Eneström–Kakeya Theorem and Some of Its Generalizations

Abstract

The study of the zeros of polynomials has a very rich history. In addition to having numerous applications, this study has been the inspiration for much theoretical research (including being the initial motivation for modern algebra). The earliest contributors to this subject were Gauss and Cauchy. Algebraic and analytic methods for finding zeros of a polynomial, in general, can be quite complicated. So it is desirable to put some restrictions on polynomials. Eneström–Kakeya theorem, which is a result in this direction, states that if $p(z)=\sum_{j=0}^n a_j z^j$ is a polynomial of degree n with real coefficients satisfying $0\leq a_0\leq a_1\leq \cdots \leq a_n$, then all the zeros of p lie in $|z|\leq 1$. Eneström–Kakeya theorem has been the starting point for considerable literature in Mathematics, concerning the location of the zeros of polynomials. In this article we begin with the earliest results of Eneström and Kakeya and conclude this by presenting some of the recent results on this subject. Our article is expository in nature.

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Appendix

Remark on a Theorem on the Roots of the Equation

$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$ Where

All Coefficients Are Real and Positive

by

G. Eneström, Stockholm, Sweden

Tôhoku Mathematical Journal, 18 (1920), 34-36

A translation of ``Remarque sur un théorème relatif aux racines de l’equation$a_nx^n+a_{n-1}x^{n-1}+\cdots a_1x+a_0=0$ où tous les coefficientes $a$ sont réels et positifs'' by G. Eneström. Translated by Robert Gardner.

In 1912 M. S. Kakeya demonstrated in a paper in this journal that the absolute value of each root of the equation above [in the title] is between the smallest and largest values of

$$\frac{a_{n-1}}{a_n}, \frac{a_{n-2}}{a_{n-1}}, \ldots, \frac{a_0}{a_1},$$

and therefore for [positive] $a_n>a_{n-1}>\cdots>a_0$, the absolute value of each root is less than 1.

This theorem has already been proposed and demonstrated by me in 1893 in a footnote to a problem on pension funds.¹ This problem leads us to the equation

$$k^{s-1}+a_1k^{s-2}+\cdots + a_{s-2}k + a_{s-1}=0 (A)$$

where all of these coefficients are real and positive and for which

$$1>a_1\geq a_2\geq \cdots \geq a_{s-1}.$$

The reference cited in the footnote is written in Swedish and at the request of Mr. Hayashi I now translate verbatim the part about the roots of this equation.

Define α₁ as the smallest of the quantities

$$a_1, \frac{a_2}{a_1}, \frac{a_3}{a_2}, \ldots, \frac{a_{s-1}}{a_{s-2}}$$

and it is then evident from this definition of α₁ that

$$a_{q+1}-\alpha_1 a_q\geq 0 \, \, \, (q=0, 1, 2, \ldots, s-2; a_0=1).$$

Multiplication of equation (A) by $k-\alpha_1$, results in

$$ k^s+(a_1-\alpha_1)k^{s-1}+(a_2-\alpha_1a_1)k^{s-2}+\cdots +(a_{s-1}-\alpha_1 a_{s-2})k-\alpha_1a_{s-1}=0 {(B)}$$

and if we substitute $\rho(\cos \phi+i\sin \phi)$ for k, where ρ is the absolute value of k, then ρ and φ must satisfy the equations

$$\rho^s \cos s\phi + (a_1-\alpha_1)\rho^{s-1}\cos (s-1)\phi + (a_2-\alpha_1a_1)\rho^{s-2}\cos(s-2)\phi $$

$$ + \cdots + (a_{s-1} -\alpha_1a_{s-2})\rho\cos \phi - \alpha_1 a_{s-1}=0,$$

$$\rho^s \sin s\phi + (a_1-\alpha_1)\rho^{s-1}\sin (s-1)\phi + (a_2-\alpha_1a_1)\rho^{s-2}\sin(s-2)\phi $$

$$ + \cdots + (a_{s-1} -\alpha_1a_{s-2})\rho\sin \phi =0. {(C)}$$

We now show that if $\rho<\alpha_1$, equation (C) can not hold, regardless of the value of φ. Indeed, all coefficients $a_1-\alpha_1, a_2-\alpha_1a_1, \ldots, a_{s-1}-\alpha_1 a_{s-2}$ are positive, so the left side can not be greater than

$$\rho^s+(a_1-\alpha_1)\rho^{s-1}+(a_2-\alpha_1 a_1)\rho^{s-2}+ \cdots + (a_{s-1}-\alpha_1 a_{s-2})\rho - \alpha_1 a_{s-1}$$

and this expression can be written as

$$\rho^{s-1}(\rho-\alpha_1)+ a_1\rho^{s-2}(\rho-\alpha_1)+ \cdots + a_{s-2}\rho (\rho -\alpha_1)+a_{s-1} (\rho-\alpha_1),$$

which is negative if $\rho<\alpha_1$. The left side of equation (C) is therefore negative for $\rho<\alpha_1$. It follows that the absolute value of each root of equation (A) is greater than or equal to α₁.

In a similar way we can show that with α₂ as the largest of the quantities

$$a_1, \frac{a_2}{a_1}, \frac{a_3}{a_2}, \ldots, \frac{a_{s-2}}{a_{s-1}},$$

the absolute value of each root of the equation (A) must be less than or equal to α₂.

For this proof, replace k with $k^{s-1}$ [in equation (A)]. Then multiply the new equation by $k-1/\alpha_2$ and we easily find that the absolute value of k can never be less than $1/\alpha_2$, from which it follows immediately that the value of k can not be greater than α₂. We now have

$$\alpha_1\leq |k_i|\leq \alpha_2, \, \, \, (i=0, 1, 2, \ldots, s-1).$$