1Unite of Biomathematics, College of Computer Science and Mathematics, University of Al-Qadisiya, Diwiniya- ... In this paper, we introduce a class of functions.
Asian Journal of Current Engineering and Maths 2: 1 Jan –Feb (2013) 10 - 14.
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SOME PROPERTIES FOR CERTAIN CLASS OF P-VALENT FUNCTIONS INVOLVING NOOR OPERATOR WITH DIFFERENTIAL EQUATION Ahmed sallal Joudah, Jumana Hekma Salman* 1Unite
of Biomathematics, College of Computer Science and Mathematics, University of Al-Qadisiya, Diwiniya- Iraq
2Department
of Statistical and Informatics, College of Computer Science and Mathematics, University of Al-Qadisiya, Diwiniya- Iraq
ARTICLE INFO
ABSTRACT
Corresponding Author Jumana Hekma Salman 2Department of Statistical and Informatics, College of Computer Science and Mathematics, University of Al-Qadisiya, Diwiniya- Iraq Key Words: Differential Subordination,Multivalent function,Cauchy-Euler differential equation. INTRODUCTION Let
In this paper, we introduce a class of functions (C,B) which are analytic and p-valent in the unit disk U. We obtain a necessary and sufficient condition for a function to be in (C,B) ,distortion bounds and convex combination. 2010 Mathematics Subject Classification: Primary 30C45, Secondary 30C50,26A33.
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denote the family of all functions ,of the form:
which are analytic and p-valent in the open unit disk point in . For functions given by:
We define the Hadamard product(or convolution) of
,and
and
by:
Suppose that and are analytic in . Then we say that the functions function in with for all ,such that is equivalent to univalent in we have that the subordination For real or complex number defined by
is a fixed
is subordinate to ,denoted or and
, the hypergeometric series is
F
2 1
We note that the series in (3) converges absolutely for all analytic function in . The authors [3] introduced a function
F
so that it represents and given by
2 1
F
F
2 1
2 1
which leads us to the following family of linear operators:
F
2 1
10
if there exists an analytic . In case is .
Salman et. al/ Some Properties For Certain Class Of P-Valent Functions Involving Noor Operator With Differential Equation
.It is evident that was define recently by Cho et al. [2], is the Noor integral operator .The operator was investigation by was introduced by Liu and Noor[4] (see also[5]), and Saitoh [8]. By some easy calculations we obtain where
where
denote the pochhammer symbol defined by:
Note that the operator
was defined recently by [1].
By using the Noor integral operator as follows:
, we now define certain Class of
Definition 1. For fixed parameters and ,with satisfies the following subordination condition:
Or equivalently ,
is in the class
,we say that
if and only if
where and is given by (6). Here, we shall obtain several interesting results on the functions which are defined by the class following non homogeneous Cauchy-Euler differential equation:
where
k n
. By using the equation(3) and the following definition
where the operator
is the saigo type fractional calculus operator ([6],see also[7]) and is defined sa follows:
Definition 2. For 2
F1
2. Necessary and Sufficient condition for Theorem 1.Let the functions
where
if it
be given by (1), then
if and only if
Then result (12) is sharp for the functions
Proof. For the sufficient condition ,suppose that the inequality(12) hold true and
11
Then we obtain
with the
Salman et. al/ Some Properties For Certain Class Of P-Valent Functions Involving Noor Operator With Differential Equation
(14)
Hence, by maximum modulus theorem for on
this implies be given by(1), then from (8), we find that
For necessary condition, let
k n
k n
Now ,since
we have
k n
k n
We can choose value of
on the real axis and all allowing
Finally, sharpness follow if we take given by (13). Corollary 1.Let the function be given by (1) .If
,through real values , so we can write (15) as
then
The result is sharp for the function given by (13). Theorem 2.If
is in the class
then
and
proof. Suppose that is given by (1).Also let the function ,occuring in the non homogeneous Cauchy-Euler differential equation (9).Then we reaeadily find from (9) that
12
Salman et. al/ Some Properties For Certain Class Of P-Valent Functions Involving Noor Operator With Differential Equation
so that
and
Next since
,the first assertion (16)of Corollary 1 yields the following coefficient inequality
which in conjunction with (20),yields
Finally ,in view of the following telescopic sum:
The first assertion (17)of Theorem (2)follows at once from (22).The second assertion (18) of Theorem (2)can be prove by similarly applying (20),(21)and (23). Theorem 3. Let a function
be in the class
.Then we have
and
proof. Using Theorem 1 we have
or
From equation (10),we get
Using (26),(27) and (19) we have
which is equivalent to(24) and
13
Salman et. al/ Some Properties For Certain Class Of P-Valent Functions Involving Noor Operator With Differential Equation
which gives (25). is closed under linear combination .
In the following theorem , we will show that the class be given by (2) .
Theorem 4.Let
and
such that
t
where
j1
Then the function
is also in the class Proof. For every
we obtain
Using (28), we get
There fore
Hence REFERENCE [1]. M.Ç ǴL R H.ORH N and E.ENIZ Majorization for certain subclasses of analytic functions involving the generalized Noor integral operator ,Filomat ,27(2013),134-139. [2] N.E. Cho, O.S. Kwon and H. M. Srivastava , Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators, J. Math. Anal. Appl. 292 (2004) 470-483. [3] X. L. Fu, M. S. Liu, Some subclasses of analytic functions involving the generalized Noor integral operator, J. Math. Anal. Appl. 323 (2006) 190-208. [4] J.L. Liu, K. I. Noor, Some properties of Noor integral operator, J. Nat. Geom. 21 (2002), 81-90. [5] J. Patel, N.E. Cho, Some classes of analytic functions involving Noor integral operator, J .Math. Anal. Appl. 312 (2005) 564-575. [6] R.K.Raina and T.S.Nahar, Informatica (Lithuanian Acad. Sci)9(1998)469-478. [7] R.K.Raina and H.M.Srivastava , compute .Math Appl.32(1996)13-19 . [8] H. Saitoh, A linear operator and its application of first order differential subordinations, Math. Japon. 44 (1996) 31-38. [9] S.G.Samko, A.A.Kilbas and O.I.Marichev, Fractional integrals and Derivatives: Theory and Applications, Gordon and Breach Science publishers, Reading Tokyo, Paris,Berlinand Langhorne(Pennsylvania)1993 .
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