gamma }_{\mu })t^{n-\mu -1}+(1-t^2)\sum _{v=\mu +1}^{s-1}{\gamma }_v{t}^{n-v-1}\hfill \\ & & +(|{\rho }_{J_2}-1||{\gamma }_s|-{\rho }_{J_2}{\gamma }_s)t^{n-s+1}\hfill \\ & & +(|{\rho }_{K_1}-1||{\delta }_r|-{\rho }_{K_1}{\delta }_r)t^{n-r-1}+(t^2-1)\sum _{v=r+1}^{\mu -1}{\delta }_v{t}^{n-v-1}\hfill \\ & & +(t^2+1)(|k_K-1||{\delta }_{\mu }|+k_K{\delta }_{\mu })t^{n-\mu -1}+(1-t^2)\sum _{v=\mu +1}^{s-1}{\delta }_v{t}^{n-v-1}\hfill \\ & & +(|{\rho }_{K_2}-1||{\delta }_s|-{\rho }_{K_2}{\delta }_s)t^{n-s+1}\hfill \\ & =& |a_0|t^{n+1}+M_r+M_s+\left(\right(|{\rho }_{R_1}-1||{\alpha }_r|-{\rho }_{R_1}{\alpha }_r)+(|{\rho }_{I_1}-1||{\beta }_r|-{\rho }_{I_1}{\beta }_r)\hfill \\ & & +(|{\rho }_{J_1}-1||{\gamma }_r|-{\rho }_{J_1}{\gamma }_r)+(|{\rho }_{K_1}-1||{\delta }_r|-{\rho }_{K_1}{\delta }_r))t^{n-r-1}\hfill \\ & & +(t^2-1)\sum _{v=r+1}^{\mu -1}({\alpha }_v+{\beta }_v+{\gamma }_v+{\delta }_v)t^{n-v-1}\hfill \\ & & +(t^2+1)\left(\right(|k_R-1||{\alpha }_{\mu }|+k_R{\alpha }_{\mu })+(|k_I-1||{\beta }_{\mu }|+k_I{\beta }_{\mu })\hfill \\ & & +(|k_J-1||{\gamma }_{\mu }|+k_J{\gamma }_{\mu })+(|k_K-1||{\delta }_{\mu }|+k_K{\delta }_{\mu }))t^{n-\mu -1}\hfill \\ & & +(1-t^2)\sum _{v=\mu +1}^{s-1}({\alpha }_v+{\beta }_v+{\gamma }_v+{\delta }_v)t^{n-v-1}+\left(\right(|{\rho }_{R_2}-1||{\alpha }_s|-{\rho }_{R_2}{\alpha }_s)\hfill \\ & & +(|{\rho }_{I_2}-1||{\beta }_s|-{\rho }_{I_2}{\beta }_s)+(|{\rho }_{J_2}-1||{\gamma }_s|-{\rho }_{J_2}{\gamma }_s)\hfill \\ & & +(|{\rho }_{K_2}-1||{\delta }_s|-{\rho }_{K_2}{\delta }_s))t^{n-s+1}\hfill \\ & =& M_2.\hfill \end{array}$$

Now by the Maximum Modulus Theorem (Theorem 7) we have $|q^{n}g(1/q)|\le M_2$ for $|q|\le t$, or $|g\left(q\right)|\le M_2{|q|}^n$ for $|q|\ge 1/t$. We now have for $|q|\ge 1/t$,

$$|p\left(q\right)|=|-q^{n-1}{a}_n+g\left(q\right)|\ge |a_n{||q|}^{n+1}-M_2{|q|}^n={|q|}^n(|a_n||q|-M_2).$$

So for ${\displaystyle |q|>max\left\{\frac{M_2}{|a_n|},\frac{1}{t}\right\}}$ we have $p\left(q\right)>0$. That is, the zeros of $p\left(q\right)$ all satisfy ${\displaystyle |q|\le max\left\{\frac{M_2}{|a_n|},\frac{1}{t}\right\}}$, as claimed. □

7. Discussion

We have given a result that restricts the location of the zeros of a quaternionic polynomial based on a monotonicity condition on the parts of the coefficients. The monotonicity condition, though complicated, is very flexible. This allows for the extraction of a number of previously known results as corollaries. There seem to be a limited number of numerical approaches to finding the zeros of a quaternionic polynomial (for some examples see [17,18,19]), so it is desirable to have some restrictions on the zeros. Future research could involve weakened hypotheses which make the results applicable to a larger class of functions. In addition, more parameters could be introduced to further improve the estimated location of the zeros. For example, additional multiplicative parameters could be added to other coefficients, and additional reversals could be added. In this way, the current (and previous results mentioned in Section 2 and Section 4) would appear as corollaries to the new results. The monotonicity conditions of this paper could also be applied to the moduli of the coefficients instead of the part of the coefficients. See ([5], Theorem 10) for an example of such a result.