PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 40, Number 1, September 1973
ON A PROPERTY OF RATIONAL FUNCTIONS. II
Q. I. RAHMAN
Abstract.
It is shown that if /-„(z) is a rational function of
degree«such that/-„(0) = 1, lim|2|_,oc|i'„(z)|=0 and all its poles lie in
|í1|^|z|^lthenmax|2|=p<|r1||rn(z)|^l/(l-p").
If rJz) is a rational function of «th degree
(1)
r„(z) = u(z)jv(z),
u, v polynomials
with
(2)
deg u(z) < deg v(z) = n,
rn(0) y¿ 0
then [1] for 1 </>5j2 and arbitrary complex z0,
(3)
lkX,VP = (^ j_K(z0 + poi» dtp)
can be estimated nontrivially from below in the maximal regularity circle
around z0 exclusively byp, \rn(z0)\, p, n and by the maximum distance of z0
from the poles. Here we obtain the sharp lower bound for
(4)
lk„IL.,..p=
max |r„(z0 + pé«)\.
Without loss of generality we may suppose z0=0, rn(0)= 1 ; further if the
poles are £,, £2, • ■• , £„ (repetition permitted), then
(5)
ö< idi< ici ^:-:<ia=í.
and
(6)
Theorem
(7)
0 < p < |C|.
1.
With the above normalizations
max
1*1-*.\rn(z)\ ^ 1 -
we have
pn
The example r*(z)= 1/(1 —zn) shows that (7) is sharp.
Received by the editors December 11, 1972.
AMS (MOS) subject classifications(1970). Primary 30A06, 30A40.
Key words and phrases. Rational functions, logarithmic derivative of a polynomial.
© American Mathematical
143
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Society 1973
144
[September
Q. I. RAHMAN
Proof of Theorem 1. Let maX|z|=p|/-„(z)| = .M(/>). For every real œ the
function u(z) —elIM(p)v(z) does not vanish in |z|<p. Hence if
u(z) =1+2
a^\
then
u(pz) -
v(z) =1+2
)t=i
eixM(p)v(pz)
= (1 +
*#
k=\
eixM(P))
(an_x
-
+ (ax -
e"M(p)bx)(pz)
eixM(p)bn_x)(pz)^
-
+ ■■■
é*M(p)bn(pzy
does not vanish in |z|<l. Consequently, the coefficient of z" does not
exceed in absolute
value the constant
term, i.e. ¡1 —eî,xM(p)|_
\—e'"M(p)b7lpn\. Choosing a appropriately,
we get M(p)(l — \bn\pn)^.l or
M(p)^l/(l
— \bn\pn). But \bn\^l since the zeros of v(z) lie in |z|^l.
Hence
M(p)^ll(l -/>»).
The condition (2) is automatically satisfied in the important special case
when rn(z) is the logarithmic derivative of a polynomial 7rn(z) of degree n
with all zeros in the unit disk; hence Theorem 1 gives a lower bound for
maX|2|=p|77^(z)/77„(z)| depending on |7r^(0)/Tr„(0)|, p, n. However, we show
that in this case there is a lower bound which is independent of |7r^(0)/7rm(0)|.
In fact, we prove
Theorem
2.
Let the zeros £,, l,2, ■■• , £„ of the polynomial
degree n be such that 0<\lx\<=\U^<(z)
max
(8)
■"álLl^lnp'
1
The example n*(z)=l— zn shows that (8) is sharp.
Proof of Theorem 2. Let 7t„(z)=">X=o ckzk and set
A = max|77;(z)/7Tn(z)|.
Then
A = max
= max
cx + 2c2pz + ■• ■+ ncnpn
= max
trn(pz)
|*|=i
ncnpn~l
+ (n-
V
c0 + cxpz + c2p2z2 + • • • + c„pnzr
l)cn.xPn-2z
+ ■■■+ 2c2pzn-2
+ cxz«
c0 + cxpz + c2p z + ■• • + cnpnzr
1+
= n
pn xmax
M-i
lc„
+ ■•• +
n
1
i
Cl
.
C2
nc„\pj
2 2 ,
l + — pz + — p Z +
Cn
Cn
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trn(z) of
Thenfor 0<¿p<\£,x\
cApi
n
+ ~P
n
Z
n
1973]
ON A PROPERTY OF RATIONAL FUNCTIONS. II
pn Vax
\'\=p
l+!=iMi\+...+íi(¿r+a(i)f
n
c„ \p I
n c„ \o /
c„ \p /
1 + — z + -z
C0
a.n
by Theorem
145
+•••+
Cg
—Z
Cq
1-p"
1. But |c„/c0|§ïl
since all the zeros of 7rn(z) lie in z^l.
Hence
A2znp»-ll(l—p»).
The problems considered in this paper were suggested to me by Professor
Paul Turan for which I am very thankful to him.
Reference
1. Q. I. Rahman and Paul Turan, On a property of rational functions, Ann. Univ. Sei.
Budapest. Eötvös Sect. Math, (to appear).
Department
Canada
of Mathematics,
University
of Montreal,
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Montreal,
Quebec,