Some Geometric Properties of a Certain Subclass of Univalent Functions Defined by Differential Subordination Property

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1 Gen. Math. Notes Vol. 20 No. 2 February 2014 pp SSN ; Copyright CSRS Publication Available free online at Some Geometric Properties of a Certain Subclass of Univalent Functions Defined by Differential Subordination Property Waggas Galib Atshan 1 A. H. Majeed 2 and Kassim A. Jassim 3 1 Department of Mathematics College of Computer Science and Mathematics University of Al-Qadisiya Diwaniya-raq waggashnd@gmail.com; waggas_hnd@yahoo.com 23 Department of Mathematics College of Science University of Baghdad Baghdad-raq 2 ahmajeed6@yahoo.com 3 kasim.hameed@yahoo.com (Received: / Accepted: ) Abstract n this paper we have studied a certain subclass of univalent functions defined by linear operator by using differential subordination property.we obtain some geometric properties like coefficient inequality neighborhoods of the class K( ) convolution properties and integral mean inequalities for the fractional integral for this class. Keywords: Univalent Function Hypergeometric Function Linear Operator Differential Subordination Convolution ntegral Mean Fractional ntegral.

2 80 Waggas Galib Atshan et al. 1 ntroduction Let S denote the class of functions of the form:- )=+ 1) which are analytic in the open unit disk U={z : <1}. Also denoted by K the subclass of S consisting of functions of the form:- which are univalent and normalized in U. )= 2) For Sand of the form 1)and 4) S given by 4)=+: We define the Hadamard product (or convolution) 4))=+ :. 3) For positive real values of. and?.? 01.C= 12.) The generalized hypergeometric function δ F m (z) is defined by DF E z) DF E ;?? ;) = ) )? )? ) ( +1; N =N P0}; Q) where ) is the Pochhammer symbol defined by J! 4) 1 J=0 ) =R +1)+2) +J1) J N. S 5) The notation δ F m is quite useful for representing many well-known functions such as the exponential the Binomial the Bessel and Laguerre polynomial.

3 Some Geometric Properties of a Certain be a linear operator defined by H[ W. ;??? W.? ]:K K H[ ;?? ])=z δ F m ( ;??? ;) ) where =[ ; ;) 6) [ ;;)= ) ] ) ]? ) ]? ) ] 1 J1)!. 7) For notational simplicity we use shorter notation n the sequel.t follows from (6) that _ [ ] for H[ ;?? ]. _ [1])=)_ [2])=`). The linear operator _ [ ] is called Dziok-Srivastava operator (see [5]) introduced by Dziok and Srivastava which was subsequently extended Dziok and Raina [4] by using the generalized hypergeometric function recently Srivastava et. al. [13] defined the linear operator as follows:- )=) )=1)_ [ ])+a_ [ ])b` = ) 8) and in general )= a )b 9) )= ] a )b0 1 +1; N =N P0} ; Q). 10) f the function f(z) is given by (2) then we see from (6)(7)(8) and (10) that where )=[ ;;;) 11)

4 82 Waggas Galib Atshan et al. [ ;;;)=e ) ] ) ] [1+J1)]? ) ]? ) ] J1)! J N\P1} N ). f 12) Unless otherwise stated.we note that when =1and =0 the linear operator would reduce to the familier Dziok-Srivastava linear operator given by(see[5]) includes (as its special cases) various other linear operators introduced and studied by Carlson and Shaffer [3] Owa [9] and Ruscheweyh [10]. For two analytic functions f4 K we say that f is subordinate to 4 written f(z) 4(z) if there exists a schwarz function w(z) which (by definition) is analytic in U with w(0)=0 and w(z) <1 for all z Qsuch that f(z)= 4(w(z))z Q. Furthermore if the function 4(z) is univalent in U then we have the following equivalence (see [8]): f(z) 4(z) f(0)= 4(0) and f(u) 4(U). Definition (1): For any function f K and 0 the Jlm4h:ophooq f is defined as N )=P4)= : r;j : }. (13) n particular for the function e(z)=z we see that N (l)={4()= : r;j : }. (14) The concept of neighborhoods was first introduced by Goodman [6] and then generalized by Ruscheweyh [11] and studied by some authors like Atshan [1] and Atshan and Kulkarni [2]. Definition (2): For fixed parameters A and B with -1 < 1 we say that f r is in class K( )if it satisfies the following subordination condition: 1+ ( st ()) ()) ( (15) n view of the definition of subordination (15) is equivalent to the following condition:

5 Some Geometric Properties of a Certain st ( ()) ()) ( v ( st ()) ()) ( For convenience we write ( Q). v<1 +() K( 12w1)=K( w) where K( w) denotes the class of functions in K satisfying the inequality: Rey1+ ( st ()) z>w 0 w<1; Q). a ()b` 2 Neighborhoods for the Class K( }~ ƒ ) The following theorem gives a necessary and sufficient condition for a function f to be in the class K( ). Theorem (1): A function f r belong to the class K( ) if and only if J(2J1) [ ( ;;;)a(1)[ t ( ;;;)+()b ) 16) for N +1 0 Jq1 < 1. Proof: Let f K( ). Then 1+ st )) )) Q. 17) Therefore there exists an analytic function w such that )= st )) st )) +) )). 18)

6 84 Waggas Galib Atshan et al. Hence () = ( st ( ()) st ()) +()( ()) st J(J1) [ ( ;;;) = () J [ ( ;;;)a[ t ( ;;;)+()b <1. Thus st J(J1) [ ( ;;;) ()+ J [ ( ;;;)([ t ( ;;;)+()) <1. Therefore st J(J1) [ ( ;;;) Šl ()+ J [ ( ;;;)([ t ( ;;;)+()) Œ <1. (19) Taking z =r for sufficiently small r with 0<r<1 since w(z) is analytic for z =1.Then the inequality (19) yields st J(J1) [ ( ;;;) p <()p+ J [ ( ;;;)a[ t ( ;;;)()b p. Equivalently J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) p ()p and (16) follows upon letting r 1. Conversely for z =r 0<r<1 we have r n <r. That is J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) p

7 Some Geometric Properties of a Certain J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) p )p. From (16) we have st JJ1)[ ;;;) st JJ1) [ ;;;) p <)p+j [ ;;;) < )+J [ ;;;) [ t ;;;)+)) p [ t ;;;)+)). This proves that 1+ st )) )) Q and hence f K( ). Theorem (2): f = 3e ) ) 1+)f? )? ) 1) ) )? )? ) 1+)) t +)Ž 20) then K( ) N e). Proof: t follows from (16) that if f K( ) then 3 [ ;;;)a1)[ t ;;;)+)bj. Hence 3 ) ) 1+)) 1)e t ) ) 1+)f + )? )? )? )? )

8 86 Waggas Galib Atshan et al. J 21) which implies that J 3 ) )? )? ) 1+)) 1) ) )? )? ) 1+) t + ) = (22) Using (14) we get the result. Definition (3): The function 4 defined by 4(z)= : is said to be a member of the class K c ( ) if there exist a function f K( ) such that 4) 1 1 Q0 <1). 23) ) Theorem (3): f f K( ) and 3 [ =1 ;;;)1)[ ;;;)+)) 6 [ ;;;)1)[ ;;;)+))) 24) then N ) K c ( ). Proof: Let 4 N ). Then we have from (13) that J : which implies the coefficient inequality : 2. Also since f K( ) we have from (16) 6 [ )1)[ t ;;;)+))

9 Some Geometric Properties of a Certain where so that [ ( ;;;)=( ( ) ( ) (? ) (? ) (1+)) [ t ( ;;;)=( ( ).( ) (? ).(? ) (1+)) t 4() () 1 = ( : ) < : 1 3 [. ( ;;;)((1)[ t ( ;;;)+()) 6 [ ( ;;;)((1)[ t ( ;;;)+())() =1-c. Thus by Definition (3) 4 K c ( ) for c given by (24). This completes the proof. 3 Convolution Properties Theorem (4): Let the functions f j (j=12) defined (C=12) (25) be in the class K( ).Then f 1 *f 2 K( ) where J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) () (+ [ t ( ;;;)) J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) () (1+ [ t ( ;;;)) Proof: We must find the largest such that J(2J1) [ ( ;;;)(a1 )[ t ( ;;;)+( )b 1. K( )(C=12)then J(2J1) [ ( ;;;)(a1)[ t ( 1 ( C=12). (26) By Cauchy Schwarz inequality we get

10 88 Waggas Galib Atshan et al. J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) š 1. (27) We want only to show that J(2J1) [ ( ;;;)((1 )[ t ( ;;;)+( )) J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) š. This equivalently to From (27) we have š š ( )((1)[ t ( ;;;)+()) ()((1 )[ t ( ;;;)+( )). Thus it is sufficient to show that J(2J1) [ ( ;;;)((1) [ t ( ;;;)+()). J(2J1) [ ( ;;;)((1) [ t ( ;;;)+()) which implies to ( )((1) [ t ( ;;;)+()) ()((1 ) [ t ( ;;;)+( )) J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) () (+ [ t ( ;;;)) J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) () (1+ [ t ( ;;;)) This completes the proof. Theorem (5): Let the functions f j (j=1 2) defined by (25) be in the class r( ).Then the function h defined by h()=( + ) (28) belong to the class K( ) where

11 Some Geometric Properties of a Certain J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) 2[ t ( ;;;)() 2() J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) +2() +2[ t ( ;;;)() Proof: We must find the largest such that J(2J1) [ ( ;;;)a(1 )[ t ( ;;;)+( )b ( + ) 1. K( ) (C=12)we get e J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) f J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) ž and e J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) f J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) ž Combining the inequalities (29) and (30) gives 1 (29) 1. (30) 1 2 ej(2j1) [ ( ;;;)((1)[ t ( ;;;)+()) f + 1. (31) But h K( ) if and only if ( J(2J1) [ ( ;;;)((1 )[ t ( ;;;)+( )) ( + ) 1. (32) The inequality (32) will be satisfied if J(2J1) [ ( ;;;)((1 ) [ t ( ;;;)+( )) (J(2J1) [ ( ;;;)((1) [ t ( ;;;)+()) 2() (J=23 ) (33)

12 90 Waggas Galib Atshan et al. so that J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) 2[ ( ;;;)() 2() J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) +2() + 2[ ( ;;;)() This completes the proof. 4 ntegral Mean nequalities for the Fractional ntegral Definition (4) [12]: The fractional integral of order s (s>0) is defined for a function f by: Ÿ ] ()= 1 Γ( ) ( ) ( ) ] q where the function f is an analytic in a simply- connected region of the complex z- plane containing the origin and multiplicity of ( ) ] is removed by requiring log(z-t) to be real when (z-t)>0. n 1925 Littlewood [7] proved the following subordination theorem:- Theorem (6) (Littlewood [7]): f f and 4 are analytic in U with f 4 then for w>0 and =pl (0<r<1) () q 4() Theorem (7): Let f r( ) and suppose that f n is defined by ()= J(2J1) [ ( ;;;)((1)[ t ( ;;;)+()) (J 2).(34) Also let (m ) ªs ()Γ(J+1)Γ( + +3) J(2J1) [ ( ;;;)((1) [ t ( ;;;)+())Γ(J+ + +1)Γ(2 ) (35) for 0 m >0 where (m ) ªs denote the Pochhammer symbol defined by (m ) ªs =(m )(m +1) m. f there exists an analytic function q defined by q

13 Some Geometric Properties of a Certain «() ] = J(2J1) [ ( ;;;)(1)[ t ( ;;;)+())Γ(J+ + +1) ()Γ(J+1) (m ) ªs _(m) ] (36) where i and _(m)= Γ(m ) ( >0m 2) (37) Γ(m+ + +1) then for z=pl ϕ and 0<p<1 Ÿ ] ]ª () Proof: Let ()= qϕ Ÿ ] ]ª () qϕ ( >0w>0). (38). For 0 and Definition (4) we get Ÿ ] ]ª ()= Γ(2) sªs Γ( + +2) (1Γ(m+1)Γ( + +2) Γ(2)Γ(m+ + +1) ] ) = Γ(2) sªs Γ( + +2) 1Γ( + +2) (m ) Γ(2) ªs _(m) ] ž where _(m)= Γ(m ) Γ(m+ + +1) ( >0m 2) Since H is a decreasing function of i we have 0<_(m) _(2)= Γ(2 ) Γ( + +3). Similarly from (34) and Definition (4) we get Ÿ ] ]ª ()= Γ(2) sªs Γ( + +2) (1 ()Γ(J+1)Γ( + +2) J(2J1) [ ( ;;;)((1) [ t ( ;;;)+())Γ(J+ + +1) ] ). For w >0and z=pl ϕ (0<p<1) we must show that

14 92 Waggas Galib Atshan et al. 1 Γ( + +2) (m ) Γ(2) ªs _(m) ] qϕ 1 (AB)Γ(J+1)Γ( + +2) J(2J1) [ ( ;;;)((1) [ t ( ;;;)+())Γ(2)Γ(J+ + +1) ] qϕ. By applying Littlewood s subordination theorem it would suffice to show that 1 Γ( sªs) Γ() (m ) ªs _(m) ] 1- (AB)Γ(J+1)Γ( + +2) J(2J1) [ ( ;;;)((1) [ t ( ;;;)+())Γ(2)Γ(J+ + +1) ]. By setting 1 Γ( + +2) (m ) Γ(2) ªs _(m) ] = (AB)Γ(J+1)Γ( + +2) J(2J1) [ ( ;;;)((1) [ t ( ;;;)+())Γ(2)Γ(J+ + +1) «() ] we find that «() ] = J(2J1) [ ( ;;;)((1)[ t ( ;;;)+())Γ(J+ + +1) (AB)Γ(J+1) (m ) ªs _(m) ] which readily yields w(0)=0.for such a function q we obtain «() ] J(2J1) [ ( ;;;)a(1)[ t ( ;;;)+()bγ(j+ + +1) (AB)Γ(J+1) (m ) ªs _(m) ] J(2J1) [ ( ;;;)a(1)[ t ( ;;;)+()bγ(j+ + +1) (AB)Γ(J+1) _(2) (m ) ªs

15 Some Geometric Properties of a Certain = J(2J1) [ ( ;;;)((1)[ t ( ;;;)+())Γ(J+ + +1) (AB)Γ( + +3)Γ(J+1) Γ(2 )(m ) ªs <1. This completes the proof. By taking τ=0 in the Theorem 7 we have the following corollary: Corollary (1): Let f r( ) and suppose that is defined by (34). Also let (AB)Γ(J+1)Γ( +3) m J(2J1) [ ( ;;;)((1)[ t ( ;;;)+())Γ(2)Γ(J+ +1) n 2. f there exists an analytic function q defined by «() ] = J(2J1) [ ( ;;;)a(1)[ t ( ;;;)+()bγ(j+ +1) (AB)Γ(J+1) m_(m) ] where _(m)= Γ(m) Γ(m+ +1) ( >0m 2). Then for z=pl ϕ and 0<p<1 References Ÿ ] () qϕ Ÿ ] () q ( >0w>0). [1] W.G. Atshan Subclass of meromorphic functions with positive coefficients defined by Ruscheweyh derivative J. Surveys in Mathematics and its Applications 3(2008) [2] W.G. Atshan and S.R. Kulkarni Neighborhoods and partial sums of subclass of k-uniformly convex functions and related class of k-starlike functions with negative coefficient based on integral operator J. Southeast Asian Bulletin of Mathematics 33(4) (2009)

16 94 Waggas Galib Atshan et al. [3] B.C. Carlson and D.B. Shaffer Starlike and prestarlike hypergeometric functions SAM J. Math. Anal. 15(1984) [4] J. Dziok and R.K. Raina Families of analytic functions associated with the Wright generalized hypergeometric functions Demonstration Math. 37(3) (2004) [5] J. Dziok and H.M. Srivastava Certain subclasses of analytic functions associated with the generalized hypergeometric function ntegral Transform Spec. Funct. 14(2003) [6] A.W. Goodman Univalent functions and non-analytic curves Proc. Amer. Math. Soc. 8(3) (1975) [7] L.E. Littlewood On inequalities in the theory of functions Proc. London Math. Soc. 23(1925) [8] S.S. Miller and P.T. Mocanu Differential subordinations: Theory and applications Series on Monographs and Textbooks in Pure and Applied Mathematics (Vol.225) Marcel Dekker New York and Basel (2000). [9] S. Owa On the distortion the theorems. Kynngpook Math. J. 18(1978) [10] S. Ruscheweyh New criteria for univalent functions Proc. Amer. Math. Soc. 49(1975) [11] S. Ruscheweyh Neighborhoods of univalent functions Proc. Amer. Math. Soc. 81(1981) [12] H.M. Srivastava and S. Owa (Eds) Current Topics in Analytic Function Theory World Scientific Publishing Company Singapore (1992). [13] H.M. Srivastava S.H. Li and H. Tang Certain classes of k-uniformly close to convex functions and other related functions defined by using Dziok-Srivastava operator Bull. Math. Anal. Appl. 1(3) (2009)