MAXIMUM MODULUS OF POLYNOMIALS HAVING SOME ZEROS AT ORIGIN Roshan Lal
Department of Mathematics V.S.K.C. Govt. P.G. College Dakpathar
Vikasnagar, Dehradun Uttarakhand, India-246162
Abstract: Let be a polynomial of degree
n
. Concerning the estimate for the maximummodulus of a polynomial on the circle , in terms of its degree and the maximum modulus on the unit circle, we have several well known results for the case R1and
r
1
respectively. In this paper, we have obtained bounds for the maximum modulus of polynomials having some zeros in the interior of a circle of radiusR1. Our result improves as well as generalizes the bounds obtained by other authors for the same class of polynomials.Key Words: Polynomials, Inequalities, Complex domain, zeros. 2000 AMS Subject Classification: 30A10, 30C10, 30C15.
1. INTRODUCTION AND STATEMENT OF RESULTS
Let be a polynomial of degree
n
. Concerning the estimate for the maximummodulus of a polynomial on the circle , in terms of its degree and the maximum modulus on the unit circle, we have the following well known results.
Theorem A If is a polynomial of degree n, then for every R1,
max
(
)
max
(
)
1z
p
R
z
p
z n R
z
. (1.1.)The result is best possible and extremal polynomial is
p
(
z
)
z
n,
0
being a complex number.Inequality (1.1.) is a simple deduction from the maximum modulus principle (for reference see [7] or [6]).
nj
j j
z
a
z
p
0
)
(
0
,
R
R
z
nj
j j
z
a
z
p
0
)
(
0
,
R
R
z
Theorem B. If is a polynomial of degree n, then for r1,
max
(
)
max
(
)
1p
z
r
z
p
z n r
z
. (1.2.)The result is best possible and extremal polynomial is ,
0
being a complex number.Inequality (1.2.) is due to Zarantonello and Varga [9].
Theorem C. If is a polynomial of degree n, having no zeros in , then for
r
1
,
max
(
)
2
1
)
(
max
1
z
p
r
z
p
z n r
z
. (1.3.)The result is best possible and equality in inequality (1.3) holds for
n
z
z
p
2
1
)
(
.The inequality (1.1) is due to Ankeny and Rivlin [1] and inequality (1.3) is due to Rivlin [8]. For the case
0
1
, we have the following result due to Aziz [2].Theorem D. Let
n jj j
z
a
z
p
0
)
(
be a polynomial of degree n, which does not vanish in. Then for
0
1
max
(
)
1
)
(
max
1
z
p
k
k
z
p
z n
z
. (1.4)
The result is sharp and equality in (1.4) is attained for
p
(
z
)
c
(
ze
i
k
)
n,
c
(
0
)
C
and
R
.The following result is due to Jain [5].
Theorem E. If be a polynomial of degree n, having all its zeros in
z
k
,
k
1
, then for2
k
R
k
,)
(
z
p
n
z
z
p
(
)
)
(
z
p
z
1
1
,
k
k
z
max
(
)
1
)
(
max
1
z
p
k
k
R
R
z
p
z s
R
z
. (1.5)where s(<n) is the order of a possible zero of p(z) at origin.
In this paper, we prove the following generalization of Theorem E by involving the coefficients of the
polynomial
n jj j
z
a
z
p
0
)
(
. In fact we prove the followingTheorem 1. Let
n jj j
z
a
z
p
0
)
(
be a polynomial of degree n, having all its zeros in1
,
k
k
z
, then fork
R
k
2,
,
)
(
min
)
1
(
)
(
)
(
)
)
(
)(
1
(
)
(
max
)
1
(
)
)(
(
2
)
)(
(
)
(
max
1 2
1 1
1 1
2 1
1 1
2 1
z
p
a
R
R
k
R
a
s
n
R
s
n
a
a
R
k
R
z
p
a
R
a
s
n
R
k
R
a
R
a
s
n
R
k
R
R
z
p
k z n s
n s
n n
n n
s n s
s
z n
s n n
s n
n s n n
s n s
n s R
z
(1.6)
where s is the order of a possible zero of p(z) at the origin .
2. LEMMAS.
For the proof of the above theorems, we need the following lemmas.
Lemma 2.1. If
na
z
z
p
0
)
(
is a polynomial of degree n, having no zeros in ,
then
max
(
)
2
)
1
(
)
(
'
max
1 1 2 0
2
1 2 0
1
k
n
a
k
a
p
z
a
k
a
n
n
z
p
z
z
. (2.1)The above lemma is due to Govil, Rahman and Schmeisser [4]. The above lemma is due to Dewan, Singh and Yadav [3]. Lemma 2.2. If
na
z
z
p
0
has no zeros in , then
1
,
k
k
z
1
,
k
k
.
)
(
min
2
)
1
(
)
)
(
max
2
)
1
(
)
(
'
max
1 2 0
2
1 0 2
1 1 2 0
2
1 2 0
1
z
p
a
k
a
n
k
a
a
n
k
n
z
p
a
k
a
n
k
a
k
a
n
n
z
p
k z n
z z
(2.2)
Lemma 2.3.If
n jj j
z
a
z
p
0
)
(
is a polynomial of degree n having all its zeros inz
k
,
k
0
,then for
r
k
R
, we have
min
.
max
2
max
z 1
2 2
2 0 2
1 0 1
z 1
2 2
2 0
1 2 0
2 2 1
z
z
p
R
r
a
k
R
r
k
r
a
n
k
a
r
a
n
r
R
r
R
r
a
k
R
r
k
r
a
n
a
k
a
n
k
r
r
z
p
k n
n n
n n
n n n
R n n n
n n r
(2.3) Proof of Lemma 2.3. Since
p
(z
)
does not vanish inz
k
,
k
1
, the polynomial)
(
)
(
z
p
rz
T
does not vanish in
,
1
r
k
r
k
z
, therefore applying Lemma 2.2 toT
(z
)
, we get
)
(
min
2
)
1
(
)
)
(
max
2
)
1
(
)
)
(
'
max
1 2 2
0 2 2
1 0
2
1 1
2 2
0 2 2
1 2 2
0
1
z
T
a
r
r
k
a
n
r
k
a
r
a
n
r
k
n
z
T
a
r
r
k
a
n
r
k
a
r
r
k
a
n
n
z
T
r k z n
z z
)
(
min
2
)
(
)
)
(
max
2
)
(
)
)
(
'
max
1 2 0
2 2
1 0
2
1 1
2 0
2 2
1 2 0
rz
p
a
r
k
a
n
k
r
a
r
a
n
k
r
n
z
p
a
r
k
a
n
k
r
a
k
r
a
n
r
n
rz
p
r
r k z n
n
z r
z
which is equivalent to
.
)
,
(
2
)
(
)
)
,
(
2
)
(
)
)
(
'
max
1 2 0
2 2
1 0
2 1
1 2 0
2 2
1 2 0
k
p
m
a
r
k
a
n
k
r
a
r
a
n
k
r
n
r
p
M
a
r
k
a
n
k
r
a
k
r
a
n
n
z
p
n n r
z
(2.4)
Again as
p
'
(
z
)
is a polynomial of degreen
1
, by maximum modulus principle [6, p. 158, problem III 269], we have
,
)
,
'
(
)
,
'
(
1
1
nn
r
r
p
M
t
t
p
M
for
t
r
(2.5) Combining inequalities (2.4) and (2.5), we have
.
)
,
(
2
)
(
)
)
,
(
2
)
(
)
)
(
'
max
1 2 0
2 2
1 0
2 1
1 2 0
2 2
1 2 0
1 1
k
p
m
a
r
k
a
n
k
r
a
r
a
n
k
r
r
p
M
a
r
k
a
n
k
r
a
k
r
a
n
r
t
n
z
p
n n n
n
.
)
,
(
2
)
(
)
)
,
(
2
)
(
)
)
,
(
2
)
(
)
)
,
(
2
)
(
)
)
(
'
)
(
)
(Re
1 2 0
2 2
1 0
2 1
1 2 0
2 2
1 2 0
1
1 2 0
2 2
1 0
2 1
1 2 0
2 2
1 2 0
1 1
k
p
m
a
r
k
a
n
k
r
a
r
a
n
k
r
r
p
M
a
r
k
a
n
k
r
a
k
r
a
n
r
r
R
dt
k
p
m
a
r
k
a
n
k
r
a
r
a
n
k
r
r
p
M
a
r
k
a
n
k
r
a
k
r
a
n
r
t
n
dt
te
p
re
p
p
n n n
n n R
r
n n n
n
R
r
i i
i
This is equivalent to
.
)
,
(
2
)
(
)
)
,
(
}
2
)
{(
]
)[
(
]
2
)
[(
,
1 2 0
2 2
1 0
2
1 2 0
2 2 1
1 2 0
1 2 0
2 2 1
k
p
m
a
r
k
a
n
k
r
a
r
a
n
k
r
R
r
p
M
a
r
k
a
n
k
r
r
a
k
r
a
n
r
R
a
r
k
a
n
k
r
r
R
p
M
n n n
n
n n n
From which the proof of Lemma 2.3 follows.
3. PROOF OF THE MAIN THEOREM
Proof of the Theorem 1. The polynomial
p
z
of degree n has all its zeros inz
k
,
k
1
, with s-fold zeros at the origin, implies that the polynomialq
(
z
)
z
np
(
1
/
z
)
is of degree (n-s) and has all its zeros in
1
,
1
1
k
k
z
.On applying Lemma 2.3 to the polynomial
q
(z
)
with , we obtain fork
r
k
1
1
2
,1
1
min
(
)
1
1
1
)
(
1
)
)
)((
1
(
)
(
max
1
1
1
1
)
(
1
2
)
(
1
)
(
max
1 1 2 2 2 2 1 1 1 1 2 2 2 1 2 2 2 1z
p
r
a
k
r
k
r
a
s
n
k
a
r
a
s
n
r
r
z
q
r
a
k
r
k
r
a
s
n
a
r
k
a
s
n
k
r
r
z
q
k z s n n s n n s n n n s n s n z s n n s n n n s n n s n r z
or equivalentlyfor
1
21
.
k
r
k
The above inequality is equivalent to
z
p
z
k
r
a
k
r
r
a
s
n
a
r
a
s
n
r
k
r
z
p
r
k
a
k
r
r
a
s
n
a
k
r
a
s
n
k
r
r
z
p
z
n k z s n n s n n n n s n s n s n z s n n s n n n n s n n r z1
min
1
1
)
(
)
)
)((
1
(
)
(
max
1
1
)
(
2
)
(
1
1
max
1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 2 2 1
1
min
(
)
for
1
21
.
k
r
k
Now replacing r by R
1
we get from inequality (3.1)
)
(
min
1
1
1
1
1
1
)
(
)
(
1
1
)
(
max
1
1
1
1
1
)
(
2
)
(
1
1
)
(
max
2 1 2
2
1 2
1
1 2
1 2
2
1 2
2 2 1
z
p
k
R
k
a
R
k
R
a
s
n
R
a
a
s
n
R
k
R
z
p
R
k
a
R
k
R
a
s
n
a
R
k
a
s
n
k
R
R
z
p
k z n s n n
s n n
n n s
n s
n s
z s
n n
s n n
n n
s
R z
for
k
R
k
2. The above inequality on simplification reduces to,
)
(
min
)
1
(
)
(
)
(
)
)
(
)(
1
(
)
(
max
)
1
(
)
)(
(
2
)
)(
(
)
(
max
1 2
1 1
1 1
2 1
1 1
2 1
z
p
a
R
R
k
R
a
s
n
R
s
n
a
a
R
k
R
z
p
a
R
a
s
n
R
k
R
a
R
a
s
n
R
k
R
R
z
p
k z n s
n s
n n
n n
s n s
s
z n
s n n
s n
n s n n
s n s
n s R
z
for
k
R
k
2. This completes the proof of Theorem 1.References
1 N.C. Ankeny and T.J.Rivlin, On a Theorem of S. Bernstein, Pacific J. Maths. , 5 (1955), 249-252.
2 A. Aziz, Growth of polynomials whose zeros are within or outside a circle, Bull. Austral. Math. Soc., 35 (1987), 247-250.
3 K.K.Dewan, Harish Singh and R.S.Yadav, Inequalities concerning polynomials having zeros in closed interior of a circle, Indian J. Pure and Appl. Math., 32 (5) (2001), 759-763.
4 N.K.Govil, Q.I.Rahman and G.Schmeisser, On the derivative of a polynomial, Illinois J. Math., 23 (1979), 319-329.
5 V.K.Jain, On polynomials having zeros in closed exterior or interior of a circle, Indian J. Pure Appl. Math., 30 (1999), 153-159.
6 G. Polya and G.
Szeg
o
, “Problems and Theorems in Analysis” Vol-1, Springler- Verlag, Berlin, 1972.7 M.Riesz,
U
ber
enin Satz der Herrn Serge Bernstein, Acta Math., 40 (1918), 337-347. 8 T.J.Rivlin, On the maximum modulus of polynomials, Amer. Math. Monthly, 67 (1960),251-253.
