 Advances in Pure Mathematics, 2011, 1, 228-234 doi:10.4236/apm.2011.14040 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM On Certain Subclasses of Multivalent Functions Associated with a Family of Linear Operators Jae Ho Choi Department of Mathematics Education, Daegu National University of Education, Daegu, South Korea E-mail: choijh@dnue.ac.kr Received March 16, 2011; revised April 7, 2011; accepted April 20, 2011 Abstract Making use of a linear operator , which is defined here by means of the Hadamard product (or convolution), we introduce some new subclasses of multivalent functions and investigate various inclusion properties of these subclasses. Some radius problems are also discussed. ,pac Keywords: Multivalent Functions, Hadamard Product (or Convolution), Linear Operators, Radius Problem 1. Introduction and Definitions Let denote the class of functions pfz of the form =1=(:{1, 2,3,})ppkpkkfzza zp  , (1) which are analytic in the open unit disk =: a <1zz ndz . We define the Hadamard product (or convolution) of two analytic functions  =0 =0=and=kkkkkkfzaz gzbz, as =0: kkkkfgzabz z . For , a0c () H. Saitoh  introduced a linear operator 0:,2,1, 0 ,:pac pp defined by  ,:,; (;ppacf zaczfzzfp) (2) where =0,; :()kpkpkkaaczz zc, (3) and k is the Pochhammer symbol defined, in terms of the Gamma function, by   1(==11(=0))kkkkk   . The operator ,pac is an extension of the Carlson- Shaffer operator (see ). In , Cho et al. introduced the following family of linear operators ,:pac pp analogous to ,pac (see also ): †0,:,;,;>;;ppacf zaczf zacpz fp . (4) where †,;pacz is the function defined in terms of the Hadamard product (or convolution) by the following condition †,;,; =1ppppzacz aczz, (5) where p is given by (3). If fz is given by (1), then from (3), (4) and (5), we deduce that  =1,=.!ppkkkkpkkac fzpczazakpz (6) It is easily seen from (6) that 11,1 =ppfzfz and 1,1=pzf zpfzp J. H. CHOI 229.  1,=, 1,pppzacfzaacfzapacfz  (7) and  1,=, ,pppzacfzpacfz acfz (8) Clearly, from (7) and (8), we have 1,Re> 01,,Re>( )1,ppppzacfzacfzac fzap apaacfz (9) and  11,Re> 0,,Re>0 .,ppppzacfzac fzac fzpac fz (10) When and 0=:anpn 0==1c, the linear operator 11,1 =pnpnp , was introduced and studied by Liu and Noor  (see also  and ). Moreover, when , was first introduced and studied by Noor  which is known as Noor Integral operator. =1p111,1 =nnLet k be the class of functions analytic in the unit disk satisfying the properties hz0=1h and 20Re d1hz k, (11) where =izre, and 2k0<1. For =0, the class k was introduced in . For 0=k =0, , we have the well known class of functions with and the class gives us the class =2k=2Reh z>0k of functions with zReh >. Also we can write, for khz  as   201121=d21ititzehz tze, (12) where is a function with bounded variation on such that t[0, 2] 2200d=2anddttFrom (11) and (12) it can be seen that ()kh if and only if there exist 12,(hh ) such that   1211=42 42kkhzh zhz   . (14) It is known  that the class k is a convex set. We also note that khz if and only if there exists kq such that =1hz qz. (15) By using the linear operator , we now define some subclasses of ,pacpap as follows: Definition 1: Let , 0c , >p, >0, 0, and 2k0<1. A function fzp,, , is said to be in the class ,,,pkac if and only if it satisfies  11,(1 )1,,1, ,,1,ppppkppacfzacgzacfza cfzacgza cgz      (16) where gzp satisfies the condition ,(0< 1;)1,ppac g zzacgz. (17) We note that g is starlike univalent in when === =1ac p in (16). Definition 2: Let 0,ac  , 0, >0, 0, and 2k0<1. A function fzp is said to be in the class ,,,,,,kacp if and only if it satisfies  111,(1 ),,, ,,,ppppkppac fzac g zac fzac fzac gzac gz  (18) where gzp satisfies the condition   1,0<1;,ppac fzzac gz. (19) k. (13) In this manuscript, we investigate several inclusion and other properties of functions in the classes ,,, , , ,pkac and ,,, , , ,pkac which are introduced above. Furthermore, some radius pro- blems are also considered. Copyright © 2011 SciRes. APM 230 J. H. CHOI 2. Main Results In order to establish our results, we require the following lemmas. Lemma 1:  Let 12=uu iu and 12=vviv and let be a complex-valued function satisfying the conditions: (,)uv1) is continuous in a domain (,)uv1,0 2,2) and . 1,0>003) whenever 21)vRe (,iu21,iu v  and 21212vu . If is analytic in , with , such that pz,zp0=1ppzz  and > 0p zRe for ,pz zz, then . Re>0pzLemma 2:  If is analytic in with , and if hz 0=1p is a complex number satisfying Re0 (0), then  Re>(0<1)hzzh z implies  1Re>121hz, (20) where 1 is given by 11R0d=1ett which is an increasing function of Re and 112 <1. The estimate (20) cannot be improved in general. Lemma 3:  Let be analytic in with and . Then, for qzR>00=1qeq z(z)=0C=1hi0AzRe 0iuh,2;kRemark: If we put and =anp==1c in Theorem 1, we have the result due to Noor and Arif [9, Theorem 3.1]. Theorem 2: Let 0. If ,,,,,,pfkac, then  ,,pkpac fac gzz, where 2=2pp , and gp satisfies the condition (18) and 020Re=||hzhz, 10,=,ppac fzhz ac gz. Proof. Let ,,,,,,pfkac and set  ,=1,ppac fzhzac g z, where is analytic in with . Then, by using same techniques as in the proof of Theorem 1, we obtain the desired result. hz 0=1hWe note that = when =0 in Theorem 1. Corollary 1: Let 1. If ,,, ,0,1,pfkac, then ,(),pkpac fzzac g z. Proof. It is clear that, for 1, ,,1, ,=(1 )1,,1, (1).1,ppppppppac fzac g za cfzacfza cgzacgzacfzacgz This implies that 12,,1, ,1=1 1, ,1,11 1=1,ppppppppac fzac g za cfzacfzacgz acgzacfz PPacgz11.     Since k is a convex set (see ), by using Theorem 1 and Definition 1, we observe that k12,PP and ,,pkpac fzac g z, which completes the proof of Corollary 1. Making use of Theorem 2 and Definition 2, we can prove the following result. Corollary 2: Let 1. If ,,,,0,1,pfkac, then  11,(),pkpac fzzac g z. Next, by using Lemma 2, we prove the following. Theorem 3: Let  be a complex number satisfying R>0e and let , >0a0c, 0 and >0. If fp satisfies the condition  11,1,1, ,ppppkppacfzzacfzacfzzz  then 1, () ()pkpacfz zz, where   1Re1110=121 with=1datt  . (26) The value of  is best possible and cannot be improved. Proof. If we set Copyright © 2011 SciRes. APM 232 J. H. CHOI  121,11==42 42ppacfzzkkhzh zhz   , then and is analytic in . By applying (7), we have 0=1hh  11,1,1, =.ppppppkacfzzac fzac fzzzhzzhza Therefore, by virtue of Lemma 2, we see that , where (= 1, 2)ihi is given by (26). Hence we conclude that kh, which evidently proves Theorem 3. By using (8) instead of (7) in Theorem 3, we have the following. Theorem 4: Let  be a complex number satisfying Re> 0 and let 0,ac , 0 and >0. If fp satisfies the condition  111,1,, ,ppppkppacfzzac fzacfzzz  then  ,()pkpac fzzz, where  is given by (26) with 1R1()10=1 dept t The value of  is best possi- ble and cannot be improved. Theorem 5. Let 20<1. Then 1,,,,0,, ,,,,0,,ppkac kac2 . Proof. If 2=0, then the proof is immediate from Theorem 1. Let 2>0 and 1,,, ,0, ,pfkac1. Then there exist two functions H, 2kH such that 11111,11,,1,=,1,ppppppacfzacgzacfza cfzHzacgza cgz     and  21, =1,ppacfzHzacgz. Then  2122212111,11,,1, ,1,=1.ppppppacfzacgzac fzac fzac g zac gzHzH z (27) Since k is convex set (see ), it follows that the right hand side of (2.8) belongs to k, which proves Theorem 5. Next, we consider the generalized Bernardi-Libera- Livingston integral operator defined by (cf. [1,8], and ) >p  10=d(;zp>)fztfttfpzp. (28) Theorem 6: Let  be a complex number satisfying R>0e and let fzp and f be given by (2.9). If ,,1ppkppacfzacfzzz  then  ,pkpacfzz , where  is given by (26) with 1Re10=1 dzptt . Proof. From (28), we obtain  ,=, ,pppzac fzpacfzacf.z (29) Copyright © 2011 SciRes. APM J. H. CHOI 233Let  ,=ppacfzhzz . Then, by virtue of (29), we have   ,,(1 )=.ppppkacfzac fzzzzh zhz p  Hence, by using Lemma 2, we obtain the desired result. Finally, we consider the converse case of Theorem 1 as follows. Theorem 7: Let ,,,,,,0pfkac,,. Then p,,,,fkac for