General Mathematics Vol. 17, No. 1 (2009), 33–37
A note on a general integral operator of the
bounded boundary rotation1
S. Latha
Abstract
In this note, we consider the classes of bounded radius rotations,
bounded radius rotation of order β, bounded boundary rotation. In
these classes we study some properties of a general integral operator.
2000 Mathematics Subject Classification: 30C45
Key words and phrases: Functions of bounded boundary rotation,
Functions of bounded radius rotation, Integral operator and Univalent
function.
1
Introduction
Let Pλk (β) denote the class of analytic functions p(z) in defined in the unit
disc U = {z : |z| < 1} with the following properties:
(i). p(0) = 1
Z 2π ℜ{eiλ p(z) − β cos λ} dθ ≤ kπ cos λ
(ii).
1−β
0
1
Received 28 december, 2007
Accepted for publication (in revised form) 13 March, 2008
33
34
S. Latha
where, k ≥ 2, λ real, |λ| < π2 , 0 ≤ β < 1 and z = reiθ for 0 ≤ r < 1.
Let Vλk (β) [4] denote the class of functions f analytic in U with the normalized properties f (0) = f ′ (0) − 1 = 0 and
1+
zf ′′ (z)
∈ Pλk (β),
f ′ (z)
z∈U
where, k, λ and β are as above. For β = 0 we get the class Vλk of functions
with bounded boundary rotation studied by Moulis [3].
Any function f ∈ Vλk (β) if and only if
√
k − k2 − 4
zf ′′ (z)
iλ
> β cos λ, for |z| <
.
1+ ′
ℜ e
f (z)
2
A function f defined in U with the normalization properties f (0) = 0 and
zf ′
f ′ (0) = 1 is said to be in the class Uλk (β) if
∈ Pλk (β).
f
From the definition of the above classes it follows that f ∈ Vλk (β) if and
only if zf ′ ∈ Uλk (β) .
Now we consider the integral operator Fn (z) [2], defined by
α
α
Z z
fn (t) n
f1 (t) 1
dt
...
(1.1)
Fn (z) =
t
t
0
and we study its properties.
Remark 1.1. We observe that for n = 1Z and α1 = 1, we obtain the
z
f (t)
dt .
integral operator of Alexander [1], F (z) =
t
0
2
Main results
Theorem 2.1. Let αi be real numbers with the properties 0 ≤ αi < 1 for
n
X
αi ≤ n + 1. If fi ∈ Uλk α1i then the integral
i ∈ {1, 2, ..., n} and
i=1
operator defined in ( 1.1) belongs to Vλk .
A note on a generalized integral ...
35
Proof. Consider,
Fn (z) =
Z
z
0
f1 (t)
t
α1
...
fn (t)
t
αn
dt.
We determine the derivatives of the first and second order for Fn .
Fn′ (z)
Fn′′ (z)
=
n
X
i=1
αi
Fn′′ (z)
= α1
Fn′ (z)
=
fi (z)
z
f1 (z)
z
αi −1
α1
...
fn (z)
z
αn
α
n
fj (z) j
zfi′ (z) − fi (z) Y
z2
z
j=1, j6=i
f1′′ (z) 1
−
f1′ (z) z
+ ... + αn
fn′′ (z) 1
−
fn′ (z) z
zf1′′ (z)
zfn′′ (z)
zFn′′ (z)
+
1
=
α
+
...
+
α
− α1 − ... − αn + 1
1
n
Fn′ (z)
f1′ (z)
fn′ (z)
′′
′
′
zFn (z)
iλ zf1 (z)
iλ zfn (z)
iλ
+1
= α1 ℜ e
+ ... + αn ℜ e
ℜ e
Fn′ (z)
f1 (z)
fn (z)
+ℜ eiλ (−α1 − ... − αn + 1)
= (n + 1) cos λ −
n
X
αi cos λ > 0.
i=1
Hence Fn ∈ Vλk .
Corollary 2.2. For parametric values k = 2, λ = 0, we get the following
result [2].
Let αi , i ∈ {1, 2, ..., n} be real numbers with the properties αi > 0 for
n
X
i ∈ {1, 2, ..., n} and
αi ≤ n + 1. We suppose that the functions fi ,
i=1
36
S. Latha
i ∈ {1, 2, ..., n} are starlike functions of order α1i , i ∈ {1, 2, ..., n}, that is
fi ∈ S∗ α1i for all i ∈ {1, 2, ..., n}. Then the integral operator defined in
( 1.1) is convex.
Theorem 2.3. Let αi be real numbers with the properties αi > 0 for
n
X
λ
αi ≤ 1 and fi ∈ Uk α1i . Then the integral
i ∈ {1, 2, ..., n} with
i=1
operator defined in ( 1.1) belongs to Vλk (α), where α = 1 −
n
X
αi .
i=1
Proof. Consider,
n
X
zFn′′ (z)
αi
=
Fn′ (z)
i=1
=
n
X
i=1
1+
iλ
ℜ e
zfi′ (z)
−1
fi (z)
zfi′ (z)
− α1 − ... − αn .
αi
fi (z)
zFn′′ (z)
zfn′ (z)
zf1′ (z)
+
...
+
α
− α1 − ... − αn + 1.
=
α
n
1
Fn′ (z)
f1 (z)
fn (z)
′
′
zFn′′ (z)
iλ zf1 (z)
iλ zfn (z)
+1
= α1 ℜ e
+ ... + αn ℜ e
Fn′ (z)
f1 (z)
fn (z)
+ℜ eiλ (−α1 − ... − αn + 1) .
But fi ∈ Uλk for all i ∈ {1, 2, ..., n}. Therefore
′
iλ zfi (z)
> 0, ∀ i ∈ {1, 2, ..., n}.
ℜ e
fi (z)
This implies,
iλ
ℜ e
Hence Fn ∈ Uλk (α).
n
X
zFn′′ (z)
+1
>1−
αi = α.
Fn′ (z)
i=1
A note on a generalized integral ...
37
Corollary 2.4. For parametric values k = 2, λ = 0, we get the following
result [2].
Let αi , i ∈ {1, 2, ..., n} be real numbers with the properties αi > 0 for
n
X
αi ≤ 1. We suppose that the functions fi , with
i ∈ {1, 2, ..., n} and
i=1
i ∈ {1, 2, ..., n} are starlike. Then the integral operator defined in ( 1.1) is
n
X
αi .
convex by order 1 −
i=1
References
[1] Alexander J. W, Functions which map the interior of the unit circle upon
simple regions, em Ann. of Maths, 17 (1915/16), 12 - 22.
[2] Breaz D. and Breaz Nicoleta, Some convexity properties for a general
integral operator, Journal of Inequalities in Pure and Applied Mathematics, Volume 7, Issue 5, Article 177, 2006.
[3] Moulis E. J, A generalization of univalent functions with bounded boundary rotation, Trans. Am. Math. Soc., 174, 369 - 381 .
[4] Moulis E. J, Generaliztion of the Robertson functions, Pacific J. Math.,
81, 169 - 174.
Department of Mathematics
Yuvaraja’s College
University of Mysore
Mysore - 570 005
INDIA.
E-mail: drlatha@gmail.com