757
Internat. J. Math. & Math. Sci.
Vol. i0 No. 4 (1987) 757-776
SOME THEOREMS ON GENERALIZED POLARS
WITH ARBITRARY WEIGHT
NEYAMAT ZAHEER
Mathematics Department
King Saud University
Riyadh, Saudi Arabia
and
AIJAZ A. KHAN
Mathematics Department
Aligarh Muslim University
Aligarh, India
(Received September 2, 1986)
The present paper, which is a continuation of our earlier work in Annali di
ABSTRACT.
[1]
Mathematica
Greece,
deals
of
vanishing
of
product
and
with
Journal
the
Math.
problem
generalized polars
abstract
homogeneous
Seminar
[2] (EE6EPIA), University of
Athens,
of determining sufficiency conditions for the non-
(with a vanishing or nonvanishing weight)
polynomials
in
the
general
case
when
the
of
the
factor
polynomials have been preassigned independent locations for their respective nullOur main theorems here fully answer this general problem and include in them,
sets.
as special cases, all the results on the topic known to date and established by Khan,
Marden and Zaheer
cited above).
Marden’s
(see Pacific J. Math. 74 (1978), 2, pp. 535-557, and the papers
one of
Besides,
general
theorem
flf2"’’fp/fp+l’’’fq f’l
KEY WORDS AND PHRASES.
the main theorems
critical
on
points
of
leads to an improved version of
rational
functions
being complex-valued polynomials of degree
Generalized
circular
regions,
of the form
n..1
circular cones,
generalized
polars, abstract homogeneous polynomials.
1980 AMS SUBJECT CLASSIFICATION CODE.
1.
Primary 30C15, Secondary 12DIO.
INTRODUCTION.
A few years ago,
the concept of generalized polars of the product of abstract
homogeneous polynomials (a.h.p.) was introduced by Marden [3] while in his attempt to
generalize to vector spaces a theorem due to Bcher [4].
use
of
hermitian
cones
[5],
a
concept
which was
first
His formulation involves the
used
by HSrmander
[6]
in
obtaining a vector space analogue of Laguerre’s theorem on polar-derivatives [7] and,
later, employed by Marden [3], [8], in the theory of composite a.h.p.’s. In all these
areas the role of the class of hermitian cones has been replaced by a strictly larger
class of the so-called circular cones.
This was successfully done
byZaheer [9], [I0],
[11] and [5] in presenting a more general and compact theory which incorporates into
A complete
it the various independent studies made by Hormander, Marden and Zervos.
account
of
the work to date on generalized polars, which fall in the category of
composite a.h.p.’s in the wider sense of the definition of the latter now in use (cf.
[5], [12], [13], [14]) can be found in the papers due
to Marden
[3], Zaheer [5], and
N. ZAHEER and A.A. KHAN
758
the authors [1], [2].
Generalized polars with a vanishing weight as well as the ones
with a non-vanishing weight have beea considered in the first two papers, while the
(resp. the fourth) deals exclusively with the ones having a vanishing (resp. a
third
But all have a common feature that the factor polynomials
non-vanishing) weight.
involved in the generalized polar of the product have been divided into two or three
groups, each of which is preassigned a circular cone containing the null-sets of all
polynomials belonging to that group.
a
with
or
vanishing
Our aim here is to consider generalized polars
where,
weight
non-vanishing
a
general,
in
no
factor
two
polynomials are necessarily required to have the same circular cone in which their
null-sets
In
lie.
must
fact,
take
we
the
up
of
problem
general
determining
sufficiency conditions for the non-vanishing of generalized polar (with a vanishing or
a
non-vanishing weight) where the factor polynomials have been preassigned mutually
independent locations for their respective null-sets.
Our main theorems fully answer
problem and include in them, as special cases, all the corresponding
general
this
results on the topic known to date and established in Marden
authors [I], [2].
[3], Zaheer [5] and the
One of the main theorems of this paper leads to a slightly improved
form of Marden’s general theorem on critical points of rational functions [7].
2.
PRELIMINARIES.
E
Throughout we let
A
zero.
istic
mapping
vector-valued a.h.p,
V
and
e
E
of degree
denote vector spaces over a field
is
n
if (for each x, y
n
r.
P(sx + ty)
k=o
where
the
Ak(x,y)
Ak(X,y
coefficients
[6],
(cf.
V
e V
called
[15],
K
[16],
of characterand
[9])
a
E)
skt n-k
s, te K,
depend only on
x
and
y.
We shall call
P an
if V is taken as K (resp. an algebra).
a. h. p. (resp. an algebra-valued a. h. p.
Pn the class of all vector-valued a.h.p.’s of degree n from E to V
We denote by
(even if
to
from
K.
V
is an
nhpo
The
E
n
algebra) and by P
n
to
V
is the unique symmetric n-linear form
P(x,x,...,x)
such that
[15]
Hille and Phillips
P, for given
P
of
x l,x2,...,x
P(x l,x
for
in
k
x
2
the class of all a.h.p’s of degree
its
P(x)
existence
for all
and
P.(.x
n
from
E
,x 2,... ,x n)
[6] and
x : E (Hormander
uniqueness).
The
kth polar of
E, is defined by
k,x)
P(x
xk,x
x).
The following material is borrowed from Zaheer [17], [I] and [2].
Given
k
: K
and
Q(x)
Qk (x)
Pk Pn k
(k
1,2
,q), we write
Pl(X)P2(x)"" Pq(x)’
(2.1)
Pl(X)’’" Pk-l(X)Pk+l(X)’’" Pq(X),
(2.2)
and
(Q;x ,x)
q
Z
k=l
mkQk(X) Pk (Xl ,x)
V x,x
(2.3)
THEOREMS ON GENERALIZED POLARS WITH ARBITRARY WEIGHT
(Q-Xl,X)
and define
algebra-valued generalized polar
as an
759
Q(x)
of the product
q
[5].
r.
scalar
The
is called its
mk
k=1
weight.
terminology [5].
that
Q e
nk-I
in
x
K, so as to conform with the existing
+ n
we ren + n +
As in Hille and Phillips [15], if n
,
,
Pn’
Qk
e
Pn_nk
degree
and of
x
in
n-1
<
I,
L
,
LP
Obviously,
LP
nomials
g
L(P(x))
k
_<
V
K
n
x
in
is an
I.
[18] and [I] and a polynomial
by
g
V x
(2.4)
E.
the product of the poly-
(LQ)k
and the corresponding partial product
LQ
is given by
(Q:x l,x)
Therefore,
q.
In the notations of (2.1) and (2.2)
P.
Pnk
k
K
E
we define the mapping LP
algebra-valued a.h.p, of degree
an
and of degree
x
in
is
Given a nontrivial scalar homomorphism
(LP)(x)
q
2
Pk(Xl,X)
and
algebra-valued a.h.p, of degree
P e P
will be
V
used in special reference to the case when
call
’generalized polar’
The Term
(achieved by deleting the kth factor in the expression for LQ) is given by
This
LQk.
immediately leads to the following
REMARK 2.1.
The algebra-valued generalized polar
(LQ;xl,x)
and the generalized polar
k’s
same
L((Q;x l,x))
K
K
L
adjoining
to
m
element
an
K
subsets of
K
and of
Izl
z
+ (a2 +
o
a+ ib e K (a,b
b2) I/2 in
[19]
K (i),
K
such that
o
E
K
o)
we
analogy with the
[5] and [2 I] achieved by
having the properties of infinity, and, by D(K),
K
We denote by
K
and
the projective
class of all generalized circular
convex
V.
For any element
K.
)/2,
(z +
ib, Re(z)
complex plane.
the
on
contain a maximal ordered subfield
must
is the unity element in
a
define
Q(x)
with the
is an algebraically closed field of characteristic zero, then we know
and [20] that
-i2
(LQ)(x),
(LQ;x l,x)
for every nontrivial scalar homomorphism
where
of the product
satisfy the relation
m
If
(Q;xl,x)
of the corresponding product
D(K)
field
K.
The notions of Kregion (g.c.r.) of
o
are due to Zervos [21], but the definitions and
a brief account of relevant details can be found in
[5].
For special emphasis in the
C of complex numbers, we state the following characterization of D(C):
The
nontrivial g. c. r. ’s of
are the open interior (or exterior) of circles or the
open half-planes, adjoined with a connected subset (possibly empty) of their boundary.
field
The
g.c.r. ’s
of
C,
with
all
or
(classical) circular regions (c. r.)
In vector space
the terms
Given
a
E
no
boundary
points
included,
over an algeraically closed field
K
N
circular cone E (N,G)
o
E (N G)
o
E2
of
and
a
circular
by
x,y
)N
TG(X
termed as
of characteristic zero,
’nucleus’, ’circular mapping’ and ’circular cone’ are
nucleus
are
of
y)
mapping
G
N
due
D(K),
to Zaheer [5].
we define the
N. ZAHEER and A.A. KHAN
760
where
{sx + ty
TG(X,y)
REMARK 2.2.
(so that
K
of
oone in
{sx
E (N,G)
o
A
for some
D(K),
g
E (N,G)
is of the form
o
A}
K; s/t
g
s,t
o
are any two linearly independent elements of
Xo,Yo
G(xo,Yo
and
E,
A.
that any two (and, hence, any finite number of)
We remark [5]
(III)
E.
+ ty
o
where
{(Xo,Yo )}
N
with
G(x,y)}.
K; s/t
g
2, then [I0] every circular cone
dim E
If
s,t
(I) [I]. If G is a mapping {tom N into the class of all subsets
G(x,y) may not necessarily be a g.c.r.), the resulting set E o (N,G)
will be termed only a
(II)
@ o
circular
cones can always be expressed relative to an arbitrarily selected common nucleus.
THE CENTRAL THEOREM.
3.
Unless
mentioned
characteristic zero,
K.
over
the
subspace
set
product
E
of
generated
L ix, y]
the
set
field
of
an algebra with identity
V
K, and
L ix,y]
We denote by
C.
E,
of
x
and
y
of
all
ordered
elemeats
by
L ix,y] (i.e.
closed
algebraically
an
denotes
a vector space over
of complex numbers is denoted by
field
The
K
otherwise,
E
2
L ix,y]
and by
the
the
pairs of elements from
[x,y] ).
In this section we establish the central theorem of this paper,
which gives
sufficiency conditions for the non-vanishing of generalized polars having a vanishing
or a non-vanishing weight
problem mentioned in the
the general
and which answers
Apart from deducing the main theorems of the authors proved earlier in
[I] and [2] the present theorem applies in the complex plane to yield an improved form
In the following theorem we take, without
of a general theorem due to Marden [7].
loss of generality (cf. Remark 2.2 (III)), circular cones with a common nucleus.
Consistently, we shall denote by Z (x,y) the null-set of an a.h.p. P (with respect
P
to given elemeats x,y
E), defined by
introduction.
Zp(x,y)
For k
3.1.
THEOREM
{sx + ty
* ols,t
1,2,
m
k
If
>
is the generalized polar
(Q;Xl,X
for
o
k
<
p(
<
q) and
m
k
linearly independent element
q
x
N
E-
n
2
u
k=1
Eo
[x,x I], x
let
q,
(k)
u
<
o
for
x,x! of
Ts(xo,Yo)
Xo + 6Yo
Pk
e
and
where
of
k
p,
then
E such that
(Xo,Yo)
E
for all
the )roduct
>
o}.
and
Pnk
ZPk(X,y) c_ TGk(X,y)
circular cones in E such that
k.
e K; P(sx + ty)
x
kjot
Eo(N,Gk)
(x,y) e N
be
and for all
Q(x) (cf. (2.1)-(2.3))with
(Q;x l,x)
q
E- u
k--I
$
0
E(k)o
is the unique element
for all
and
in
THEOREMS ON GENERALIZED POLARS WITH ARBITRARY WEIGHT
q
{0 e
S(Xo,Yo
E=I
q
I
k=l
Ok)
S(Xo,Yo
m must belong to
OkeGk(Xo,Yo)}.
y/6).
mk)/(O
Y/6
in the case when
(k)’
q
q
u
m
k
Let us note that 0
REMARK.
and
Kml
mk/(O
761
Gk(Xo,Yo). Also the hypothesis ’x nk=l E o
would be all of K and the theorem would
S(x ,yo
o
# m
For
is necessary.
k=l
otherwise,
Let
PROOF.
q
(u
x
x,x!
E(k))o u Ts(xo,Yo)
,B,Y,
scalars
independent
and
o
x
q
E
(k)o
BY
(with
6
o
q
k=l
S
BY
and
(Xo,Yo)
0).
q
(u
m
says that
m.
y/
case
m
m
q
due to
y/
(since
Gk(Xo,Yo),
k=l
So that
(Xo,Yo)"
m
and
YXo + Yo"
x
u S(x o,yo ),
Gk(X o,yo ))
y/
and m
S
must belong to
and
This is trivial when
It is obvious also when
However, in
Gk(X o,yo)
/
We claim that
is the unique element
Xo + BYo
k=l
the notation in (2.5).
that
such
Then there exists a unique set of
x
a/B
E
(Xo,Yo)
where
such that
implies that
x
Obviously, the choice of
O)
become uninteresting.
of
elements
(cf. definition of nucleus [5]).
[x,x I]
in
linearly
be
belongs to
definition of
the
/B
m
m
in all cases
The fact that K is algebraically closed allows us to write, for each k-- 1,2,...,q,
n
Pk(SX
N k
+ tx I)
j=l
y
(jk s
J kt).
n
k
3=I jk
Pk (x)
Since
for all
j
#
If we set
1,2,...,n k.
we have that for each
k,
for all
0
Yjk/jk
Ojk
e U
(Gk(Xo,Yo))
U(Gk(Xo,Yo))
and, further, that
homographic transformation [21]
fore
(3. I) and the
are
Ko-COnvex
j=l
U(Ok)
Let us write
g.c.r.’s of
U(Gk(Xo,yo ))
give
U(Gk(Xo,Yo))
(0
for
Pk e Gk(Xo,Yo)
(Ok Y)/(-BOk
+ )
q)
jk
<k<q
k,
U(O)
(I/nk) Ojk
<
0
[5], we conclude that
m
for
(3I)
U
is the
Y)/(-BO + a).
There-
where
K,
k
E
This implies that there exist elements
k
n
given by
Km
of
o
k(say)
.....
j
K -convexity of
n
1,2
for
k
then, using the same technique as
in the beginning of the proof of Theorem 2.5 due to Zaheer
Ojk
<
k (I
k =1,2
.....
(3.2)
q.
such that
k=l,2
(33)
q.
n
k
mk k
We now claim that
simultaneously.
k
l
j=l
(mk/nk) 0jk
I + v2 +’’’+
q
O.
V
k
(3.4)
1,2,..,q.
First we notice that the
For, otherwise, Y/ would belong to all the g.c.r.’s
k’S
cannot vanish
Gk(Xo,Yo)
N. ZAHEER and A.A. KHAN
762
for
k=l,2,...,q,
which in turn would imply that
Yxo +
x
q
T
n
G
k
k=l
6Yo
This contradicts the fact that
(Xo,Yo)
q
f n
x
k;1
said
n
q
k=l
E(k)o"
E
(k)
o
Therefore, in order to establish the
claim, it remains only to deal with the case when at least any two of the
do not vanish (since the claim is obvious otherwise).
I+2
suppose on the contrary that
+ q
+
O.
k’S
Now, with thts assumption,
Then equations (3.3) amd (3.4)
would imply that
q
l
mk(6
k=l
,
a/8
Since
O
Y)/(-8 O + a)
k
k
that
see
we
8
0
0.
and,
the last equation can be
consequently,
written as
q
r.
m [-618 +
k
k=l
{(a618) -‘(}1(-8
+ =)]
k
0.
Therefore,
q
(alS)
where
A
sd
8‘(
q
l
k=l
z mkI(-8 Pk +
k=l
O.
mk/(a/8
6
ink)
6
or
0
q
(kE=
8"(
(a/8)
O,
-‘(
(kE=l ink)"
we get
q
z mkl(al8
Z
pk
k=1
k=1
mk)l(alS-‘(l),
Gk(Xo,yo) for k 1,2,...,q, (XoYo) e N o L 2 [x,x I} and
‘(Xo + 6y o. This implies that a/8 e S(Xo,Yo) and, hence, that
aXo + 8Yo Ts(Xo,Yo ), contradicting the choice of x already made.
+ v2 +
+
O. But we know ([5] or [3]) that
q
nk
q
q
#(Q;xl,x)
].
(mk/nk)
Pk(X)
Pjk
[k=II j=II
where
x
mk’
q
68
pk
q
V
k=l
Or,
That is, irrespective of whether
x
q
r.
(618)
a)
0
k
e:
Therefore
k
q
I
k=l
Since
Pk(X)
0
If we take
when
x.t
varies
for all
q
l m
k
k=l
freely
k
0
in
q
k).
and since
k=l
v
Pk(X)
+
(due to (3.4)).
+ v
q
O,
in the above theorem, the set
i[x
O
,yo
the proof is complete.
S(xo,Yo)
subject to the condition that
remains unchanged
x.i
q
C=I
(k)
E0
THEOREMS ON GENERALIZED POLARS WITH ARBITRARY WEIGHT
763
.
In order to get a simpler and more interesting version in
xl varies freely
q k
We do precisely
n Eo
over all of E it is desirable to further assume that
k=l
this to obtain the following theorem which deals exclusively with generalized polars
whicht
having a vanishing weight.
o
q
q
(k)
E
k=!
r.
and
o
m
k=!
k
0
(Q;x ,x)
then
q+!
elements
the cone
E such that
of
x,x
x
E-u
g
-
defined by
for all
(x
,yo
o
PROOF.
x,x
q:u"+l E o-k- then
k=l
x!_ YXo + Yo and,
are
k=l
from Theorem 3.1.
(k)o
0;
(x
E (N
o
’Gq+l
is
Gk(Xo,Yo)}
gk e
E
k=l
(k))o
o
yo)
u
Ts(Xo,Yo ).
(since
E
x
[.2 [X,Xl]
e N n
Gq+l(Xo’Yo)
S(xo,Yo
q
(u
x
and
(q+l)
independent elements such that
linearly
in the present set up,
n q E
x
where E o
there exists a unique element
x
That is,
any
E(k)
o
mk/(P -Pk
k=I
N.
If
for all linearly independent
q
KI
O
Gq+l(Xo,Y o)
k=l
0
and
such that
q
k=IE
0).
m
k
Now the proof follows
As application of Theorem 3.1 in the complex plane we prove the following
corollary which, apart from generalizing the two-circle theorem and the cross-ratlo
theorem of Walsh [22] (cf. also [7], Theorems 20,1 and 22,21, improves upon Marden’s
general theorem on critical points of rational functions ([7] or [23] and [24]).
the following, Z(f) denotes the set of all zeros of f.
For each k
COROLLARY 3.3.
to )
o" degree
p
and if
k=o
Ck,
n
k.
then every
P+I
k=o
n
PROOF.
k
in
--@
x
o
k
Z(fk
such that
Ck
for
finite zero of the derivative of the
(q
fq+l(Z)fq+2(z)’’’fp(Z)
or
-
P
{p
<
(from
p
0,I
k
rational
funon
p)
E mk/(p
[ k=o
k
according as
0;
pk
<_
q
or
k
Pk
>
e
Ck}
q.
In view of Remark 2.2 (II), the sets
Eo(k)
where
D()
be a polynomial
Ck, where
Cp+
and mk
C
fk (z)
let
fo(Z)fl(z)’’’fq(z)
f(z)
lies in
If
O,l,...,p,
In
(I,0)
such that
y
o
Xo
Eo(N,Gk)
(0 I)
(I,0)
N
{SXo + tYo Ols,t C; s/t
((Xo,Yo)} and Gk(Xo,Yo)
p
k=o
E
"
ko"
Letting
fk(z
k}, 0 _< k _< p,
C
Ck,
are circular cones
.k ajkzj,
j =o
0
_< k_<
p,
we
N. ZAHEER and A.A. KHAN
76
define the mappings
Pk(X)
Then
Pk
by
C
Pk:
+
Pk(S xo
is an a.h.p, of degree
ZPk(Xo,Y o) c_C TGk(xo,Y o)
and because Z(f
k)
__c
k
2.
such that
C
to
(s,t) e C
V
fk(s/t)
(s,t)
Vx
(3.7)
0
(Xo,Yo).
T
C
t
nk-j
This is so because
O,l,...,p.
+ ty o)
Pk(SX
2
C
from
k
n
Pk(X)
ajk t
j=o
n
k
for
Enk
tYo
Now we consider the generalized polar
k
of the product Q(x) of these a.h.p.’s, with m
n or-n
according as k < q or
k
k
k
k > q.
If we take x
and t
(I,0) (so that s
o), we see as in
G
k
(Q;Xl,X
Xo
[5], that
(I/nk)3Pk/S
Pk(Xo,X)
fr
0,I
k
,p.
If we set
Fk(Z)
equations (3.7) and
n
t
P
r.
m.
k=o
t
t
x
O
n
p
k=o
and 6
(with Y
the element x
+
n
+ n
m
p
(3.8)
and define
(k)
EO
P
7.
Eo(P+I)
and
r.
Fk(S/t
fv (s/t)
[f
Fk(S/t)]
k=q+l
q+l
(s/t) ..f
y/6
),
Eo(N,Gp+
(s/t)j2
(3.9)
(Q;Xl,X)
it implies that
is linearly independent to x
=_
p
Theorem 3.1 is applicable in the present set up
since
so that
0,
(s,t)
m
(mk/nk) Fk(S/t
q
m
k=o
x
+
nk-I f(s/t)
fo(Z)fl(z)’’’fk-l(z)’f(z)’fk+l(Z)’’’fp(Z)’
(3.8) imply that, for x
(s,t) e C2,
#(Q;xo,x)
Since
o
(I/nk)t
o
such that
x ff u
p+l
k= O
{SXo + tYo ojs,t
0
E
whenever
(k)
o
where
Gp+ l(xo,yo)}
C; s/t
and whe re
Gp+l(Xo,Yo)
CO
CO
P
e
Ii
[ k=o
mk/(O
is,
#(Q;x O ,x)
0
k
O; O k e
Gk(Xo,Yo)}
k
O; O k
C
P
E mk/(O
[ k=o
Cp+
That
-O
for all elements
-O
k}
(due to the choice of m
k
x
(s,t)
for which
made above).
t # 0
and for which
765
WEIGHT
THEOREMS ON GENERALIZED POLARS WITH ARBITRARY
p+l
u
s/t
C
k=O
s,t e C such that t
for all
0
0
p+l
u
s/t
and
(3.9) says that f’(s/t)
k. Finally,
C
This establishes the corollary.
k.
k=o
In the special case when the g.c.r.’s
are specialized as
C
k
the above corolthat
we
claim
the closed interior or the closed exterior of circles,
(I)
REMARK 3.4.
lary
essentially
reduces
following
Marden’s Theorem 21,1,
the
C
of Marden
21,1
Theorem
to
If
arguments:
[7].
This
the
regions
taken to be
are
k
then Lemma 21,1
[7]
of Marden
upheld by
is
OkCk(Z) <
the
of
0
and the succeeding arguments
p+l
the
that
show
therein
region
C
u
k=o
corollary is precisely the region
in our
k
satisfying the p+2 inequalities 21,3 in Marden’s theorem.
placed by
In what follows we show that Corollary 3.3 holds as such when C is reK, provided the term ’derivative’ is replaced by ’formal derivative’. We
know
([5]
(II)
n
by
[12])
or
the
that
f’(z)
polynomial
k
k=l
the polynomial
formal derivative of
the
ak zk
f(z)
k=o
from
K
ak zk-I
to
K
called
is
and that
n
).’
(flf2...fn)’
fl/f2 (fl
quotient
flf2...fk_l f fk+l’’’fn
If we now define the formal derivative of the
are polynomials [12].
f.l
where the
k--1
being
polynomials)
to be given by
(fl/f2)
)2
(fl,f2_f f,)/(f2
2
then the formal derivative of the quotient
(z)...f (z)
f (z)f (z)...f (z)/f
p
q+l
o
q
is given by equation
(3.9).
q
P
).’
Fk(Z
k=o
)’.
k=q+l
Fk(Z)]/[fq+l(Z)...fp(Z)] 2.
f’ (z)
In view of the definition of the formal derivative
from K to K and of formal partial deruatives %P/Ss
2
(3.10)
of a polynomial
f(z)
P(s,t) from
of a polynomial
K [5], we can easily show that Corollary 3.3 stll holds when C is replaced by
K. The proof proceeds exactly on the lines of the proof of Corollary 3.3, except only
that we replace C by K all along. Let us point out that the expression (3.10) is
K
to
precisely the formal derivative of the function
f(z)
in (3.5) and it justifies the
validity of steps (3.8) and (3.9) in the proof of Corollary 3.3.
We remark that in Corollary 3.3, we must add the hypothesis
(Ill)
n
P C
k=o
order
in
n
o
+
have
to
+ n
then fixing
n
q
P
q+l
a
+
P
o
P,:
k=o
and, hence, that
p
p
f(z)
with
Ck’S
for
the
rational
is
P
mkl( k
Cp+ C.
’,
common to all the
For, if a point
p
>-’ m
0)
in (3.6) we see that (since
k
k=o
result
nont rivial
+ n
k
k=o
mk)l(
)
0 V
C
function
N. ZAHEER and A.A. KHAN
766
It has been shown by Marden [7] that Walsh’s cross-ratio Theorem 22,2, is a
(IV)
special case of Marden’s general Theorem 21,1, but only in terms of closed interior or
D(C)).
closed exterior of circles (a proper subclass of
Whereas, our Corollary 3.3
g.c.r.’s. In fact, applying our corollary in
C.’s taken as g.c.r.’s, we conclude that every
validates Walsh’s theorem in terms of
the set up of Walsh’s theorem with
fl(z)f2(z)/f3(z)
finite zero of the derivative of the function
u
lies in
where
C
IV
4
e
CInl(p pl) + n2/(p p2) n3/(p p3)
In veiw of Lemma 4.2 of the next section, we see that
G
HA
4
0;
for
K
Ci,
Pi g Ci}.
,
A
n2/n
and that
Ci,
i
C
4
i=I
(C
{})
4
u
3
C
{’})
(RA
i
i=l
3
u
C i,
i=!
where
m,CI(P P3’92’Pl
IV
RA
n2/nl;-
P
i
C
i
Consequently, every finite zero of the derivative of the said function lies in C
C3U C,
(V)
This shows that 6In
improved ve60n
0
uC2u
W0/h’ cao-azut/0
It may be observed that an improved form of Walsh’s two-circle theorems in
its complete form
[5],
([5], Corollaries 2.8 and 4.3) may also be obtained from the above
To this effect we apply Corollary 3.3 in the set up of Zaheer’s Corollary
corollary.
4.3
{}.
RA
C
where
D.’s
the
with
by g.c.r.’s
replaced
C
i
such
m e C
that
conclude that every finite zero of the derivative of the function
3
in
where
C
n C
2,
fl(z)/f2(z)
and
lies
i
C
3
{a
Inll(P
-pi
CIP
(nlP 2
IV
of Zaheer
[5], the region
In the case when
C
n
3
2
Pl)/(n 2
C
and
2
O; P i
Ci}
ni) Pie C.}..
are taken as the regions
is precisely
n
holds, but in this case the region
in [5].
4.
n
C.’s
We point out that, in case the
done.
-n21(P -p2
C
3
D(c3,r 3)
n C
,
D.1
of Corollary 4.3
(cf. notation there) and we are
the conclusion just drawn still
i
2
is empty, and we are done with Corollary 2.8
THE CASE OF ALGEBRA-VALUED GENERALIZED POLARS.
Our aim in this section is to obtain a more general formulation of Theorem 3.1
that
could answer the corresponding problem for algebra valued generalized polars
having an arbitrary weight.
In fact, it will be shown that, whereas the main theorem
of this section does include in it the main theorem of the preceding section, it also
incorporates
into
it
a variety
of
other known
results.
First, we describe some
concepts and establish some results that we need in this section.
and [2] for the following material.
We refer [17], [I]
THEOREMS ON GENERALIZED POLARS WITH ARBITRARY WEIGHT
767
A subset of M of V is called fully supportable(initially termed as ’A-supportable’
by Zaheer [9]) if every point
V
subspace of
M
outside
is contained in some
is a unique nontrivial scalar homomorphism L on V
for every v e M [18].
such that
ideal maximal
e V
In other words, for every
M.
which does not meet
L()
M, there
L(v) $ 0
but
0
If M is a fully supportable subset of V, then M is a supportable
(regarded as a vector space), but not conversely (for definition of
supportable subsets see [6]). We remark that the complement in V of every ideal maximal
of V
subset
subspace of V
fully supportable subset of
a
is
M
supportable subset
REMARK 4.1.
(take V
K,
K- {o}
M
the set
Zp(x,y)
K
K
to
P
of
p e
K
E,
M}.
(4.1)
is the only nontrivial scalar
Ep(x,y),
as given by
(P’
K
(4.1),
stands for the cross-
PI’ P2’ P3
with respect to given distinct elements
K(for
>
A
Given an element
and a fully
n
as defined in the beginning of Section 3.
and it designates a unique element in
LEMMA 4.2.
P P
is the only fully supportable subset of
next few lemmas, the notation
the
ratio of an element
let us
x,y
K in the definition) and the corresponding set
becomes the null-set
In
Given
e K; P(sx + ty)
Since identity map from
on
homomorphism
01s,t
{sx + ty
Ep(x,y)
V.
V, we shall write, for given
of
PI’P2’P3
K
e
definition and other relevant details see [5]).
0 in K and g.c.r.
’s G.
e
D(K)for
i
1,2,3
define
HA
{p e
Kmll/(p -pl)
RA
{P e
Km I(p’ P3’ P2’PI
+ A/(p
(I+A)/(p
-p2)
O;
-p3
Pi
e
Gi}
and
If
,
G n G n G
3
2
PROOF.
u G
2
u
3
In
order
to
u {m}
and
l/(p
(RA
Gi
i=l
u G
3
G’}’l
Pi
then
{m})
(H A
A;
prove
Pl
A/(p
pl +
the
Gi
(i
p2)
{m}
u
3
i=1
lemma
it
1,2,3),
(I + A)I(p
G..l
sufficient
is
to
show
that,
if
the equation
p3
o
(4.2)
holds true if and only if the equation
(P’
holds true.
P3’ P2’
of
cross-ratio.
coincide
K
PI’ P2’ P3
This is obvious in case of (4.3) due to the definition
K
case of (4.2), this follows from the fact that if any two of the
In
and
G n G
dicting that
elements of
(4.3)
First, we claim that none of these equations can hold unless
are distinct elements of
P.’s
1
-A
P
if
2
n G
.
(4.2)
3
holds
then
all
the
three must coincide, contra-
Therefore, we assume that
PI’ P2’ P3
and so we divide the proof into the following two cases:
are distinct
N. ZAHEER and A.A. KHAN
768
Case (i).
In this case, since 0,
01 02
0
are distinct
3
the equation (4.2) holds if and only if
K,
elements of
.
01, 02, 03
03)1(0
(01
0
I)(0
+
03
(02 03)1(0
02)(03
0
I)1(0
(0,
This is true if and only if
01 )(03
03 02 0
(4.3) holds.
Case (ii).
One of the
are distinct elements of
0
0
0
3
.
or, if and only if
(0
02)(0
2
-.
That is, (4.2) holds if and only if
.
In this case,
Ol.’s is
K with only one of the
.
let us point out that
O.’s
being
0,01,02,0 3
Therefore, the
equation (4.2) is equivalent to the equation
Xl(0
0
/Co
01
I/(0
01
2)
(I + X)l(0
(4.4)
( + )/(o
(4.5)
o
according as
(4.6)
(03
are,
I(0
, 02
01
or
pl)/(p
,
pl
m,
03
equivalent
respectively,
under consideration.
(4.6)
02
the
to
02)/(0
(p
and
,
Now, the equations (4.4)
equations (0
02
respectively.
0
,
I)
02)/(03
the
in
respective
cases
The definition of cross-ratio implies that each of the equations
(4.2) holds if and only if
(4.3) holds.
(0,
0
0
3
That is (4.2) holds if and only if
k.
2,01
Cases (i) and (ii) complete our proof.
q
any r,
<
r
k
1,2,3,
< q)
i
for k_< p(P
0
K,
and a
p,
,
>
m
such that
<
D(K) for
Let G i e
LEMMA 4.3.
1,2
we
e K- {0}
0
for
k
>
for
k
p. Giuen
q
q
Pr+
k
m
k
<
define
KI kZ=l mkl(P
{0
R
and Zet
and m
(k=IZ mk)/(P -);
pk)
e
,0p
G2;
0
.,0
p+l
q
e G
GI;
01 ..,Or
3},
and
KI
{P e
H
r
mk,
kl
then
If
2
(})
0
E
ink,
3
k=p+l
(H*
G.
iffil
{})
(H
u
3
i--I
(i=1,2,3) such that
Al/(p -pl)
0
+
G
i=l
Ai)l(p
-);
q
u
,
E
AilCP -01)
i=l
l
k=r+l
A
(R
PROOF.
3
r.
P
Z
with A
3
u G
2
mk.
{})
Gi,
u G 3 u {}
then
n
With
3
i=l
3
U
G
iffil
there
01
E
Gi}
G i,
i.
exist
elements
0
i
G
i
and
A2/(p -p2) +A3/(p -p3)
(A + A +
2
A3)/(p
-r.).
(4.7)
THEOREMS ON GENERALIZED POLARS WITH ARBITRARY WEIGHT
Obviously,
O
"
Pl’ P2 P3’
If we choose elements
for k
1,2,
for k
O
1,2
q, such that
r
,p
r+l,.
for k
k
Ok,
769
,q,
p+l,
then equation (4.7) can be written as
r
p
r.
mk/(p -pk
k=l
+
q
)".
mk/CP
k=r+l
-p
q
r.
+
k)
Z mk)/Cp -)
k=l
mk/CP -pk)
k=p+l
or
q
q
Z
mk/(p
k=l
r.
p)
{})
p e (R*
This implies that
k)l (p
m
k=l
u
3
).
Hence
G..
i=l
(H*
{})
3
u
(R*
G.
{))
u
i=l
For
the
reverse
3
(4.8)
G..
i=l
if
containment,
p e
(R*
{})
u
3
i=I
O e G
for
u G
k
2
r,
<
{}
G
Pk
3
e G
2
and there exists elements
<
r
for
k
<
q
p
Pk
e G
for
3
O
p
(I
k
<
k
<
<
k
<
q
G i,
then
q)
such that
O
k
eG
and such that
q
r.
k=l
r.
mklCP pl)
k=l
ink)/( p
;).
p) +
Z
k=r+l
Therefore,
q
r
E
mk)/(
k=l
B
Since
that
G
i
e
@p(Gi)
D(Km)
is
p
Z
k=l
p)
p
and
Ko-cOnvex
mk/(p
+ B
G
2
+ B3
p(G2)
p)
p(G 3
we conclude that
such hat
Bi/Ai
Bi/Ai
I/(P
#p(Gi)
i
O) +
Z
k=p+l
mk/(O
(say).
p)
(4.9)
for i
1,2,3, we see from the definition of g.c.r.’s
i
for i
1,2,3 [5]. In view of this and the fact that
p(G
1/(P.
q
mk/(O
-O).
for
for
for
k-- 1,2,...r
k
r+l,...,p
k
p+l
q,
1,2,3. Therefore, there exist elements p e G
i
q
i
Now, (4.9) implies (since A + A + A
mk) that
3
2
for i
k
Z=
N. ZAHEER and A.A. KHAN
770
3
3
l A I (p
i
i=l
p
e (H*
which says that
r.
i
Ai) / (
i=l
-{})- u 3 G
)
Hence
i=l
(R*
u 3
{})
(H*
Gi
i=l
u 3
{})
(4.10)
G..x
i=l
Finally (4.8) and (4.10) prove our lemma.
Next, we take up the most general theorem of this paper, which we establish via
application of Theorem 3.1.
THEOREM 4.4. Let M be a fully supportable subset
let
Pk
e P*
n
TGk(X,y)
and
k
<
x, of
for
0
V
k
where
PROOF.
>
S(xo,Yo
f
L()
0
If #(Q;x 1,x)
k.
is the
>
e M
1,x)
q
EPk(X,Y)
algebra-valued
0
for
k
_<
L(v)
*
x e E- (u
q
E
k=l
defined
in Theorem
for all v e: M.
0
<
p (p
(k))o
q)
u
3.1.
L
there is a unique nontrivial scalar homomorphism
but
_c
for all linearly independent elements
and
is the set as
1,2,...,q,
k
for
E such that
(cf. (2.1)-(2.3)) with
E’k’o(
k=l
e E-
x.i
Now,
Pk (x,y) _c k(x,y) _c TGk(X ,y)
can be easily shown that
all
for all
then 0(Q;x
p,
e V- M,
If
such that
and
V and,
of
be circular cones in
of the product Q(x)
E such that
Ts(X o,yo ),
Eo(N’Gk)
all (x,y) e N
for
generalized polar
and m
k
E(k)
o
LP
Pnk
k
for all
on
(2.4) and it
(x,y) e N and for
In view of remark 2.1 and the discussion immediately preceding it (with the
k.
notations therein), we have
both
sides
(LQ;Xl,X)
using
the
k’s.
same
of the product
(4.11)
(LQ;x 1,x),
L(’ (Q;x ,x))
m
LQ
Applying
of the polynomials
for all linearly independent elements
x,x
of
LPk,
E
we see that
as claimed.
for all x,x
(Q;x l,x) #
(in V-M) completes the proof.
relations (4.11) implies that
arbitrary nature of
to the generalized polar
3.1
Theorem
-
(LQ;x l,x)
as claimed.
q
The following version of Theorem 4.4 for the case when
k=l
0
Finally, the
q
0 and n
k=l
m
k
#
Consequently, the
E(k)o
is a result exclusively in terms of algebra-valued generalized polars with a vanishing
weight.
The proof is immediate as in the case of Theorem 3.2.
THEOREM 4.5. Under the notations and
q
k--I
E(k)
0
x,x
elements
i8 the cone as
hypotheses of Theorem 4.4 if we assume that
q
and
Z
k--I
of
defined
m
k
0
then
(Q;x l,x) e M
q+
E
such that
in Theorem 3.2.
x e E-
Uk=l
’k’j
Eo
for all linearly independent
where
Eo
(q+l)
=- Eo(N,Gq+ I)
THEOREMS ON GENERALIZED POLARS WITH ARBITRARY WEIGHT
Since Theorem 4.4 reduces to Theorem 3.1 on taking
V
K
771
M
and
Remark 4.1), it becomes the most general result of this paper.
{o} (c.f.
K
Besides, it leads to
the following corollary, which combines two earlier results due to the authors [I] and
[2] which includes in it (as a natural consequence) a number of other known results
due to Zaheer [5], [17], Marden [3], Walsh [22], and to Bcher [4].
COROLLARY 4.6.
Let
Eo(N’G’i )’
Eo,i
i=l ,2,3,
E.
be rculcm cones in
4.4,
if the circular cones
the notations and hypotheses of Theorem
Eo(N’Gk) e given by
E(k)o
E
(k)
Eo
E
x
e E-
x
xo
for
Eo,2
then (Q;x ,x) e M
<
for
for
o,3
<
r
k
3
3
x e E-
i=lU
<
p)
(4.12)
<
q,
Eo,i) u Ts’(Xo,yo ),
x,x
E such that
of
where(xo,yo) e N o L 2 [x,x
,yo
o
3
I
{p e K
r
i
E
i
i--I
p
E
m
k
k=r+l
If x,x
PROOF.
A /(p -p
i=l
E ink,
I--I
A
Ai)/(p
I
-y/); pie G
yo)}
(x
q
an
A
E
3
mk.
k=p+l
E
are linearly independent elements of
3
such that x
i=l
u
and x
3
i--I
(see
E
O)
8y
U
o,i
definition
a
Ts,(X yo ),
N)
of
3
,
a6
(u
3
i=l
S’(xo,Yo)
8Y
G’
i
0
x
for
implies that
(Xo’Yo) )"
alS
Since
H
n3 E
i=l
,
R
o
,yo
is the unique element of
a unique set of scalars
exists
aXo BYo’ Xl Yxo Yo
+
x
(Ui=l
where (x
there
then
such that
a/8 d
where
I],
Yo and
/
3
with
r (r
for all linearly independent elements
i=In Eo,i
S’(x
<
k_< p
<
p
k
Under
G
+
i’ (Xo,Yo))
u
S’(xo,Yo
N
o,i’
a,8,Y,6
(with
(cf. (2.5))
and
Y/
Gi= G.(Xo,Yo) (cf. Lemma 4.3). Note that
a/8
Y/. This implies that a/8
H u {y/6} u
{16))
we see that
u 3
G(xo,Yo)
y/6
n 3 G
i=l
q
{p e
Lx,x I]
and
Therefore,
(H*
Eo" i
Kl.r-
K=
mkl(P -pk)
q
I:
k--I
(x o,yo ),
mk)l(P
where
-Y/();
Pl
"’’P r e
N. ZAHEER and A.A. KHAN
772
Pp G2 (Xo’Yo); Pp+l’
GI (Xo’Yo); Pr+l’
G (xo,yo)}.
Pq
That is,
R* u {y16} u
=/
e/8
Consequently,
R.
3
U
G’.l (Xo’Yo))"
i=l
of Theorem 4.4 in the present set up are
Since
the
fpr
<
k
<
r
for r
<
k
<
p
fpr p
<
k
<_ q,
G
k
given by
G
G
G
4
k
2
G
have
we
(4.12)).
S(x
Therefore
linearly
u
R*
that
,yo ),
E(k)
o
u q
k=l
E
of
elements
u 3
S(x
E
i=l
,yo ).
O
q
n
and
o,i
such that
)(Q;x l,x)
By Theorem 4.4,
E
i=l
/B
we see that
independent
Ts(Xo,Yo).
o
(4.13)
(k)
n
x
E
(k)
and
x
(cf.
o,i
x
and
x
q
u
k=l
k-I
as was to be proved.
M,
E
i=l
Consequently,
q
3
n
o
are
(k))
Eo
q
l
m # 0, Corollary (4.6) is Theorem 4.3 of a paper due to the authors [2],
k
k=l
and if (in addition) V
K and M=K
{o}, it is Theorem 3.1 in the same paper.
q
3
l m
In case when
0 and n
E
#, Corollary 4.6 leads to the following cork
o,i
k=l
i=l
ollary, which is again a result due to the authors [I] and which reduces to Theorem
If
K- {o}.
V
K and M
Under the notations and hypotheses
3.1 [I] when specialized for
COROLLARY 4.7.
and n
3
E
i=l
of
(Q;x ,x)
then
O,i
E such that
G
4
x E E
4
i--Iu Eo,i,
{P e
(Xo,Yo)
(x
PROOF.
o ,yo
N
with
Eo,4 -= Eo(N,G 4’)
where
__I
l
k=
then there exist a unique element
4
i=l
i
BY
Gi(Xo,Yo)
O)
where
l m
k
k=l
e
0
x,x
defined by
Gi (Xo’Yo)}
p
m
A
k and
l
2
k=r+
mk.
are linearly independent elements of E such that x
If x,x
(with 6
is the cone
zA"/AI; Pi
,I<I(’3’P2’Pl
A
4.6
for all linearly independent elements
M
r
for all
q
of CoroIL2y
(Xo,Yo)
such that
G4(xo,Yo)
x
N
N
SXo
[2
[X,Xl]
+ By
Rx for X
u 4 E
o,i’
i=l
and a unique set of scalars
and
h/Al
Xl
YXo + 6Yo"
and G
i
Then
Gi(xo,Yo)
THEOREMS ON GENERALIZED POLARS WITH ARBITRARY WEIGHT
We divide the proof into the following two cases:
(Lemma 4.2).
.
18
Case (i).
a/6
and
so
773
a/8
/6
+ A + A
3
2
0,
u 3
u () u
R%
Gi(xo,Yo)),
i=l
u 3
{})
(R%
implies that
In this case
{})
(H-
,
Lemma
n Gi(xo,yo)
Gi(xo,Yo). Since i=l
3
/A
Gl(Xo,Yo) where (since
i=l
i=l
4.2
and
q
A
A
where
r.
3
mk)
k=p+l
3
{p
HI
KI E=
g
S (x
O
Gi(xo’Yo)}
O; P i
Ai/(p -Pi
i
(cf. Corollary (4.6).
,yo
Therefore,
a/8
and (hence) a/8
S
(Xo,Yo)
u 3
u {m} u
Gi(xo,Yo))
i=l
S’(Xo,Yo).
Ts.(Xo,Yo).
x
That is,
linearly independent elements of E such that
n3 E
x
i=l
Ts,(Xo,Yo).
m.
/8
and x
x
3
u
x
and
o,i
are
E
i=l
#(Q;Xl,X)
Finally, Corollary 4.6 says that
Case (ii).
Consequently,
6
u
as was to be proved.
M.
In the case under consideration
o,i
0
and
a/6
If
L()
E
V
M,
0
but
m
g
v ( 3
Rk
(4.14)
G.(Xo,Yo ))"
iffil
there exists a unique nontrivial scalar homomorphism L on
L(v)
0
v E M.
for
hypotheses
the
Since
(4.12),
satisfied for the choice of circular cones given by
hypotheses of Theorem 3.1 with the
,
Pk
fact that
E
G
and the
O
n
of degree
is an a.h.p,
k
k
E
from
such that
4.4
Theorem
are
we proceed as in the proof
say) satisfy the
given by (4.12) and (4.13). The
of Theorem 4.4 and again observe that the polynomials
(k)
of
V
to
LPk (--Pk’
allows us to write it in
K
the form
,
Pk
n
k (6jk s Tjkt),
j=l
(sx + tx I)
3
Since x is not in the set
E
iffil
for all
q
o,i
Pk (x)
k, we see that
kffil
J=l
6Jk
E
(k)
for all k.
the lines of proof of Theorem 3.1 (except that we replace
all
along)
and
Pk Gk(Xo’Yo)
using
the
such that
vanish simultaneously (since
notations,
same
v
ink(6 Pk-
k
q
k=l
E
(k)
o
Y)/
3
i=1
Z
and since
o
0
1,2,...,q.
k
we
a"
Eo,i
find
(x
k o,yo
TGk(Xo,Yo)
Now, proceeding exactly on
Pk
by
that
Pk
and take
there
exist
Note that all the
)"
Now,
Vk
8
0
elements
s
cannot
774
N. ZAHEER and A.A. KHAN
q
r
E
k=l
(/a) [B
are
Gi(xo,Yo)
Since
Bi/A i e Gi(xo,Yo)
0
i
e
Gi(x yo)
such
ko-cOnvex
where
A3
because A
+ A + A
2
B
that
q
l v
k
k=l
(6/a)
LQ
Gi(xo,Y o)
A2/A
(4.16)
02,
I,
m
0)
k
say.
conclude
Now,
mk.
k=p+l
A 0
i i
for
03
+
from
(4.13)
that
there
must
elements
exist
1,2 3, and (hence)
i
A2(02
q
g
k=l
(LQ;Xl,X
0
3
+
Pk"
A2(02 -03
03)]
(4.15)
ffq
k).
k=l
Pk(X)
Since
Pk(X)
0
#
(4.16)
0,
for all
k, (4.15) would
0.
must be distinct elements of
and so a/
and
E
mk0k(since k=l
q
r.
and (4.14) holds).
holds
q
E
k=p+l
0)
AI(0 I- 03
0 I,
B3],
(4.14), we
is the product of a.h.p.’s
imply that (since
Note that
+
2
[A1(P
L((Q;x l,x))
where
i
+ B
+
Ok]
From Remark 2.1 and equation (3.4) [5] we have
0.
3
q
p
E m 0 +
E
m
k k
k
k=l
k=r+!
/a
k
m
(Q;x l,x)
RX
#
Therefore, (,
(since
K
03, 02, 01)
A I, A
2
>
for any
M.
That is,
3
i=l
(03 01)/(03 02
This contradicts equation (4.14).
e V
O, n
Consequently,
(Q;x l,x)
e M,
as was
to be proved.
Finally, cases (i) and (ii) complete the proof.
The following Corollaries 4.8 and 4.9 can be proved directly from Theorem 4.4,
via applications of a suitably modified form of Lemmas 4.2 and 4.3, exactly in the
manner in which Corollaries 4.6 and 4.7 have been derived with the help of Lemmas 4.2
and 4.3.
But it would neither be necessary nor worthwhile to do so.
This is because
it has already been proved in earlier papers due to the authors Corollary 4.5
[I] and
Corollary 4.4 [2], that Corollary 4.8 (resp. Corollary 4.9) follows from Corollary 4.6
(resp. Corollary 4.7). We, therefore, state these without proof.
COROLLARY 4.8
Eo(N,Gi ), i 1,2, be circular cones in
[17].LetEo,i
the notations and hypotheses
(k)
Eo =- Eo (N’Gk) are
given by
of Theorem 4.4, if
q
Z m
k
k--!
#
0 and
if
E.
Under
the circular cones
THEOREMS ON GENERALIZED POLARS WITH ARBITRARY WEIGHT
p (p
k
1,2
k
p+l,...,q,
o,I
<
775
q)
(k)
Eo
o,2
then
x!
(Q’x ,x) e M
E
o,
n E
/_2[x,x I]
x
and
o,2
E
E
o,I u o,2
o
{P e K (p,/6
co
,yo
r.
A
and
o
Yo
p ,p
A /A
2
pi
e
G’.(x o, yo)}
Under the same notations and hypotheses as in Corollary 4.8,
we have that
k__E1 mk 0 and
all Zinearly independent elements x,x of E such that
q
except that this time
(Q;x l,x) e M for
u E
x E- E
V
(x
where
mk
k=p+l
COROLLARY 4.9 [17].
For
TS* (xo Yo
E such that
q
P
l m
k
k=l
with A
U
of
Yx o + 6x o and
x
S * (x
x,x
for all lnearly independent elements
,2
o,l
K and M
K
{o},
’
Eo,2
Eo,l
the above Corollaries 4.8 and 4.9 are known results
[5].
At the end it emerges that Theorem 4.4 of this paper happens to be the most
general result known thus far on (algebra-valued) generalized polars, whether having a
due to Zaheer
vanishing
a nonvanishing weight,
or
and
it
includes
in
it
all
the
results that have been established earlier in the papers due to Marden
[5],
and to the authors [I] and [2].
[II], Zaheer
It also includes improved versions of some well-
known classical results, such as: Walsh’s two-circle theorems
theorem [7]
corresponding
[5], Marden’s general
expressed in Corollary 3.3, and Bocher’s theorem [5].
To sum up: apart
from the fact that all the previously known results [3], [5], [2], have been jacketted
into Theorem 4.4, the present study answers in full generality the type of problem on
generalized polar pursued since 1971, and, it unifies the hitherto unnecessary and
separate treatments traditionally meted out to the cases of the vanishing and the
nonvanishing weight.
scope left for further
With Theorem 4.4 in view,
stu.dies
it may be pointed out there is no
in this subject area,
except possibly when different
new concepts are developed on some other lines.
ACKNOWLEDGMENT:
This
paper
was
supported
by
the
Re search
Project No.
Saudi Arabia.
Center
Math/1406/29 of the Faculty of Science, King Saud University, Riyadh,
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