IJMMS 29:3 (2002) 183–186
PII. S0161171202006920
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
A NOTE ON RUSCHEWEYH TYPE OF INTEGRAL OPERATORS
FOR UNIFORMLY α-CONVEX FUNCTIONS
M. ANBU DURAI and R. PARVATHAM
Received 7 March 2001
We prove that the class of uniformly α-convex functions introduced by Kanas is closed
under the generalized Ruscheweyh integral operator for 0 < α ≤ 1.
2000 Mathematics Subject Classification: 30C45.
We denote by Ꮽ the class of functions f (z) = z + a2 z2 + · · · which are analytic in
∆ = {z ∈ C : |z| < 1}. Let S denote the class of functions in Ꮽ that are univalent in ∆.
The subclasses of S containing functions which are uniformly convex and uniformly
starlike, introduced by Goodman [1, 2], are denoted by UCV and UST, respectively.
The class of uniformly α-convex functions was introduced by Kanas [3] and she
gave an analytic condition for such functions as follows: f (z) is a uniformly α-convex
function if and only if
(z − ζ)f (z)
(z − ζ)f (z)
>0
+α 1+
Re (1 − α)
f (Z) − f (ζ)
f (z)
(1)
for all z, ζ ∈ ∆ and 0 ≤ α ≤ 1. For ζ = 0, this class of functions reduces to Mocanu’s
class M(α) of α-convex functions [4].
In this note, for α > 0, we consider the integral operator
F (z) =
Fα (z, ζ) − Fα (0, ζ)
,
Fα (0, ζ)
(2)
where
Fα (z, ζ) =
c + 1/α
(z − ζ)c
z
ζ
1/α
(t − ζ)c−1 f (t) − f (ζ)
dt
α
(3)
for all z ∈ ∆ and for fixed ζ ∈ ∆ with z ≠ ζ. We prove that this normalized function
F (z) is a uniformly α-convex function when f (z) is a uniformly α-convex function in
the sense of Kanas [3].
For ζ = 0 the operator F (z) reduces to Ruscheweyh’s integral operator [5]. It is well
known that Mocanu’s class M(α) of α-convex functions is closed under Ruscheweyh’s
integral operator for α > 0.
184
M. A. DURAI AND R. PARVATHAM
Theorem 1. Let f (z) = z + a2 z2 + · · · be a uniformly α-convex function in ∆ and
let c > 0. Then, for 0 < α ≤ 1, the function
F (z) =
Fα (z, ζ) − Fα (0, ζ)
,
Fα (0, ζ)
z, ζ ∈ ∆,
(4)
is uniformly α-convex where Fα (z, ζ) is defined as in (3).
Proof. We have from (3) that
Fα1/α (z, ζ) =
c + 1/α
(z − ζ)c
z
ζ
1/α
(t − ζ)c−1 f (t) − f (ζ)
dt.
(5)
Differentiating with respect to z, we have
(z − ζ)c
1 1/α−1
F
(z, ζ)Fα (z, ζ) + c(z − ζ)c−1 Fα1/α (z, ζ)
α α
1/α
1
(z − ζ)c−1 f (z) − f (ζ)
= c+
α
(6)
and again differentiating with respect to z we get
2
1
1
− 1 Fα1/α−2 (z, ζ) Fα (z, ζ)
(z − ζ)Fα1/α−1 (z, ζ)Fα (z, ζ) + (z − ζ)
α
α
c
+ Fα1/α−1 (z, ζ)Fα (z, ζ) + Fα1/α−1 (z, ζ)Fα (z, ζ)
α
1/α−1
1 1 ;
f (z) f (z) − f (ζ)
= c+
α α
(z − ζ)Fα (z, ζ)
Fα (z, ζ)
Fα1/α−1 (z, ζ)Fα (z, ζ) α
+
(1
−
α)(z
−
ζ)
+
α(1
+
c)
Fα (z, ζ)
Fα (z, ζ)
1/α−1
.
= (αc + 1)f (z) f (z) − f (ζ)
(7)
Thus we get
(z − ζ)Fα (z, ζ) (z − ζ)Fα (z, ζ)
−
Fα1/α−1 (z, ζ)Fα (z, ζ) α 1 +
Fα (z, ζ)
Fα (z, ζ)
(z − ζ)Fα (z, ζ)
+
+ cα
Fα (z, ζ)
1/α−1
= (cα + 1)f (z) f (z) − f (ζ)
.
(8)
From (2) we have
F (z) =
Fα (z, ζ)
,
Fα (0, ζ)
(9)
showing that F (0) = 0 and F (0) = 1.
Considering
(z − ζ)Fα (z, ζ)
(z − ζ)F (z)
=
F (z) − F (ζ)
Fα (z, ζ)
(10)
A NOTE ON RUSCHEWEYH TYPE OF INTEGRAL OPERATORS . . .
185
and differentiating with respect to z, we have
Fα (z, ζ) Fα (z, ζ)
1
F (z)
1
F (z)
+
−
=
+
−
;
F (z) z − ζ F (z) − F (ζ)
z − ζ Fα (z, ζ) Fα (z, ζ)
(z − ζ)Fα (z, ζ) Fα (z, ζ)(z − ζ)
(z − ζ)F (z)
(z − ζ)F (Z)
+1−
= 1+
−
.
F (z)
F (z) − F (ζ)
Fα (z, ζ)
Fα (z, ζ)
(11)
(12)
Substituting (10) and (12) in (8), we obtain
(z − ζ)F (z)
(z − ζ)F (z)
(z − ζ)F (z)
+
1
−
+
cα
+
Fα1/α−1 (z, ζ)Fα (z, ζ) α
F (z)
F (z) − F (ζ)
F (z) − F (ζ)
1/α−1
= (cα + 1)f (z) f (z) − f (ζ)
;
(z − ζ)F (z)
(z − ζ)F (z)
+
cα
Fα1/α−1 (z, ζ)Fα (z, ζ) (1 − α)
+α 1+
F (z) − F (ζ)
F (z)
1/α−1
= (cα + 1)f (z) f (z) − f (ζ)
.
(13)
Setting
P (z, ζ) = (1 − α)
(z − ζ)F (z)
(z − ζ)F (z)
+α
+1 ,
F (z) − F (ζ)
F (z)
(14)
equation (13) becomes
1/α−1
Fα1/α−1 (z, ζ)Fα (z, ζ) P (z, ζ) + cα = (cα + 1)f (z) f (z) − f (ζ)
.
(15)
Taking logarithmic differentiation with respect to z, we get
(1 − α)(z − ζ)
Fα (z, ζ)
F (z, ζ) α(z − ζ)P (z, ζ)
+ α(z − ζ) α
+
+α
Fα (z, ζ)
Fα (z, ζ)
P (z, ζ) + cα
(z − ζ)f (z)
(z − ζ)f (z)
+ (1 − α)
;
f (z)
f (z) − f (ζ)
F (z, ζ)
(z − ζ)Fα (z, ζ)
(z − ζ)Fα (z, ζ) α(z − ζ)P (z, ζ)
α (z − ζ) α
+1−
+
+
Fα (z, ζ)
Fα (z, ζ)
Fα (z, ζ)
P (z, ζ) + cα
(z − ζ)f (z)
(z − ζ)f (z)
+ (1 − α)
= α 1+
.
f (z)
f (z) − f (ζ)
= α+α
(16)
Equations (10) and (12) give
α
(z − ζ)F (z)
(z − ζ)F (z)
(z − ζ)F (z) α(z − ζ)P (z, ζ)
+
1
−
+
+
F (z)
F (z) − F (ζ)
F (z) − F (ζ)
P (z, ζ) + cα
(z − ζ)f (z)
(z − ζ)f (z)
+ (1 − α)
= α 1+
.
f (z)
f (z) − f (ζ)
(17)
186
M. A. DURAI AND R. PARVATHAM
That is
(z − ζ)F (z)
α(z − ζ)P (z, ζ)
(z − ζ)F (z)
+
(1
−
α)
+
α 1+
F (z)
F (z) − F (ζ)
P (z, ζ) + cα
(z − ζ)f (z)
(z − ζ)f (z)
+ (1 − α)
.
= α 1+
f (z)
f (z) − f (ζ)
(18)
Hence, we have
P (z, ζ) +
(z − ζ)f (z)
(z − ζ)f (z)
α(z − ζ)P (z, ζ)
= α 1+
+ (1 − α)
.
P (z, ζ) + cα
f (z)
f (z) − f (ζ)
Since f (z) is uniformly α-convex, we have
α(z − ζ)P (z, ζ)
Re P (z, ζ) +
>0
P (z, ζ) + cα
(19)
(20)
for all z, ζ ∈ ∆, 0 ≤ α ≤ 1.
We show that Re P (z, ζ) > 0. Suppose that there exists a point ζ0 ∈ ∆ such that the
image of the arc Γ : z(t) = ζ0 + r eit is tangent to the imaginary axis. Let w0 be the
point of contact and let z0 ∈ ∆ such that w0 = P (z0 , ζ0 ). Then Re P (z0 , ζ0 ) = 0 and
therefore P (z0 , ζ0 ) = i x, where x ∈ R. Hence the outer normal to F (Γ ) is
(21)
z0 − ζ0 P z0 , ζ0 = y < 0.
For such ζ0 , we have
α z 0 − ζ0 P z 0 , ζ0
αy
Re P z0 , ζ0 +
= Re ix +
P z0 , ζ0 + cα
cα + ix
αy(cα − ix)
= Re ix +
c 2 α2 + x 2
=
cα2 y
<0
(cα)2 + x 2
(22)
for c > 0
which contradicts (20) and hence Re P (z, ζ) > 0 in ∆ showing that F (z) is a uniformly
α-convex function.
Acknowledgement. This work was carried out when the first author was under
the Faculty Improvement programme of University Grant and Commission of IX plan.
References
[1]
[2]
[3]
[4]
[5]
A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), no. 1, 87–92.
, On uniformly starlike functions, J. Math. Anal. Appl. 155 (1991), no. 2, 364–370.
S. Kanas, Uniformly alpha convex functions, Int. J. Appl. Math. 1 (1999), no. 3, 305–310.
P. T. Mocanu, Une propriété de convexité généralisée dans la théorie de la représentation
conforme, Mathematica (Cluj) 11 (34) (1969), 127–133 (French).
S. Ruscheweyh, Eine Invarianzeigenschaft der Basileviv Funktionen, Math. Z. 134 (1973),
215–219 (German).
M. Anbu Durai: The Ramanujan Institute for Advanced Study in Mathematics,
University of Madras, Chennai-600 500, India
E-mail address: anbuduraim@yahoo.co.in
R. Parvatham: The Ramanujan Institute for Advanced Study in Mathematics,
University of Madras, Chennai-600 500, India