American International Journal of Research in Science, Technology, Engineering & Mathematics
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New Classes Containing Ruscheweyh Derivative and a New Generalized Multiplier Differential Operator S R Swamy Department of Computer Science and Engineering, R V College of Engineering, Mysore Road, Bangalore-560 059, India Abstract: New subclasses of analytic functions, containing the linear operator obtained as a linear combination of Ruscheweyh derivative and a new generalized multiplier transformation have been considered. Sharp results concerning coefficients, distortion theorems of functions belonging to these classes are determined. Furthermore, Functions with negative coefficients belonging to these classes are also discussed. 2010 Mathematics Subject Classification: 30C45, 30A20, 34A40. Key words and phrases: Analytic function, differential operator, multiplier transformation. I.
Introduction.
U {z C : z 1}. Let H (U ) be the space of holomorphic functions in U . Let A denote the family of functions in H (U ) of the form Denote by
U
the open unit disc of the complex plane,
f ( z) z ak z k .
(1.1)
k 2
The author has recently introduced the following new generalized multiplier differential operator in [12]. Definition 1.1 Let m N 0 N {0}, 0, a real number such that 0 . Then for f A , a new generalized multiplier operator
I m, was defined by
I 0 , f ( z ) f ( z ) , I 1 , f ( z )
f ( z ) zf ' ( z ) , …, I m, f ( z ) I , ( I m,1 f ( z )) .
Remark 1.2 Observe that for f (z ) given by (1.1), we have
I m, f ( z ) z Ak ( , , m)a k z k ,
(1.2)
k 2
where m
k . Ak ( , , m) m m We note that: i) I1 , , 0 f ( z ) D f ( z ), 0
(1.3) (See
F.
M.
Al-Oboudi
[1]),
ii)
I lm1 , ,0 f ( z ) I lm, f ( z ), l 1, 0 (See A. Catas [3] and he has considered for l 0 ) and iii) I m,1 f ( z ) I m f ( z ), 1 (See Cho and Srivastava [4]) and Cho and Kim [5]). D1m f ( z ) was introduced by Salagean [8] and was considered for Definition
1.3
([7])
m 0 in [2].
For m N 0 , f A ,
the
operator
R m is
defined
by
Rm : A A ,
R 0 f ( z ) f ( z ) , R 1 f ( z ) zf ' ( z ) ,…, (m 1) R m 1 f ( z ) z ( R m f ( z )) ' mR m f ( z ), z U . Remark 1.4 If
f ( z ) z k 2 a k z k A , then R m f ( z ) z k 2 Bk (m)a k z k , z U , where
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Bk (m)
(m k 1)! . m! (k 1)!
(1.4)
The author has introduced the following operator in [13]. Definition 1.5 Let f A, m N 0 N {0}, 0, [0,1), 0,
0.
Denote
by
RI
m , ,,
,
the
operator
a real number such that
given
RI m, , : A A,
by
RI m, , f ( z ) (1 ) R m f ( z ) I m, f ( z ), z U . The operator was studied also in [10] and [11]. Clearly RI , ,, 0 R m
Remark 1.6 If
m
and RI , ,,1 I , . m
m
f ( z ) z k 2 a k z k , then from (1.2) and Remark 1.4, we have
RI m, , f ( z ) z k 2 {(1 ) Bk (m) Ak ( , , m)}a k z k , z U ,
where Ak ( , , m) and B k (m) are as defined in (1.3) and (1.4), respectively. m
We introduce new classes as below by making use of the generalized operator RI , , . Definition
1.7
f A, m N 0 N {0}, 0, [0,1), 0,
Let
that 0 . Then f (z ) is in the class
that 0 . Then f (z ) is in the class
m , , ( ) if and only if
z ( RI m, , f ( z )) ' Re RI m, , f ( z )
z ( RI m, , f ( z )) '' 1 ( RI m, , f ( z )) '
1.9
that 0 . Then f (z ) is in the class
number
such
m , ,
2 z[ z ( RI m, , f ( z )) ' ] ' Re ( RI m f ( z )) ' , ,
(1.5) a
real
number
such
, z U .
f A, m N 0 N {0}, 0, [0,1), 0,
Let
real
m, , ( ) if and only if
[ z 2 ( RI m, , f ( z )) ' ] ' , z U . Re 2 RI m f ( z ) , , Definition 1.8 Let f A, m N 0 N {0}, 0, [0,1), 0,
Definition
a
(1.6)
a
real
number
such
( ) if and only if
, z U .
(1.7)
In section 2 we study the characterization properties for the function f A to belong to the classes , , m
,
( )
m , , ( ) and m, , ( ) by obtaining the coefficient bounds. Distortion theorems of functions belonging
to these classes are obtained in section 3. Analytic functions with negative coefficients belonging to the above classes are considered in section 4. II. Gene1ral Properties. In this section we study the characterization properties following the paper of M. Darus and R. Ibrahim [6]. Throughout this paper, unless otherwise mentioned we shall assume that Ak ( , , m) and B k (m) are as defined in (1.3) and (1.4) respectively,. Theorem 2.1 Let f A, m N 0 N {0}, 0, [0,1), 0,
a real number such that
0 . i) If
(k
2
k 2
then
k 2 ){(1 ) B k (m) Ak ( , , m)} a k 2(1 ),
(2.1)
f ( z ) m, , ( ) .
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ii) If
{(2k
2
k 2
then
(k 1) )}{(1 ) B k (m) Ak ( , , m)} a k 2(1 ),
(2.2)
f ( z ) m , , ( ) .
iii) If
{(k k 2
then
2
)}{(1 ) B k (m) Ak ( , , m)} a k 1 ,
(2.3)
f ( z ) m, , ( ) . The results (2.1), (2.2) and (2.3) are sharp.
Proof. i) It suffices to show that the values of
[ z 2 ( RI m, , f ( z )) ' ] ' / 2 RI m, , f ( z ) lie in a circle centred at
1 with radius 1 . We have
[ z 2 ( RI m, , f ( z )) ' ] ' 2 RI m, , f ( z )
1
[ z 2 ( RI m, , f ( z )) ' ] ' 2 RI m, , f ( z ) 2 RI m, , f ( z )
k 2
(k 2 k 2){(1 ) B k (m) Ak ( , , m)} a k
2[1 k 2 {(1 ) B k (m) Ak ( , , m)} a k ]
.
The last expression is bounded above by 1 if
k 2
(k 2 k 2){(1 ) B k (m) Ak ( , , m)} a k
2(1 ) 1 k 2 {(1 ) Bk (m) Ak ( , , m)} a k , which is equivalent to (2.1). Hence
[ z 2 ( RI m, , f ( z )) ' ] ' 2 RI m, , f ( z )
1 1 , and the theorem is proved.
The proof of the second and third part of the theorem is similar and so omitted. The assertions (2.1),(2.2) and (2.3) are sharp and extremal functions are given by
2(1 ) z k , z U , k 2 ( k k 2 ){(1 ) Bk ( m) Ak ( , , m)} 2(1 ) f ( z) z z k , z U , 2 k 2 {( 2k ( k 1) )}{(1 ) B k ( m) Ak ( , , m)}
f ( z) z
and
2
1 z k , z U ( k ){( 1 ) B ( m ) A ( , , m )} k 2 k k
f ( z) z
2
respectively. Theorem 2.2 Let the hypothesis of the Theorem 2.1 i) ,ii) and iii) be satisfied. Then
i)
ii)
iii)
2(1 ) , k 2. (k k 2 ){(1 ) B k (m) Ak ( , , m)} 2(1 ) ak , k 2. 2 {( 2k (k 1) }{(1 ) B k (m) Ak ( , , m)} 1 ak 2 , k 2. (k ){(1 ) Bk (m) Ak ( , , m)} ak
2
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The following inclusion results can be proved using Theorem 2.1. Theorem 2.3 Let 0 1 2 1 . Then i)
m, , ( 2 ) m, , ( 1 ) , ii) m , , ( 2 ) m , , ( 1 ) and iii) m, , ( 2 ) m, , ( 1 ) . III.
Distortion theorems
Theorem 3.1 Let f A, m N 0 N {0}, 0, [0,1), 0,
0 .i) If
[( k
2
k 2
z
ii) If
[( 2k
2
k 2
(k
2
k 2
k 2 )]{(1 ) B k ( m) Ak ( , , m)} a k 2(1 ), then
2(1 ) z 5 2
2
RI m, , f ( z ) z
2(1 ) 2 z , z U . 5 2
(k 1) )]{(1 ) B k (m) Ak ( , , m)} a k 2(1 ), then z
iii) ) If
a real number such that
2(1 ) z 8 3
2
RI m, , f ( z ) z
2(1 ) 2 z , z U . 8 3
){(1 ) B k ( m) Ak ( , , m)} a k 1 , then
z
1 2 1 2 z RI m, , f ( z ) z z , z U . 4 4
Proof i) Note that (5 2 )
(1 ) B k 2
k
(m) Ak ( , , m) a k
(k
2
k 2
by Theorem 2.1. Thus
(1 ) B k 2
k
k 2 )(1 ) B k (m) Ak ( , , m) a k 2(1 ) ,
(m) Ak ( , , m) a k
2(1 ) .Hence we obtain 5 2
RI m, , f ( z ) z (1 ) B k ( m) Ak ( , , m) a k z
k
k 2
zz
(1 ) B
2
k 2
z
k
( m) Ak ( , , m) a k
2(1 ) 2 z . 5 2
Similarly
RI m, , f ( z ) z (1 ) B k ( m) Ak ( , , m) a k z
k
k 2
zz
2
(1 ) B k 2
z
k
(m) Ak ( , , m) a k
2(1 ) 2 z . 5 2
This completes the proof of (i). (ii) and iii) can be proved on similar lines. Theorem
3.2
f A, m N 0 N {0}, 0, [0,1), 0,
Let
that 0 . i) If
[( k k 2
2
a
real
number
such
k 2 )]{(1 ) B k ( m) Ak ( , , m)} a k 2(1 ), then
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f ( z) z
2(1 ) 2 z , z U (5 2 ){(m 1)(1 ) A2 ( , , m)}
f ( z) z
2(1 ) 2 z , z U . (5 2 ){(m 1)(1 ) A2 ( , , m)}
and
ii) If
[( 2k
2
k 2
(k 1) )]{(1 ) B k (m) Ak ( , , m)} a k 2(1 ), then
f ( z) z
2(1 ) 2 z , z U (8 3 ){(m 1)(1 ) A2 ( , , m)}
f ( z) z
2(1 ) 2 z , z U . (8 3 ){(m 1)(1 ) A2 ( , , m)}
and
iii) If
[( k k 2
2
)]{(1 ) B k ( m) Ak ( , , m)} a k 1 , then f ( z) z
1 2 z , z U (4 ){(m 1)(1 ) A2 ( , , m)}
f ( z) z
1 2 z , z U . (4 ){(m 1)(1 ) A2 ( , , m)}
and
Proof i) In virtue of Theorem 2.1, we have
(5 2 ){(m 1)(1 ) A2 ( , , m)} a k k 2
(k
2
k 2
Thus
a k 2
k
k 2 )(1 ) B k (m) Ak ( , , m) a k 2(1 ) .
2(1 ) .So we get (5 2 ){(m 1)(1 ) A2 ( , , )} 2(1 ) 2 2 f ( z) z z a k z z . (5 2 ){(m 1)(1 ) A2 ( , , m)} k 2
On the other hand
f ( z) z z
2
a k 2
k
z
2(1 ) 2 z . (5 2 ){(m 1)(1 ) A2 ( , , m)}
This completes the proof. The proof of (ii) and iii) are similar.
IV.
Functions with negative coefficients
k Let T denote the subclass of A consisting of functions of the form f ( z ) z a k z , a k 0 . k 2
m , ,
We denote by T
( ) , T , , ( ) and T m
m , ,,
( ) , the classes of functions f ( z ) T satisfying
(1.5),(1.6) and (1.7) respectively. We study the coefficient estimates, distortion theorems and other properties of these classes, following the paper of H. Silverman [9]. For functions in T , the converses of Theorem 2.1 is also true. Theorem 4.1 i) A function f ( z ) T is in
Tm, , ( ) if and only if
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(k
2
k 2
k 2 )(1 ) B k (m) Ak ( , , m)a k 2(1 ).
ii) A function f ( z ) T is in
( 2k k 2
2
(k k 2
2
T m , , ( ) if and only if
(k 1) )(1 ) B k (m) Ak ( , , m)a k 2(1 ). .
iii) A function f ( z ) T is in
(4.1)
(4.2)
Tm, , ( ) if and only if
)(1 ) B k ( m) Ak ( , , m)a k 1 .
(4.3)
The results are sharp. Proof i) In view of Theorem 2.1, it suffices to prove the only if part. Assume that
2 z {(1 ) B k (m) Ak ( , , m)}k (k 1)a k z k [ z 2 ( RI m, , f ( z )) ' ] ' k 2 Re Re k 2 RI m f ( z ) , , 2 z k 2 {(1 ) B k (m) Ak ( , , m)}2a k z
.
(4.4)
letting z 1 through real values in (4.4), we obtain 2 {(1 ) B k (m) Ak ( , , m)}k (k 1)a k 2 1 {(1 ) B k (m) Ak ( , , m)}a k . k 2 k 2
Hence we obtain (4.1), and the proof is complete. The proof of ii) and iii) are similar. Finally, we note that assertions (4.1), (4.2) and (4.3) of Theorem 4.1 are sharp, extremal functions being
1 z k , z U , k 2 ( k k 2 ){(1 ) B k ( m) Ak ( , , m)}
f ( z) z
2
1 z k , z U , k 2 {( 2k ( k 1) )}{(1 ) B k ( m) Ak ( , , m)}
f ( z) z and
2
1 z k , z U , 2 k 2 ( k ){(1 ) B k ( m) Ak ( , , m)}
f ( z) z respectively.
Our coefficient bounds enable us to prove the following. f A, m N 0 N {0}, 0, [0,1), 0, Theorem 4.2 Let that 0 .i) If A function f ( z ) T is in
m , ,
T
a
real
2(1 ) 2 z , z U (5 2 ){(m 1)(1 ) A2 ( , , m)}
f ( z) z
2(1 ) 2 z , z U . (5 2 ){(m 1)(1 ) A2 ( , , m)}
T m , , ( ) then
ii) If A function f ( z ) T is in
f ( z) z
2(1 ) 2 z , z U (8 3 ){(m 1)(1 ) A2 ( , , m)}
f ( z) z
2(1 ) 2 z , z U . (8 3 ){(m 1)(1 ) A2 ( , , m)}
and
such
( ) then
f ( z) z and
number
iii) If A function f ( z ) T is in
Tm, , ( ) then
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f ( z) z
1 2 z , z U (4 ){(m 1)(1 ) A2 ( , , m)}
f ( z) z
1 2 z , z U . (4 ){(m 1)(1 ) A2 ( , , m)}
and
Remark 4.3 The bounds of i) and ii) of Theorem 4.2 are sharp since the equalities are attained for the functions
2(1 ) z 2 ( z r ) , (5 2 ){(m 1)(1 ) A2 ( , , m)} 1 f ( z) z z 2 ( z r ) and (8 3 ){(m 1)(1 ) A2 ( , , m)} 1 f ( z) z z 2 ( z r ) ,respectively. (4 ){(m 1)(1 ) A2 ( , , m)}
f ( z) z
References [1] [2] [3]
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Int. J. Math. Math. Sci., 27(2004), 14291436. S. S. Bhoosnurmath and S. R. Swamy, On certain classes of analytic functions, Soochow J. Math., 20 (1) (1994), 1-9. A. Catas, On certain class of p-valent functions defined by new multiplier transformations, Proceedings book of the an international symposium on geometric function theory and applications, August, 20-24, 2007, TC Istanbul Kultur Univ., Turkey, 241-250. N. E, Cho and H. M. Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Modeling, 37(1-2) (2003), 39-49. N. E, Cho and T. H. Kim, Multiplier transformations and strongly Close-to Convex functions, Bull. Korean Math. Soc., 40(3) (2003), 399-410. M. Darus and R. Ibrahim, New classes containing generalization of differential operator, Applied Math. Sci., 3 (51) (2009) , 2507-2515. St. Ruscheweyh, New citeria for univalent functions, Proc. Amer. Math. Soc., 49(1975), 109-115. G. St. Salagean, Subclasses of univalent functions, Proc. Fifth Rou. Fin. Semin. Buch. Complex Anal., Lect. notes in Math., Springer -Verlag, Berlin, 1013(1983), 362-372. H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51(1) (1975),109-116. S.R. Swamy, On analytic functions defined by a new generalized multiplier differential operator and Ruscheweyh derivative, preprint. S.R. Swamy, Subordination and superordination results for certain subclasses of analytic functions defined by the Ruscheweyh derivative and a new generalized multiplier transformation, J. Global Res. Math. Arch., 1 (6) (2013), 27-37. S. R. Swamy, Inclusion properties of certain subclasses of analytic functions, Int. Math. Forum, 7(36) (2012), 1751-1760. S. R. Swamy, A note on a subclass of analytic functions defined by the Ruscheweyh derivative and a new generalized multiplier transformation, J. Math. Computational Sci., 2(4) (2012), 784-792.
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