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research
Integral mean estimates for the polar derivative of a polynomial
Authors
Suhail Gulzar
N. A. Rather
Publication date
27 February 2013
Publisher
View
on
arXiv
Abstract
Let
P
(
z
)
P(z)
P
(
z
)
be a polynomial of degree
n
n
n
having all zeros in
∣
z
∣
≤
k
|z|\leq k
∣
z
∣
≤
k
where
k
≤
1
,
k\leq 1,
k
≤
1
,
then it was proved by Dewan \textit{et al} that for every real or complex number
α
\alpha
α
with
∣
α
∣
≥
k
|\alpha|\geq k
∣
α
∣
≥
k
and each
r
≥
0
r\geq 0
r
≥
0
n
(
∣
α
∣
−
k
)
{
∫
0
2
Ï€
∣
P
(
e
i
θ
)
∣
r
d
θ
}
1
r
≤
{
∫
0
2
Ï€
∣
1
+
k
e
i
θ
∣
r
d
θ
}
1
r
M
a
x
∣
z
∣
=
1
∣
D
α
P
(
z
)
∣
.
n(|\alpha|-k)\left\{\int\limits_{0}^{2\pi}\left|P\left(e^{i\theta}\right)\right|^r d\theta\right\}^{\frac{1}{r}}\leq\left\{\int\limits_{0}^{2\pi}\left|1+ke^{i\theta}\right|^r d\theta\right\}^{\frac{1}{r}}\underset{|z|=1}{Max}|D_\alpha P(z)|.
n
(
∣
α
∣
−
k
)
⎩
⎨
⎧
​
0
∫
2
Ï€
​
​
P
(
e
i
θ
)
​
r
d
θ
âŽ
⎬
⎫
​
r
1
​
≤
⎩
⎨
⎧
​
0
∫
2
Ï€
​
​
1
+
k
e
i
θ
​
r
d
θ
âŽ
⎬
⎫
​
r
1
​
∣
z
∣
=
1
M
a
x
​
∣
D
α
​
P
(
z
)
∣.
\indent In this paper, we shall present a refinement and generalization of above result and also extend it to the class of polynomials
P
(
z
)
=
a
n
z
n
+
∑
ν
=
μ
n
a
n
−
ν
z
n
−
ν
,
P(z)=a_nz^n+\sum_{\nu=\mu}^{n}a_{n-\nu}z^{n-\nu},
P
(
z
)
=
a
n
​
z
n
+
∑
ν
=
μ
n
​
a
n
−
ν
​
z
n
−
ν
,
1
≤
μ
≤
n
,
1\leq\mu\leq n,
1
≤
μ
≤
n
,
having all its zeros in
∣
z
∣
≤
k
|z|\leq k
∣
z
∣
≤
k
where
k
≤
1
k\leq 1
k
≤
1
and thereby obtain certain generalizations of above and many other known results.Comment: 8 page
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Last time updated on 30/10/2017