A well known result due to Ankeny and Rivlin [1] states that if
is a polynomial of degree
n
satisfying
p
(
z
) ≠ 0 for 
z
 < 1, then for
R
≥ 1
It was proposed by Professor R.P. Boas Jr. to obtain an inequality analogous to this inequality for polynomials having no zeros in 
z
<
k, k
> 0. In this paper, we obtain some results in this direction, by considering polynomials of degree
n
≥ 2, having all its zeros on the disk 
z
 =
k, k
≤ 1.
AMS Mathematics Subject Classification : 30A10, 30C10, 30C15.
1. Introduction
For an arbitrary entire function
f
(
z
), let
M
(
f, r
) = max
_{z=r}

f
(
z
). As a consequence of Maximum Modulus Principle [6, Vol. I, p. 137, Problem III, 269] it is known that if
p
(
z
) is a polynomial of degree
n
, then
The result is best possible and equality holds for polynomials having zeros at the origin.
Ankeny and Rivlin
[1]
considered polynomials not vanishing in the interior of the unit circle and obtained refinement of inequality (1.1) by proving that if
p
(
z
) ≠ 0 in 
z
 < 1, then
The inequality (1.2) is sharp and equality holds for
p
(
z
) =
α
+
βz^{n}
, where 
α
 = 
β
.
While trying to obtain inequality analogous to (1.2) for polynomials not vanishing in 
z
 <
k, k
≤ 1, recently the authors
[2]
proved the following result.
Theorem 1.1
(
[2]
).
is a polynomial of degree n having all its zeros on

z
 =
k, k
≤ 1,
then for every positive integer s
By involving the coefficients of
p
(
z
), Dewan and Ahuja
[2]
in the same paper obtained the following refinement of Theorem 1.1.
Theorem 1.2
(
[2]
).
is a polynomial of degree n having all its zeros on

z
 =
k, k
≤ 1,
then for every positive integer s
In this paper, we restrict ourselves to the class of polynomials of degree
n
≥ 2 having all its zeros on 
z
 =
k, k
≤ 1 and obtain improvement and generalization of Theorems 1.1 & 1.2. More precisely, we prove
Theorem 1.3.
is a polynomial of degree n
≥ 2
having all its zeros on

z
 =
k, k
≤ 1,
then for every positive integer s and R
≥ 1
and
The following result immediately follows by choosing
s
= 1 in Theorem 1.3.
Corollary 1.4.
is a polynomial of degree n
≥ 2
having all its zeros on

z
 =
k, k
≤ 1,
then for R
≥ 1
and
Our next result is a refinement of Theorem 1.3, the proof of which is omitted as it follows on same lines as proved in
[2]
by the same authors.
Theorem 1.5.
is a polynomial of degree n
≥ 2
having all its zeros on

z
 =
k, k
≤ 1,
then for every positive integer s and R
≥ 1
and
Remark 1.1.
The above theorem also generalize as well as improves upon Theorem 1.2 for
n
≥ 2.
If we choose
s
= 1 in Theorem 2, we get the following result.
Corollary 1.6.
is a polynomial of degree n
≥ 2
having all its zeros on

z
 =
k, k
≤ 1,
then for R
≥ 1
and
2. Proof of Theorems
For the proof of these theorems, we need the following lemmas. The first lemma is due to Govil
[5]
.
Lemma 2.1.
is a polynomial of degree n having all its zeros on

z
 =
k, k
≤ 1,
then
Lemma 2.2.
is a polynomial of degree n having all its zeros on

z
 =
k, k
≤ 1,
then
The above lemma is due to Dewan and Mir
[3]
.
Lemma 2.3.
is a polynomial of degree n, having all
R
≤ 1,
and
The above result is due to Frappier, Rahman and Ruscheweyh
[4]
.
Proof of Theorem 1.3.
We first consider the case when
p
(
z
) is of degree
n
> 2. Note that for every
θ
, 0 ≤
θ
< 2π and
R
≥ 1, we have
Then
Since
p
(
z
) is of degree
n
> 2, the polynomial
p
’(
z
) is of degree (
n
− 1) ≥ 2, hence applying inequality (2.15) of Lemma 2.3 to
p
’ (
z
), we have for
r
≥ 1 and 0 ≤
θ
< 2π
Inequality (2.18) in conjunction with inequalities (2.17) and (1.1), yields for
n
> 2 and
R
≥ 1
On applying Lemma 2.1 to the above inequality, we get for
n
> 2
This gives
from which proof of inequality (1.5) follows.
The proof of inequality (1.6) follows on the same lines as that of (1.5), but instead of using (2.15) of Lemma 2.3 we use (2.16) of the same lemma.
Proof of Theorem 1.5.
We first consider the case when polynomial
p
(
z
) is of degree
n
> 2, then the polynomial
p
’(
z
) is of degree (
n
−1) ≥ 2, hence applying inequality (2.15) of Lemma 2.3 to
p
’(
z
) we have for
r
≥ 1 and 0 ≤
θ
< 2π
Now for every
θ
, 0 ≤
θ
< 2π and
R
≥ 1, we have
Inequality (2.20) in conjunction with inequality (1.1) and (2.19), gives for
n
> 2
which on combining with Lemma 2.2, yields for
n
> 2
from which we get the desired result.
The proof of inequality (1.10) follows on the same lines as that of inequality (1.9), but instead on using (2.15) of Lemma 2.3, we use inequality (2.16) of the same lemma.
BIO
K. K. Dewan received her D.Sc. from Dr. B.R.Ambedkar University, Agra and Ph.D. at I.I.T. Delhi. Her research interests centre on Complex Analysis, Approximation Theory, Mathematical Modeling and Transportation.
Presently, she is a Professor of Mathematics, Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia (Central University), New Delhi, INDIA. She was formerly the Dean, Faculty of Natural Sciences and Former Head, Department of Mathematics, Jamia Millia Islamia (Central University), New Delhi, INDIA.
email: kkdewan123@yahoo.com
Arty Ahuja received her Ph.D. and Masters from Jamia Millia Islamia (Central University), New Delhi and Graduation from University of Delhi. Her research interests centre on Complex Analysis and Approximation Theory
Presently, she is working in Govt. Girls Sr.Sec. School, Vivek ViharII, Delhi under Directorate of Education Govt. Of National Capital of Delhi, INDIA.
email: aarty ahuja@yahoo.com
Dewan K.K.
,
Ahuja Arty
(2011)
Growth of polynomials with prescribed zeros
J. Math. Inequalities
5
(3)
355 
361
DOI : 10.7153/jmi0531
Dewan K.K.
,
Mir A.
(2007)
Note on a Theorem of S. Bernstein
Southeast Asian Bulletin of Math
31
691 
695
Govil N.K.
(1980)
On the Theorem of S. Bernstein
J. Math. and Phy. Sci
14
183 
187
PÓlya G.
,
SzegÖ G.
1925
Aufgaben and Lehrsatze aus der Analysis
SpringerVerlag
Berlin