Advances in Pure Mathematics, 2011, 1, 23-27
doi:10.4236/apm.2011.12006 Published Online March 2011 (http://www.SciRP.org/journal/apm)
Copyright © 2011 SciRes. APM
Inequalities for the Polar Derivative of a Polynomial
Gulshan Singh1, Wali Mohammad Shah2, Yash Paul1
1Bharathiar University Coimbatore, Tamil Nadu, India
2Department of Mathematics, Kashmir University, Srinagar, India
E-mail: {gulshansingh1, wmshah, yashpaul2011}@rediffmail.com
Received February 20, 2011; revised March 1, 2011; accepted March 5, 2011
Abstract
If

=0
:n
j
j
j
Pz az is a polynomial of degree n, having all its zeros in ,zK 1K, then it was pro-
vied by Aziz and Rather [2] that for every real or complex number
with K
,
=1z
MaxD Pz


=1
1z
n
nK
M
axP z
K
. In this paper, we sharpen above result for the poly nomials

Pz of degree >3n.
Keywords: Polynomial, Inequality, Polar Derivative
1. Introduction
Let

=0
:= n
j
j
j
Pz az
be a polynomial of degree n and

P' z its derivative, then
 
=1 =1zz
M
axPznMaxP z
(1)
Inequality (1) is a famous result due to Bernstein and
is best possible with equality holding for the polynomial

=n
Pz z
, where
is a complex number.
If we restricted ourselves to a class of polynomial
having no zeros in <1z, then the above inequ ality can
be sharpened. In fact, Erdös conjectured and later Lax [6]
proved that if

0Pz in <1z, then
 
=1 =1
2
zz
n
axPzMaxPz
(2)
On the other hand, it was proved by Turán [10] that if

Pz is a polynomial of degree n having all its zeros in
1z, then
 
=1 =1
2
zz
n
axPzMaxPz
(3)
The inequalities (2) and (3) are also best possible and
become equality for polyno mials which have all zeros on
1z.
For the class of polynomials having all the zeros in
zK, Malik [7] (See also Govil [5]) proved that if

Pz is a polynomial of degree n having all zeros lie in
zK, then
 
=1=1 , if 1,
1
zz
n
MaxPzMaxP zK
K

(4)
where as Govil [5] showed that
 
=1=1 , if 1
1
zz
n
n
MaxPzMaxP zK
K

(5)
Both the inequalities are best possible, with equality in
(4) holding for
 
=n
Pzz K and in (5) the equality
holds for the p olynomial


=nn
P
zzK.
Let
DP z
denote the polar derivative of the
polynomial
Pz of degree n with respect to
, then
 
=DP znP zz Pz
 .
The polynomial
DP z
is of degree at most 1n
and it generalizes the ordinary derivative in the sense that
 
=.
lim DP zPz

Aziz and Rather [2] extended (5) to the polar deri-
vative of a polynomial and proved the following:
Theorem 1: If the polynomial

=0
:= n
j
j
j
Pz az
has
all its zeros in zK
, 1K, then for every real or
complex number
with
K
,




=1 =1
1
zz
n
nK
M
axDP zMaxPz
K
(6)
In this paper, we prove th e following result which is a
refinement as well as generalization of Theorem 1.
G. SINGH ET AL.
Copyright © 2011 SciRes. APM
24
Theorem 2: Let

=0
:= n
j
j
j
P
zaz
, 00
n
aa be a po-
lynomial of degree >3n, having all its zeros in zK
,
1K, then for every real or complex number
with
K
,



 







1
=1=1 =
2
2
1
21
1
1
11 121
2, if >3.
123
n
n
zzzK
n
nn
n
K
nK a
MaxDP zMaxP zMinP zK
nn
K
KnKK nK
an
nnn n






 
 
 


 


 
 

(7)
Remark 1: For =1
K
, Theorem 2 provides a refi-
ment of a theorem proved by Shah [9].
Remark 2: For >1
K
, and for >1y,



11
1
y
KyK
yy

 

and
1
y
K
y
are both increa-
sing functions of y and so the expressions






2
11 121
123
nn
KnKK nK
nnn n

 
 
 

 


 
 

and

11
n
KK
n

are always non-negative so that for polynomials of
degree >3n, Theorem 2 is an improvement of Theo-
rem 1.
Dividing both sides of (7) by
and letting
,
we get the following:
Corollary 1: Let

=0
=n
j
j
j
P
zaz
, 00
n
aa be a po-
lynomial of degree >3n, having all its zeros in zK
,
1
K
, then


  







1
=1=1 =
2
2
1
21
1
1
11 121
2, if >3.
123
n
n
zzzK
n
nn
n
K
a
n
MaxPzMaxP zMinP zK
nn
K
KnKK nK
an
nnn n






 
 
 


 


 
 

(8)
2. Lemmas
We need the following lemmas.
Lemma 1: Let
Pz be a polynomial of degree n,
then for 1R.
 
==1
.
n
zR z
M
axPzRMaxPz
The above lemma is a simple consequence of the
maximum modulus principle [8].
Lemma 2: If

=0
:= n
j
j
j
Pz az
, 0
n
a, is a poly-
nomial of degree n having all its zeros in 1z, then
 

=1=1 =1
2
zzz
n
M
axPzMaxPzMinP z
.
This lemma is due to Aziz and Dawood [1].
Lemma 3: If

=0
:= n
j
j
j
Pz az
is a polynomial of
degree n having no zeros in 1z, and
=1
=z
mMin Pz, then for 1R and >3n,













=1
2
111
20
,1
221
11 121
0123
nnn
z
nn
RRR
P
MPRMax PzmR
nn
RnRR nR
Pnnn n



 
 
 



 


 
 

.
The above result is a special case of a result due to
Dewan, Singh and Mir [4, Theorem 1] with =1K and
=1
.
Remark 3: Here we note that for the proof of this
result an additional hypothesis that

00P is required.
A simple counter example in this case is

=n
Pz z.
G. SINGH ET AL.
Copyright © 2011 SciRes. APM
25
3. Proof of Theorem 2
Since

Pz has all its zeros in zK, therefore
 
=Gz PKz has all its zeros in 1z and hence by
applying lemma 2 to the polynomial

Gz, we get
  

=1=1=1 .
2
zzz
n
M
axGzMaxG zMinG z
 (9)
Let

1
=n
Hz zGz



. Then it can be easily verified
that
 
=, for =1.HznGzzGzz

(10)
The polynomial
H
z has all its zeros in 1z
and
 
=
H
zGz for 1z
, therefore, by result of
a de Bruijn [3]

for =1Hz Gzz

(11)
Now for every real or complex nu mber α with
K
,
we have
  
 
=
K
DGznGzzG zGz
K
GznGzzGz
K




For this, we get by using (10) and (11)
 
=1 =1zz
K
K
M
axD G zMaxGz
K
(12)
Using (9) in (12), we get

 

=1=1 =1
2
zzz
K
Kn
M
axDG zMaxGzMinG z
K
.
Replacing

Gz by

PKz, we have



||=1
=1 =1
2z
zz
K
nK
M
axDP KzMaxP KzMinPKz
K

.
This gives
 
 

=1 =1 =1
2
zzz
nK
M
axnPKzzKPKzMaxP KzMinP Kz
KK

 

 .
Equivalently

 

===
.
2
zKzK zK
nK
M
axDP zMaxP zMinP z
K

(13)
Since the polynomial P(z) has all its zeros in
zK, 1K. If

1
=n
QzzP z



be the reciprocal
polynomial of
Pz. Then the polyn omial z
Q
K



has
all its zeros in 1z. Hence applying lemma 3 to the
polynomial z
Q
K



, 1
K
, we get









1
==1=1
2
2
11 1
21
221
11 121
2123
nn n
n
zK zz
nn
n
KK K
a
zzz
MaxQMax QMin QK
KKKnn
KnKK nK
annn n
 
 


 

 
 
 


 


 
 

This in particular gives


 







1
=1 ==
2
2
11 1
21
1
22
11 121
2123
nn n
n
zzKzK
nn
nn
n
KK K
a
MaxP zMaxP zMinP zK
nn
KK
KnKK nK
annn n
 


 
 
 


 


 
 

G. SINGH ET AL.
Copyright © 2011 SciRes. APM
26
which is equivalent to









 





1
==1=
2
2
11
24
1
1
111
11 121
4
123
1
nn
n n
n
zK zzK
nn n
nn
n
n
n
KK
a
KK
MaxP zMaxP zMinP zK
nn
KK K
KnKK nK
Kannn n
K

 


 
 
 


 


 
 

(14)
Using (14) in (13), we get










 




 
1
==1=
2
2=
11
24
{ 1
21
111
11 121
4
123
1
nn
n n
n
zKz zK
nn n
nn
n
nzK
n
KK
nK a
KK
MaxDP zMaxP zMinP zK
Knn
KK K
KnKK nK
KaMinPz
nnn n
K








 
 
 


 


 
 

, if >3.n
Equivalently



 







11
==1=
2
2
1
21
1
1
11 121
2, if >3.
123
n
n
n
zKz zK
n
nn
n
K
nKK a
MaxDP zMaxP zMinP zK
nn
K
KnKK nK
an
nnn n


 
 
 


 


 
 

(15)
Since

DP z
is a polynomial of degree 1n
and 1K, therefore by using Lemma 1, we get
1
==1
n
zK z
M
axDP zKMaxDP z

(16)
Combining (16) and (15) we have



  







1
=1=1 =
2
2
1
21
1
1
11 121
2||, if >3.
123
n
n
zzzK
n
nn
n
K
nK a
MaxDP zMaxPzMinP zK
nn
K
KnKK nK
an
nnn n






 
 
 


 


 
 

This completes the proof of Theorem 2.
4. Acknowledgements
The authors are extremely grateful to the referee for his
valuable su ggestions.
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Copyright © 2011 SciRes. APM
27
doi:10.1090/S0002-9939-1973-0325932-8
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