Mathematics. - On the zeras of a palynamial and of its derivative. By
N. G. DE BRUIJN. (Natuurkundig Laboratorium der N.V. Philips'
Gloeilampenfabrieken, Eindhoven-Nederland. ) (Communicated by
Prof. W. VAN DER WOUDE.)
(Communicated at the meeting of October 26, 1946.)
1. Considering polynomials f (z) with real coefficients, we observe that,
very roughly speaking, there exists a tendency for the zeros of its derivative
f' (z) to !ie closer to the rea I axis, than those of f (z). This is illustrated by
the following well-known facts:
A. The number of imaginary l) zeros of f' (z) does not exceed that of
[(z). This is a consequence of ROLLE's theorem.
B. If the zeros of f (z) !ie in a strip I Im z I <: a, the same can be said
concerning the zeros of f' (z). This is a special case of a famous theorem
of GAUSS, expressing that the zeros of f' (z) all !ie in any convex domain,
containing all the zeros of f (z) .
C. JENSEN'S Cirdes Theal\em. 2) If av -+- ib. denote the imaginary
zeros of f(z), then any imaginary zero of f'(z) lies inside at least one of
the circles I z-av I <: I b.I·
In this paper we prove another theorem, illustrating the same tendency.
Theorem 1. Let the real 3) palynamial f (z) of degree n (n > 1) have
the zeras 4) al' ... , an, and lel fJl' ... , fJn-l be thase of f'(z). Then we have
n-t
n-l •.~ I Im
1
n
,8,·1 :S; n ,,2-: IIm a. I·
(1)
There is equality if, and anly if, all the zeros of f(z) are realo
In section 2, a genera!ization of Theorem 1 is stated and proved. Section
3 contains some applications of Theorems 1 and 2.
We do not know, whether Theorem 1 remains va!id if the condition
"bout the rea!ity of the coefficients of f (z) is omitted. The cases n
2, 3
are easy to deal with, but the general case seems to be difficult. We are,
however, able to prove it if all the zeros of f(z) are purely imaginary,
but not necessarily conjugated by pairs. This will be carried out in section
4, which is independent of sections 1, 2, 3. As a corrolary, we will obtain
=
+
1) We call a complex number ct (rt = Re ct
i Im a) imaginary, if Im (t op 0; and
purely imaginary, if Re ct = O.
2) Acta Math., 36, 181-195 (1913).
a) Throughout the paper, a polynomial or rational function of z is called a real
function, if it is real for real z.
4) A double zero is counted twice, etc.
1038
a simple and more general inequality. containing an arbitrary convex
function (Theorem 7).
2. The results concerning the zeros of f'(z) and f(z) can also be
expressed in terms of zeros and poles of the function f'(z)/f(z). We
extend our considerations to rational functions of the more general type
rp(z)=az
+ b +j=1
Î ~
+ J=I
Z (---.!L
+~).
z-aJ
z-eJ
Z-ej
a::::; O. b. al' .... ak real; SI
}
(
~ O..... Sk ~ 0; tI ~ 0 ..... t/~~. el' :... el (' (2)
Jmagmary)
=
Henceforth. we con si der the point z
co as a possible pole or zero of
rational functions. For instance. if a ~ O. z
co represents a simple pole
of (2); we call-a its residue 5). IE a = b = O. z = co is a zero of rp(z).
Moreover. z
co is considered as a point on the real axis; thus we always
take Im z to be zero. if z represents the point at infinity.
We can now describe a function of the type (2) as a real rational
function. all of whose pol es are simpIe. with positive residues. In analogy
to the behaviour of f'(z)/f(z). referred to above. we observe the tendency
for the zeros of rp(z) to lie closer to the real axis than the poles of rp(z).
Proper ties analogous to A. B. C hold true. and also their proofs are
analogous 6) .
Our generalization of theorem 1 reads:
=
=
Theorem 2. Let rp (z) be of the type (2). and iet al ..... an denote the
poies of rp(z). and {3l ..... {3n its zeros. Then we hav.e
n
n
v=1
~=I
;E IIm {3~ I ::::; ;E IIm
lf, moreover, a
n
= b = 0, we have
n
k
.'=\
I
;E IIm {3v I ~ ;E IIm a v1-2 (2' Sj
v=1
a.I·
I
I
+ 2;E tj)-I ;E tJ IIm eJ I.
I
I
In both cases, there is equaZity if, and only if, the poies of rp (z) all Zie on
tfle reai axis.
Theorem 1 follows immediately from the second part of Theorem 2.
For. on taking rp(z) = f'(z)/f(z). we obtain a = b = O. tj = 1.
ZSj+2Ztj=n.
Our proof of Theorem 2 is relatively simple when rp (z) has neither
real poles. nor real zeros. In the general case. however. we need an auxiliary
function 1p(Z). to be constructed in Lemma 2.
5) With this (rather unusual) convention. the sign of the residue of a pole on the
rea! axis remains invariant with respect to transformations z = (az'
b)/(cz'
d)
(ad-bc>O; a, b, c, d rea!).
0) The ana!ogue of JENSEN 's Orcles Theorem was proved by J, v. Sz, NAGY. Jhrsber.
D, Math. Ver. 31, 238-251 (1922).
+
+
1039
Lemma 1.
lf
n
f(z)
= II (z-a,.),
n
g (z)
~=l
= II (z-fJ,·),
v=l
then
V. P.
joo~ log Igf(z)(z) I dz -_
n
n
n
I film fJ,· 1- film a" 11.
-00
J= J.
A
00
where V.P. means 'valeur principale': V.P.
lim
A-++oo
-00
Proof.
-A
It is easily verified, that, for a arbitrary and A
~
+ 00
A
JIOg Iz-al dz= 2 (A log A-A) + n Ilm al + 0 (A-I).
-A
Now our lemma is evident.
Lemma 2. Let cp (z) be a teal ratiolnal lunction (not identically zero)
ot the type (2). Then we can construct a real rational luncHon ljJ(Z), all of
whose poles are simple and real, with negative residu es, such that
and
Proof.
a)
b)
0 <: cp(z) ljJ(Z) <
cp(z) ljJ(Z)
1
=
00
for real z,
lor z
00.
=
A point a on the real axis is called a positive change of sign for
(p(z), if the behaviour of the sign of cp(z) in theneighbourhood of a is the
same, as it is for a real function, which has a simple pole at z = a with
positive residue. So if a is fini te, this means (z-a) cp(z) > 0 for
0<lz-al<é;ifa=ooitmeanszcp(z)<OforO<lz-11<é. Nega~
tive changes of sign are defined accordingly. Any change of sign represents
a pole or zero of odd order of cp (z ) .
Now construct a real rational function ljJ(Z). which has its (simpie)
zeros in the positive changes of sign for cp (z), and its (simpie) poles in
the negative ones. Moreover, we take care, that cp(z) ljJ(Z) :> 0 for all
real values of z. It follows, that the poles of lp (z) are negative changes of
sign for lj.' (z); hence all the residues of lp(Z) are negative.
Any pole or zero of lp (z) is a pole or zero of cp (z). The residues of
(p (z) being positive, any real pole of cp(z) is a positive change of sign for
'I' (z), and hence it is a zero of lp (z). So if a is a pole of one of the functions
(p(z) or rp(z), it is a zero of the other one. cp(z) and lp(Z) having simp Ie
zeros only, we now conclude, that the product cp(z) lp(Z) has no poles on
the real axis.
We shall prove, th at cp(z) 'lp(Z) has no zero at z = 00. lf z = 00 is a
pole of cp(z), it will be a zero of ljJ(Z). Both pole and zero are simpie, and
hence 0 ~ cp( (0) ljJ( (0) ~ 00. H, in the second pi ace, 0 ~ cp( (0) ~ 00,
we have also 0 ~ 1jJ( (0) ~ 00, whence it followsO ~ cp( (0) lp( (0) ~ 00.
If,lastly, cp(oo) = 0, we represent cp(z) in the form (2), with a = b = O.
1040
=
From Sj :> O. tj :> O. we infer. that z
co is a simple zero. and moreover.
th at it is a negative change of sign for cp. Hence z
co is a pole of 'ljJ (z).
and again 0 -:::j:- cp( co) 'ljJ( co) -:::j:- co.
Now having obtained cp( co) 'ljJ( co)
C> O. we take lP1 (z) C-1 'ljJ(z).
which is easily seen to satisfy the conditions of our lemma.
=
=
=
Proof of Theorem 2. Let cp (z) be given by (2). and let 'ljJ (z) satisfy
the conditions of Lemma 2. Then we have. by Lemma 1:
v. p.fCl?IOg Icp (z) 'ljJ (z) Idz = nl 1'1 Im {l,·I- il Im a,·II.. .
I
I
-Cl?
(3)
where al' .... an and {ll' .... (ln are the poles and zeros of cp(z). respectively.
For. the zeros and poles of 'ljJ(z) are real. and do not contribute to the
right-hand side of (3).
It follows from the inequality
log u ::s; u-I
(u
> 0).
.
.
.
.
(4)
::s; cp (z) 'ljJ (z) -1.
(5)
th at. for z real.
log Icp (z) 'ljJ (z) I = log cp (z) 1f1 (z)
In the upper half plane. the function cp(z) 'ljJ(z) -1 has the pol es ej
(we may. of course. suppose Imej>O>lmej). with residues tj1Jl(ej).
At z
co this function behaves like clZ-l + c2z-2 + .... with Cl real.
Now contour integration shows that
=
Cl?
v. p.f' cp (z) 'ljJ (z) -11 dz =
-
,
2n j~IIm I tJ 'ljJ (eJ) I· . .
(6)
By Lemma 2. lp (z) is a function of the type
'ljJ (z)
=
Az
+B -
wh ere B. dl ..... d p are real. and A
Im 1Jl(ej) :> Alm ej. hence
- Im I tj
Rld - ... -
z-
I
:>
O. Rl
lp
(ej) 1 ::s;
:>
R Pd
z-
O..... R p
. (7)
P
:>
O. This implies
o. .
(8)
and even
(9)
The first part of Theorem 2 follows from (3). (5). (6) and (8). N ow
take a
b
O. In this case. we have
= =
= SI + ... + Sk + 2 (tl + ... + t,).
to cp ( co ) 'ljJ ( co) = 1. we infer from (7). that
A = [SI + . " + Sk + 2 (tl + ... + t,)J-I.
lim z cp (z)
Z~Cl?
According
N ow using (9) instead of (8). the second part of Theorem 2 is readily
proved.
1041
=
In (4), the sign
occurs only, if cp (z)
no imaginary pol es.
=
only holds for u
1. Hence, in Theorem 2, equality
1 identically in z. This means, that cp (z) has
1jJ (z)
=
3. An interesting specialization of Theorem 1 is obtained by taking all
the zeros of f (z) to lie on the imaginary axis . This leads to:
Theorem 3. Let F(y) be a real palynamial, whase zeras )' 1' ... ,;'n
> 0) are all > 0, but nat all 0, and let 01' ... , On_l be the zeros af
F' (y). Then we have the inequality
=
(n
n-I
I
I
'"~j,,
u,. <n-y
n
I
Proof.
-+- i Y
The
real
polynomial
11
~
~
I
i
(10)
y ••
= F(-z2)
f(z)
has
the 2n
zeros
t ' .... ± i y A . lts derivative f' (z) = -2zF' ( - z2) having the zeros
iot ..... ± ióLI ' Theorem 3 follows by applying Theorem 1 to f(z).
O. ±
An inequality in the opposite direction was given by KAKEYA 7) for
general exponents. He proved. under the assumptions of Theorem 3, putting
1
1
n
n-I
Dp=-;J;yP_-n 1 JO
n-1
th at D p > 0 for p > 1 or p <: O. and D p
only in three cases: p
O. p
1. and
p
t, KAKEYA's result reads
=
=
=
n-I
1:
1
n-l
1:1
ÓP
,. '
<:
0 for 0
)' 1
= ... = J'n
<:
P
1. with equality
(p arbitrary) . For
<:
11
ät ~ -n- 1: y! .
1
In section 4, KAKEYA' s inequality for D p will appear to be a special case
of Theorem 7.
We give two more applications of Theorem 2.
=
Theorem 4. Let the real palynamial f(z)
ao + a1z + ... + anZ n
have the zeros al' ... , an , and let P(y) be a palynamial, all af whase zeras
are <: O. FlIrthermare, let ~1' ... , fin be the zeros of
g (z)
= ao P (0) + alP (1) z + ... + a n P (n) zn.
We then have
n .
P (n-1) n
.
"~I I Im fi,· 1:S; -P~ '~I I Im a,· I ·
Theorem 1 follows from this one by taking P(y)
=
Proof. It is sufficient to prove the case P(y)
= y +a
y.
(a > 0), the
7) Proc. Ph,ys. M ath. Soc. Japan (3) 15. 149-154 (1933). The cas co that p is a
natura1 number. was considered before by H . E. BRA \' . Am. J. M a th. 53. 864- 872 (1931).
KAKEYA use s BRA y's result.
1042
general case being obtained by iteration. Now g(z)
hence Pl' ...• P" and 00 are the zeros of the function
lP (z)
f' (z)
= f(z)
= zf'(z) + af(z);
a
+ --.z-'
whose poles are al' ...• an. O. Applying the second part of Theorem 2. we
obtain ~ Sj + 2 ~ tj
n + a. t 2
tk
1. and. consequently
=
"
2;' IIm
I
p,.I ~
= ... = =
(1
)".2' I
n a
1- ~
I
I
Im a,. I
= P P(n-l)
() .2'" I Im a.I.
n
I
The following theorem is obtained by a similar iteration process.
Theorem 5. Let al' .... an be the zeros of the Teal polynomial f (z),
and suppose that g(y)
b o + b 1 y + ... + bmym has real zeros only. lf.
furthermore. Pl ..... pn arc the zeros of bof(z) + bd'(z) + ... + bmf(ml(z).
then we have
=
" Im Py I ~ II
" Im a.I.
.2'1
I
. . . . . . .
(11)
1
Pro of. It is sufficient to consider the linear function g(y) = b o + y.
Taking cp(z)
bo + f'(z)/f(z) and applying the first part of Theorem 2.
we immediately obtain (11).
=
4. We do not know whether the inequality (1) holds true for poly~
nomials with complex coefficients. We can however prove it. if all the
zrros of f (z) are assumed to lie on the imaginary axis. Introducing a
rotation z
ix. our result reads
=
Theorem 6. Let the polynomial f (x) of degree n> 1 have the zeros
an. and let Pl' .... P,,_l denote the roots of f'(x) = O. Supposing
Cl' .... an to be real (which implies the reality of Pl' ... , pn _1). we have
al' ....
1
TI-I
- 1 .2'
n- 1
1
TI
I p,. I :::; -.2'
Ia I.
n 1
y
with equaiity oniy if
or if
b)
Pro of.
al? O. a2 ? O•...• a ? O.
Tl
Since we have
1 TI-I
- 1 .2' p,.
n- I
1
(both sides equalling -an_l/nan. if f(x)
sufficient to prove
D (f)
1
TI
TI
= -n
1
2;'a,..
.
.
.
.
•
.
(12)
= anxn + an_1X n- + ... ) it is
= -.2'IP
(a,.)- - 1
n
nI
.
1
1
TI-I
.2'
1
lP (p,.)?
o. . . . .
(13)
1043
where
~
cp (x)
x if x;:: O.
= t (x + Ix I) = ~ 0
if x ~ O.
In case a) we have also f3l <: O..... f3n_l <: O. and consequently D
The same holds true in caSe b). Now suppose
0< k
< n.
al :>
O..... ak
:>
O.
aHI
=
O.
< O..... an < O.
Stlppose furthermore. that at least one f31 is positive. and not equal to a
multiple root of [(x)
O. for otherwise (13) is trivial. Put
=
n
k
fe (x) = II (x- ay) II (x- e(ly).
k
I
f(x)
+I
= fl (x).
The zeros f3de). ...• f3n-l(e) of f'f!(x) are continuous functions of e for
for 0 < e <: 1. Let f3l' .... f3 j (j :> 1 )
he the positive zeros of ['(x). and suppose h :> 1. [(f3l) "# o..... [(f3h) "# O.
[(f3h+l)
[(f3j) = O. The zeros f3de) • .... f3h(e) are increasing. if
(! decreases from 1 to O. For. by differentiation of the relations
o <: e <: 1. and differentiable at least
= ... =
with respect to
e. we obtain
whence it follows
c;:;/ < 0 (i = 1. .... Tt; 0 < e <: 1). Furthermnre
f3h+1 (e) •...• f3 j (e)
élre constant. Hence
J
J
1=1
1=1
2 Pdl) < 1,' f3dO)
~
n-I
:E
cp (f3y (0)). .
.
.
.
.
(14)
~=I
Now considering the expressions D{fdx)} and D{fo(x}}. we observe
the contributions of the a's to be equal for both. 50 by (14). we have
D If(x)l
> D Ifo (x)l·
.
.
.
.
.
.
.
The polynomial [o(x) belonging to case b). we have D{fo(x)}
Now (15) proves our theorem.
(15)
= O.
Theorem 7. Let the palynamial [(x) af degree n> 1 have the n real
zeras al <: a2 <: ... <: an. and let f3l' .... f3n-1 denate the ze ros af ['(x).
Let the functian 1jJ(X) be convex in the interval al <: x <: an. Putting
D
(tp;
f)
1
= -n
n
1
n-I
:E tp (a v ) - --1 2
I
n-
I
tp
(f3).).
1044
we have D(ljJ; f)
x <: an 8).
::>
O. D(lf; f)
=0
holds only i[ ljJ(X) is linear [or
al <:
Proof. Let)ll < /'2 < ... < ym denote the set ar. ... , an, {Jl' ... , {Jn-l,
arranged in ascending order (each number of this set is taken only once).
It follows th at )11
al' )lm
an. We can evidently construct a convex
function lf* (x). satisfying lfo ()lï)
lf ()lï) (i
1. ... , m), which is linear
and continuous upon the intervals
=
=
=
=
x::S; 1'2' 1'2 ~ X ~ 1'3, •..• rm-2 ~ x ~ rm-l. rm-l ~ x.
This function can be represented as
'Po (x)
=
m-l
~ À. i I X-ri
i=2
1
+ C x.
À. i ~
O.
. (16)
According to Theorem 6, we have
Dllx-ril: f(x)I=Dllxl:
and by (12) D(x; f)
f(x+ri)I~O.
= O. Hence
m-l
D ('P; f) = D (Vl". f) =
~ À. i
2
Dil X-ri I; fl
+ CD (x; f) ~ O.
(17)
Now suppose al < an, and suppose th at lf(X) is not linear up on the
interval al <: x <: an. The open interval al < x < an containing at least
one root of f'(x)
O. we observe that m > 2. and that lfO(X) is not linear
upon al <: x <: an. Hence at least one term on the right~hand side of (16).
say the term with i
k, is non~linear, and consequently À.k > O. By theorem
6 it follows. that }.kD(1 X-)lk I; f) > 0, the roots of f(x) occurring on both
sides of )Ik. Now (17) yields D(lf; f) > O. which proves our theorem.
KAKEYA'S result. mentioned in section 3, is contained in Theorem 6. This
follows from the convexity of yP for y > 0, p ::> 1 or p <: 0, and of - yP
for y > 0, 0 <: P <: 1.
=
=
We are indebted to Mrs. T. VAN AARDENNE-EHRENFEST for some
valuable remarks.
8)
This includes the case al =a2
Eindhoven, September 1946.
= ... = a n .