Annals of Mathematics

An Inequality for the Roots of An Algebraic EquationAuthor(s): J. L. WalshSource: Annals of Mathematics, Second Series, Vol. 25, No. 3 (Mar., 1924), pp. 285-286Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1967916 .

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AN INEQUALITY FOR THE ROOTS OF AN ALGEBRAIC EQUATION.

BY J. L. WALsH.

This note gives an inequality for the roots of an algebraic equation which can be proved from the following theorem:

If the roots of the equation

Xm + CtXm-1+ *+ Cin= 0

lie on or within a circle C whose radius is r and whose center is the point x = a, then all the roots of the equation

(1) Xm + Clxm1 + **+ Cm=X

lie on or within the m circles whose centers are the m points a + Mm and whose common radius is r.*

In the application of this theorem we shall use merely the fact that the roots of (1) lie on or within the circle whose center is a and radius

r+ IMrl r+1 ? A6" All roots of the equation

x2+aix = 0

lie on or within the circle whose center is the on and radius a1 a. Then all roots of

XI+alx+a2 = 0

lie on or within the circle whose center is the origin and radius I at i + VTa. Hence all roots of

Xs+ax'+ax = 0

lie on or within the same circle, so all roots of

x?+aIX2+aX+ag = 0

I Walsh, Trana. Amer. Math. Soc., vol. 24 (1922). 285

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286 J. L. WALSH.

lie on or within the circle whose center is the origin and radius 3

a, +VTW+ VYI7 Continuation of this reasoning shows that all roots of the equation

(3) x++ajxn-l'+a 1-2x + * **+an = 0

lie on or within the circle whose center is the origin and whose radius is

a (4) Ia,+I+/+ VI + .. +FIanI.

This upper limit (4) for the moduli of the roots of (3) is actually attained when all but one of the coefficients as vanish, and is thus attained not merely for a single type of equation (3) but for n types of such equations. The upper limit (4) is particularly useful when all but one of the coefficients a1 are small when compared with the remaining coefficient.

If we use the fact that all the roots of equation (2) lie on or within the circle whose center is - a1/2 and radius aj/2 1, we find that all the roots of (3) lie on or within the circle whose center is a1/2 and whose radius is

I al , 1, -,,,-, , ,3

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