### THE KOMATU INTEGRAL OPERATOR AND STRONGLY

```Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 3 Issue 3(2011), Pages 209-219.
THE KOMATU INTEGRAL OPERATOR AND STRONGLY
CLOSE-TO-CONVEX FUNCTIONS
(C O M M U N IC AT E D B Y C L A U D IO C U E VA S )
M. K. AOUF
Abstract. In this paper we introduce some new subclasses of strongly closeto-convex functions de…ned by using the Komatu integral operator and study
their inclusion relationships with the integral preserving properties.
Theorem[section] [theorem]Lemma [theorem]Proposition [theorem]Corollary Remark
1. Introduction
Let A1 denote the class of functions of the form :
f (z) = z +
1
X
ak z k
(1.1)
k=2
which are analytic in the open unit disc U = fz : jzj < 1g. If f (z) and g(z) are
analytic in U , we say that f (z) is subordinate to g(z), written f g or f (z) g(z),
if there exists a Schwarz function w(z) in U such that f (z) = g(w(z)).
A function f (z) 2 A1 is said to be starlike of order if it satis…es
( 0
)
zf (z)
Re
>
(z 2 U )
(1.2)
f (z)
for some (0
< 1). We denote by S ( ) the subclass of A1 consisting of
functions which are starlike of order in U . Also a function f (z) 2 A1 is said to
be convex of order if it satis…es
(
)
00
zf (z)
>
(z 2 U )
(1.3)
Re 1 + 0
f (z)
for some (0
< 1). We denote by C( ) the subclass of A1 consisting of all
functions which are convex of order in U .
2000 Mathematics Subject Classi…cation. 30C45.
Key words and phrases. Komatu integral operator, strongly close-to-convex.
c 2011 Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted May 20, 2011. Published July 5, 2011.
209
210
M . K. AOUF
It follows from (1:2) and (1:3) that
0
f (z) 2 C( ) , zf (z) 2 S ( ) :
(1.4)
The classes S ( ) and C( ) are introduced by Robertson  (see also Srivastava
and Owa ).
Let f (z) 2 A1 and g(z) 2 S ( ). Then f (z) 2 K( ; ) if and only if
)
( 0
zf (z)
>
(z 2 U ) ;
(1.5)
Re
g(z)
where 0
< 1 and 0
< 1. Such functions are called close-to-convex functions
of order and type . The class K( ; ) was introduced by Libera  (see also
Noor and Alkhorasani  and Silverman ).
If f (z) 2 A1 satis…es
0
arg(
zf (z)
f (z)
) <
(z 2 U )
2
(1.6)
for some (0
< 1) and (0 <
1), then f (z) is said to be strongly starlike
of order and type in U . We denote this by S ( ; ).
If f (z) 2 A1 satis…es
00
arg(1 +
zf (z)
f 0 (z)
) <
2
(z 2 U )
(1.7)
for some (0
< 1) and (0 <
1), then we say that f (z) is strongly
convex of order and type in U . We denote this class by C( ; ) (see also Liu
 and Nunokawa ). In particular, the classes S ( ; 0) and C( ; 0) have been
extensively studied by Mocanu  and Nunokawa .
It follows from (1.6) and (1.7) that
0
f (z) 2 C( ; ) , zf (z) 2 S ( ; ) :
(1.8)
Also we note that S (1; ) = S ( ) and C(1; ) = C( ).
Recently, Komatu  introduced a certain integral operator Ia (a > 0;
de…ned by
Z1
a
1
Ia f (z) =
ta 2 (log ) 1 f (zt)dt
( )
t
0)
(1.9)
0
(z 2 U ; a > 0;
0; f 2 A1 ) :
Thus , if f (z) 2 A1 is of the form (1.1), it is easily seen from (1.9) that
Ia f (z) = z +
1
X
k=2
a
a+k
1
ak z k
(a > 0;
0) :
(1.10)
(a > 0;
0) :
(1.11)
Using the above relation, it is easy verify that
0
z(Ia +1 f (z)) = aIa f (z)
(a
1)Ia +1 f (z)
We note that :
(i) For a = 1 and = n (n is any integer), the multiplier transformation I1n f (z) =
n
I f (z) was studied by Flett  and Salagean ;
(ii) For a = 1 and
= n (n 2 N0 2 f0; 1; 2; :::g), the di¤erential operator
I1 n f (z) = Dn f (z) was studied by Salagean ;
THE KOM ATU INTEGRAL OPERATOR AND STRONGLY CLOSE-TO-CONVEX...
211
(iii) For a = 2 and = n (n is any integer), the operator I2n f (z) = Ln f (z) was
studied by Uralegaddi and Somanatha ;
(iv) For a = 2, the multiplier transformation I2 f (z) = I f (z) was studied by
Jung et al. .
For a > 0 and
0, let K a ( ; ; ; A; B) be the class of functions f (z) 2 A1
satisfying the condition
!
0
z(Ia f (z))
<
arg
(0
< 1; 0 <
1; z 2 U ) ;
(1.12)
Ia g(z)
2
for some f (z) 2 S a ( ; A; B), where
(
a
S ( ; A; B) =
0
z(Ia g(z))
Ia g(z)
1
g 2 A1 :
1
!
1 + Az
1 + Bz
)
(0
< 1; 1 B < A 1; z 2 U ) :
(1.13)
a
We note that
; 1; ; 1; 1) = K( ; ). We also note that K0 (0; ; 0; 1; 1) is
the class of strongly close-to-convex functions of order in the sense of Pommerenke
. Also the class S0a ( ; A; B) = S( ; A; B) was studied by Aouf .
In the present paper, using the technique of Cho , we give some argument
properties of analytic functions belonging to A1 which contain the basic inclusion
relationships among the classes K a ( ; ; ; A; B). The integral preserving properties in connection with the operators Ia de…ned by (1.10) are also considered.
Furthermore, we obtain the previous results given by Bernardi  and Libera  as
special cases.
K0a (
2. Main Results
In proving our main results, we need the following lemmas.
Lemma 1. . Let h(z) be convex univalent in U with h(0) = 1 and Ref h(z) +
g > 0 ( ; 2 C). If p(z) is analytic in U with p(0) = 1, then
0
zp (z)
p(z) +
p(z) +
h(z)
(z 2 U );
implies
p(z)
h(z)
(z 2 U ):
Lemma 2. . Let h(z) be convex univalent in U and w(z) be analytic in U with
Re w(z) 0. If p(z) is analytic in U and p(0) = h(0), then
0
p(z) + w(z)zp (z)
h(z)
(z 2 U ) ;
implies
p(z)
h(z)
(z 2 U ) :
Lemma 3. . Let p(z) be analytic in U with p(0) = 1 and p(z) 6= 0 in U . If
there exist two points z1 ; z2 2 U such that
1
2
for some
1;
2
(
0
1;
2
= arg p(z1 ) < arg p(z) < arg p(z2 ) =
2
2
> 0) and for all z (jzj < jz1 j = jz2 j), then we have
z1 p (z1 )
=
p(z1 )
i
1
+
2
2
0
m
and
z2 p (z2 )
=i
p(z2 )
1
+
2
2
m;
(2.1)
(2.2)
212
M . K. AOUF
where
1 jcj
2
1
and c = i tan (
):
1 + jcj
4 1+ 2
At …rst, with the help of Lemma 1, we obtain the following :
m
(2.3)
Proposition 4. Let a 1 and h(z) be convex univalent in U with h(0) = 1 and
Re h(z) > 0. If a function f (z) 2 A1 satis…es the condition
!
0
1
z(Ia f (z))
h(z) (0
< 1; z 2 U );
1
Ia f (z)
then
!
0
z(Ia +1 f (z))
Ia +1 f (z)
1
1
h(z) (0
Proof. Let
p(z) =
!
0
z(Ia +1 f (z))
Ia +1 f (z)
1
1
< 1; z 2 U ) :
(z 2 U ) ;
(2.4)
where p(z) is analytic function in U with p(0) = 1. By using (1:11), we get
a
1 + + (1
)p(z) = a
Ia f (z)
:
Ia +1 f (z)
(2.5)
Di¤erentiating (2.5) logarithmically with respect to z and multiplying by z, we
obtain
!
0
0
zp (z)
1
z(Ia f (z))
p(z) +
=
(z 2 U ):
a 1 + + (1
)p(z)
1
Ia f (z)
By using Lemma 1, it follows that p(z)
1
1
0
+1
z(Ia f (z))
Ia +1 f (z)
h(z), that is,
!
h(z)
(z 2 U ) :
1 + Az
( 1 B < A 1), in Proposition 1, we have
1 + Bz
Corollary 5. The inclusion relation, S a ( ; A; B)
S a+1 ( ; A; B), holds for any
a > 0 and
0.
Taking h(z) =
Proposition 6. Let h(z) be convex univalent in U with h(0) = 1 and Re h(z) > 0.
If a function f (z) 2 A1 satis…es the condition
!
0
1
z(Ia f (z))
h(z) (0
< 1; z 2 U );
1
Ia f (z)
then
1
1
0
z(Ia L f (z))
Ia L f (z)
!
h(z) (0
where L (f ) is the integral operator de…ned by
Zz
+1
L (f ) = L f (z) =
t 1 f (t)dt
z
0
< 1; z 2 U ) ;
(
0) :
(2.6)
THE KOM ATU INTEGRAL OPERATOR AND STRONGLY CLOSE-TO-CONVEX...
213
Proof. From (2.6), we have
0
z(Ia L f (z)) = ( + 1)Ia f (z)
Ia L (f )(z) :
Let
p(z) =
!
0
1
z(Ia L f (z))
Ia L f (z)
1
(2.7)
(z 2 U ) ;
where p(z) is analytic function in U with p(0) = 1. Then, by using (2.7), we have
+ + (1
)p(z) = ( + 1)
Ia f (z)
:
Ia L (f )(z)
(2.8)
Di¤erentiating (2.8) logarithmically with respect to z and multiplying by z, we have
!
0
0
zp (z)
1
z(Ia f (z))
p(z) +
=
(z 2 U ):
+ + (1
)p(z)
1
Ia f (z)
Therefore, by using Lemma 1, we obtain that
!
0
z(Ia L f (z))
1
1
Ia L f (z)
Taking h(z) =
:
1 + Az
( 1
1 + Bz
B<A
h(z) (z 2 U ):
1), in Proposition 2, we have immediately
Corollary 7. If f (z) 2 S a ( ; A; B), then L (f ) 2 S a ( ; A; B), where L (f ) is the
integral operator de…ned by (2.6).
We now derive:
Theorem 8. Let f (z) 2 A1 and 0 <
1
2
1; 2
0
z(Ia f (z))
Ia g(z)
< arg
for some g(z) 2 S a ( ; A; B), then
2
where
8
>
>
<
1 =
>
>
:
and
2
=
8
>
>
<
>
>
:
1
and
1
+
2
2
(0 <
tan
1
1;
< arg
2
0
z(Ia +1 f (z))
Ia +1 g(z)
(
(1
!
< 1. If
<
!
<
2
2
2
2
;
1) are the solutions of the equations
1
2(
1; 0
)(1+A)
+
1+B
1+
2 )(1
jcj) cos
+a 1)(1+jcj)+(
2 t1
1+
2 )(1
jcj) sin
2 t1
2
1;
for B = 1;
(2.9)
1
2
for B 6=
+
2
tan
(
1
2(
(1
)(1+A)
+
1+B
1+
2 )(1
jcj) cos
+a 1)(1+jcj)+(
2 t1
1+
2 )(1
jcj) sin
2 t1
for B 6=
1;
for B = 1;
(2.10)
214
M . K. AOUF
where c is given by (2.3) and
2
t1 =
sin
1
(1
)(1
(1
)(1 B)
AB) + ( + a 1)(1
Proof. Let
p(z) =
1
(2.11)
1)Ia +1 f (z) :
(2.12)
!
0
z(Ia +1 f (z))
Ia +1 g(z)
1
:
B2)
:
Using the identity (1.11) and simplifying, we have
)p(z) + ]Ia +1 g(z) = aIa f (z)
[(1
(a
Di¤erentiating (2.12) with respect to z and multiplying by z, we obtain
0
0
)zp (z)Ia +1 g(z)+[(1
0
0
(a 1)z(Ia +1 f (z)) :
(2.13)
Since g(z) 2 S a ( ; A; B), from Corollary 1, we know that g(z) 2 S a+1 ( ; A; B). Let
!
0
z(Ia +1 g(z))
1
(z 2 U ) :
q(z) =
1
Ia +1 g(z)
(1
)p(z)+ ]z(Ia +1 g(z)) = az(Ia f (z))
Then, using the identity (1.11) once again, we have
(1
)q(z) + + a
From (2.13) and (2.14), we obtain
0
z(Ia f (z))
Ia g(z)
1
1
!
1=a
Ia g(z)
:
Ia +1 g(z)
(2.14)
0
= p(z) +
(1
zp (z)
)q(z) + + a
1
;
while, by using the result of Silverman and Silvia , we have
q(z)
1
1
AB
(A B)
<
(z 2 U ; B 6=
B2
1 B2
1) ;
(2.15)
and
Refq(z)g >
1
A
2
Then, from (2.15) and (2.16), we obtain
(1
(z 2 U ; B =
)q(z) + + a
1 = ei
1) :
'
2
(2.16)
;
where
(1
)(1 A)
1 B
+ + a 1 < < (1
t1 < ' < t1
for B =
6
1;
)(1+A)
1+B
+ +a
1;
when t1 is given by (2.11), and
(1
)(1 A)
2
+ +a 1< <1 ;
1<'<1
for B = 1 :
Here, we note that p(z) is analytic in U with p(0) = 1 and Re p(z) > 0 in U
1
by applying the assumption and Lemma 2 with w(z) =
.
(1
)q(z) + + a 1
Hence p(z) 6= 0 in U . If there exist two points z1 ; z2 2 U such that the condition
THE KOM ATU INTEGRAL OPERATOR AND STRONGLY CLOSE-TO-CONVEX...
215
(2.1) is satis…ed, then (by Lemma 3) we obtain (2.2) under the restriction (2.3). At
…rst, for the case B 6= 1, we obtain :
!
0
z1 p (z1 )
arg p(z1 ) +
(1
)q(z1 ) + + a 1
=
=
2
1
2
1
2
1
2
1
+ 2
m( ei 2 ) 1
2
( 1 + 2 )m sin 2 (1 ')
tan 1
2 + ( 1 + 2 )m cos 2 (1 ')
8
<
( 1 + 2 )(1 jcj) cos 2 t1
tan 1
: 2 (1 )(1+A) + + a 1 (1 + jcj) + ( +
1
1+B
+ arg 1
;
1
i
and
0
arg p(z2 ) +
2
2
1
+ tan
(1
8
<
z2 p (z2 )
)q(z2 ) + + a
1
(
:2
(1
)(1+A)
1+B
2 )(1
9
=
jcj) sin 2 t1 ;
!
+
1
+ +a
2 )(1
jcj) cos 2 t1
1 (1 + jcj) + (
1
+
2 )(1
jcj) sin 2 t1 ;
2 ;
2
where we have used the inequality (2.3), and 1 ; 2 and t1 are given by (2.9), (2.10)
and (2.11), respectively. Similarly, for the case B = 1, we obtain
!
0
z1 p (z1 )
arg p(z1 ) +
1
(1
)q(z1 ) + + a 1
2
=
and
0
arg p(z2 ) +
z2 p (z2 )
)q(z2 ) + + a
(1
1
!
2
2
:
These are contradiction to the assumption of Theorem 1. This completes the proof
of Theorem 1.
Taking
1
=
2
=
in Theorem 1, then we obtain :
Corollary 9. The inclusion relation, K a ( ; ; ; A; B)
for any a > 0 and
0.
Taking
= 0; a = 1 and
1
=
2
=
K a+1 ( ; ; ; A; B) holds
in Theorem 1, we obtain :
Corollary 10. Let f (z) 2 A1 . If
0
zf (z)
arg(
g(z)
) <
for some g 2 S( ; A; B), then
arg(
2
f (z)
I11 g(z)
(0
) <
< 1; 0 <
2
;
1);
9
=
216
M . K. AOUF
where
(0 <
=
1) is the solution of the equation :
8
>
>
<
+
2
cos
1
tan
(
(1
)(1+A)
+
1+B
2 t1
)+ sin
2 t1
; for B 6=
1;
for B =
1;
>
>
:
where t1 is given by (2.11) with a = 1.
Putting
obtain
=
= 0; a = 1; B ! A (A < 1), and g(z) = z in Theorem 1, we
Corollary 11. Let f (z) 2 A1 and 0 <
2
then
1
1
and
2
(0 <
1;
1
=
1
+
2
=
2
+
and
2
2
< arg
tan
1
(
1
tan
1
(
1
Next, we prove
Theorem 12. Let f (z) 2 A1 and 0 <
2
1
+ 2 )(1 jcj)
:
2(1 + jcj)
1; 0
0
z(Ia f (z))
Ia g(z)
< arg
< arg
+ 2 )(1 jcj)
2(1 + jcj)
1; 2
for some g(z) 2 S a ( ; A; B), then
2
;
2
2
f (z)
<
2 ;
2
z
2
1) are the solutions of the equations :
2
1
1. If
0
< arg f (z) <
1
where
1; 2
!
< 1. If
<
!
0
z(Ia L (f )(z))
Ia L (g)(z)
2
2
<
where L (f ) is de…ned by (2.6), and 1 and 2 (0 < 1 ;
of the equations :
8
( 1 + 2 )(1 jcj) cos t2
>
>
< 1 + 2 tan 1 2 (1 )(1+A) + + (1+jcj)+( +2 )(1
( 1+B
)
1
2
1 =
>
>
:
2
2
;
1) are the solutions
2
jcj) sin
2 t2
1
and
2
=
8
>
>
<
>
>
:
2
+
2
tan
(
1
2(
(1
)(1+A)
+
1+B
1+
+
2 )(1
jcj) cos
)(1+jcj)+(
1+
2 t2
2 )(1
jcj) sin
2 t2
2
for B 6=
1;
for B =
1;
for B 6=
1;
for B =
1;
where c is given by (2.3) and t2 is given by
t2 =
2
sin
1
(1
)(1
(1
)(A B)
AB) + ( + )(1
B2)
:
(2.17)
THE KOM ATU INTEGRAL OPERATOR AND STRONGLY CLOSE-TO-CONVEX...
Proof. Let
p(z) =
!
0
1
z(Ia L (f )(z))
Ia L (g)(z)
1
217
(z 2 U ) :
Since g(z) 2 S a ( ; A; B), we have from Corollary 2 that L (g) 2 S a ( ; A; B). Using
(2.7) we obtain
[(1
)p(z) + ]Ia L (g)(z) = ( + 1)Ia f (z)
Ia L (f )(z):
Then, by a simple calculation, we get
0
(1
)zp (z) + [(1
)p(z) + ][(1
)q(z) + + ] =
0
( + 1)
z(Ia f (z))
;
Ia L (g)(z)
where
q(z) =
1
Hence we have
!
0
1
z(Ia f (z))
Ia g(z)
1
!
0
z(Ia L (g)(z))
Ia L (g)(z)
1
:
0
= p(z) +
zp (z)
)q(z) + +
(1
:
The remaining part of the proof of Theorem 2 is similar to that of Theorem 1 and
so we omit it.
Taking
1
=
2
=
in Theorem 2, we have
Corollary 13. Let f (z) 2 A1 and 0
arg
< 1; 0 <
1. If
!
0
z(Ia f (z))
<
Ia g(z)
2
for some g(z) 2 S a ( ; A; B), then
0
arg
z(Ia L (f )(z))
Ia L (g)(z)
!
<
2
;
where L (f ) is de…ned by (2.6), and (0 <
1) is the solution of the equation:
8
cos
t
2
1
2 2
>
< + tan ( ( (1 )(1+A) + + )+ sin t2 ); for B 6= 1;
1+B
2
=
>
:
for B = 1;
where t2 is given by (2.17).
From Corollary 6, we see easily the following corollary.
Corollary 14. f (z) 2 K a ( ; ; ; A; B) =) L (f ) 2 K a ( ; ; ; A; B), where
L (f ) is the integral operator de…ned by (2.6) and
is the solution of equation
in Corollary 6.
Taking
= 0;
= 1; A = 1 and B =
1 in Corollary 7, we obtain :
218
M . K. AOUF
Corollary 15. Let f (z) 2 A1 . If
( 0
)
zf (z)
Re
>
g(z)
then
Re
(
0
z(L (f )(z))
L (g)(z)
)
(0
>
< 1);
(0
< 1) ;
where L (f ) is the integral operator de…ned by (2.6) and g(z) 2 S ( ) (0
< 1).
Remark 1. Taking = = = 0; A = = 1 and B = 1 in Corollary 7, we
obtain the classical result obtained by Bernardi , which implies the result studied
by Libera .
2.1. Acknowledgements. The author would like to thank the referee of the paper
References
 M. K. Aouf, On a class of p-valent starlike functions of order , Internat. J. Math. Math.
Sci. 10 4 (1987) 733-744.
 S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc. 35 (1969)
429-446.
 N. E. Cho, The Noor integral operator and strongly close-to-convex functions, J. Math. Anal.
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Department of Mathematics,, Faculty of Science,, Mansoura University,, Mansoura
35516, Egypt.