 Advances in Pure Mathematics, 2011, 1, 193-200 doi:10.4236/apm.2011.14034 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Inclusion and Argument Properties for Certain Subclasses of Analytic Functions Defined by Using on Extended Multiplier Transformations Oh Sang Kwon Department of Mathematics, Kyungsung University, Busan, Korea E-mail: oskwon@ks.ac.kr Received March 28, 2011; revised April 27, 2011; accepted May 5, 2011 Abstract Making use of a multiplier transformation, which is defined by means of the Hadamard product (or convolu-tion), we introduce some new subclasses of analytic functions and investigate their inclusion relationships and argument properties. Keywords: Subordination, Starlike Functions, Convex Functions, Closed-to-Convex Functions, Multiplier Transformation, Multivalent Functions, Argument Principle 1. Introduction Let pA denote the class of functions f normalized by =1=(:{1, 2,3,})pkpkpkfzza zp (1.1) which are analytic and -valent in the open unit disk p=: and<1Uzz z If f and g are analytic in U, we say that f is subordinate to g, and write  or( )fgfzgzzU if there exists a Schwarz function , analytic in with and zU0=0<1z in , such that zU =fzg z for . zUWe denote by *pS and pC the subclasses of pA consisting of all analytic functions which are, respectively, -valent starlike of order p (0 0()zzU Making use of the aforementioned principle of subordination between analytic functions, we define each of the following subclasses of pA:  *;1:Re :and(0< ;;)ppSzf zffA zpfzpz UM  (1.2)  ;1:Re :and1(0< ;;)ppKzf zffA zpfzpz UM  (1.3) For 0:{0,1, 2,}m , we define the multiplier transformation ,,mJpl of functions pfA by   *1,;, :Re:and;..(0,<;; ,)pppzf zCffAgSstpgzpz UM   z (1.4) O. S. KWON Copyright © 2011 SciRes. APM 194 =1,, =(>0; 0;)mmpkpklk kpJplfz zazllzU (1.5) Put ,,=1=(;>0;0;)mmpplklkzz zlml zUkp (1.6) The operators ,,mpl and ,1,mpl, are the multiplier transformations introduced and studied earlier by Sarangi and Uralegaddi  and Uralegaddi and Somanatha ( and ), respectively. Correspending to the function ,,mplz defined by (1.6), we introduce a function ,,,mplz given by the Hadamard product (or convolu- tion): ,,, ,,*= (>1pmmpl plpzzzz  )p Then, analogous to ,,mJpl, we have define a new multiplier transformation ,, :mppIplA A as follows:  ,,,,, =*mmplIplfz zfz (1.7) We note that  012,1,1=and1,1, 2=pIpfzfz Ifzzfz It is easily verifed from the above definition of the operator ,,mIpl, that  1,,=,, ,,mmmzIpl f zpIpl fzIplf z (1.8) and  1,,=,, ,,mmmzIplf zlIpl f zplIplf z  (1.9) The definition (1.6) of the multiplier transformation ,,mpl is motivated essentially by the Choi-Saigo- Srivastava operator  for analytic functions, which includes a simpler integral operator studied earlier by Noor  and others (cf. [4-6]). Next, by using the operator ,,mIpl defined by (1.7), we introduce the following subclasses of analytic functions: *=: and ,,;(;,,>0;;0<1mppff AI p lfzSMl m,,, ;mplS)  (1.10) ,,, ;=:and ,,;(;,,>0;;0<1mplmppKff AI p lfzKMl m)  (1.11) and ,, ,;,=:and ,,,;,(,;,, >0;;0, <1)plmpff AI p lfzCMl m,mC  (1.12) We also note that  ,, ,,pl plzf zS,m,;;mfz K (1.13) In particular, we set ,,,, ,,1mmAz;=;,(1<<1)1pl plSSABBABz (1.14) and ,,,, ,,1;= ;,(1<<1mmpl plAzKKABBBz  1)A(1.15) In the present paper, we investigate some increlusion lationships and argument properties associated with such multivalent functions in the class pA as those be- longing to the subclasses ,,, ;mplS, ,,;,mKpl and ,,, ,;,mplC defined(1.12), by (1.10), .11) and (1respectively. 2. Inclusion Properties emma 2.1: Let L be convex univalent in with U01 and Re> 0z  (,). Ip is n U f analytic iwith 0=1p, then  ()zp zpzzz Uz  implies that pz z2.2: Let (zU). Theorem M withRe>max,minzUlpzpp O. S. KWON195 then ; ,1 ,1,,,,, ,,;;mmmplpl plSSS  . Proof. First of all, we show that  ,1 ,,, ,,;;mmpl plSS. Let ,1,, ;mplfS and set  ,,1mzIpl f z=,,mpz pIp fzl (2.1) where the function pz obtain is analytic in with . ng (2.1), we U0=1pApplyi1,,mpppzIplf,, =mIplfzz (2.2) tiating both sides of (2.2) and multiplying the reseulting equation by , we have By logarithmically differenz ,,1=()mzIplf zzp zpzz UppzSince , by appma 2.1 to (2.3), it follows that in , that is, that ,,mpIplfz (2.3) Re> 0pz  lying Lem-  pz zU ,,, ;mplfz S. prove the second To parm t of Theore 2.1, letz S ,,, ;mplf and put  11=,,mqz pIplfz,,1mzIplf z where the function is analytic in with . precisely the sanner, we can ult that in , that is, that qzme maUfind the res0=1qIn  qz zU 1,,, ;mplfz S unhe hypothesis der tRe> 0pzp  t zM with lTheorem 2.3: Le>max ,minzUlpRe zpp then ; ,1 ,1,,,,, ,,;;mmmplpl plKKK   ,1 ,1,, ,,,,,, ,,;;;;mmpl plmmpl plfzKzfzSzfzSf zK  and  ,,,,,,;;mmpl plfzKzfzS 1, 1,,, ,,;;mmpl plzfzSf zK  which evidently prove Theorem 2.3. By setting 1=(1<<1;1AzzBABz)zU in Theorems 2.2 and 2.3, we deduce the focorollary. Corollary 2.4: Suppose that llowing . Proof. Applying (1.11) and Theorem 2.2, we observe that 1>max ,1lpABpp Then, for the function classes defined by (1.12) and (1.13), 1,,,,,,, ;,mplpl plSA Band ,1 ,;, ;,mmSABSAB,1 ,1,,,,, ,,;, ;,;,mmmplpl plKAB KAB KAB   Theorem 2.5: Let with ,M>max ,minzUlpRe zpp then ,1 ,,, ,,,,,;, ,;,;,mmpl plplCCC1, ,m  Proof. We begin by proving that ,1 ,,, ,,,;, ,;,mmpl plCC , which is tfirst inclusion relationship asserted by Theorem 2.5. he Let ,1,, ,;,mplfz C. Then there exists a function ;pkz S* such that  1,,1()mzIpl f zzzUpkzChoose the function  gz such that 1,, =mpIplgzk *;zS Copyright © 2011 SciRes. APM O. S. KWON Copyright © 2011 SciRes. APM 196 Then  ,,,;;plS ,1,,mmplgz S, and  11,,1(),,mmzIpl f zzzUpIplgz (2.4) Now let ,,=,,mmzIplf zIplgzpz (2.5) where the function pz is analytic in with U0=1p. nd th Using (1.9), we fiat  112,, ,,,,11=,, ,, ,,1,, ,,1=,, ,,1=mmmmmmmmmmzzI p lfzI p lfzzIpl f zppIplgzzI p lgzI p lgzzIplfzzIplfzpzI p lgzI p lgzp   2,, ,,1,, ,,,,,,mmmmmmzIplfz zIplfzIplgz IplgzzI p lgzIplgz Since  ,,, ;mplgz S, then we set  ,,1=,,mmzIplf zqz pIplgz  ,,,,=1mmzI p lfzIplfzpzpzpqz ppz (2.8) Hence (2.6) w in Uumption that here q with the ass z zM. By (2.5), 2,,,,=1 mmzI plfzIplgzpzpzpqzppz,, =,,mmzIpl f zppzIplgz and mu, we obtain (2.7)    Differentiating both side of (2.7) with respect to z ltiplying by zComputing the above equations, we can obtain     11,,1,,=mmzIpl f zpIplgzpzppqzzp zpz pqz11=ppz pz 1pqz ppz     O. S. KWON197 Since , applying Lemma 2.1 witRe> 0pz h  1=wz pz , we can show that pzz in U, so that ,,, ,;,mplfz C. 3. Argument Properties emma 3.1: Let L be convex univalent in and U betic in with analy URe z0. If pz is analytic in U and 0= 0, thepn )pzz U  (zzp zzimpli z (zes that pz U). Lemma 3.2: Let be analytic in with p U0=1p and =0pz for o points ,zz U such that all . If there exist zUtw 1211 22=arg 0) and fr all oz12(<= )zz z. 112 212 1212=and=22zp zzpzim ipz pz m(3.2) where 11bmb and 2112=tan4bi. Theorem 3.3: Let pfA. 120< ,1. 0<< p. If 1121,, 0pz (), and hence RzU=0pz (zU). By using 12zz ULem if there exist ma 3.2,two points , such thwe obtain (3.2at the hcondition (3) is ) under te constraint (3.2). A .1satisfied, then nd we obtain    1111111211212 111arg=arg1 exp222211cos 12tan22cos121cos2tan2121zp zpz pqziimmbtpAB     m  112 1=211cos2bbt    and  12 122 122 2212 11cosπ2arg =tan221211cos12btzp zpz pqz pA bbtB       which would obviously contradict the assertion of Theorem 3.3. We thus complete the proof of Theorem 3.3. If we let 12= onsequein Theorem 3.5, we easily obtain the following cnce. rollary 3.4: Let CopfA. 0< 1. 0<< p. If ,,arg< 2,,mmzIpl f zIplgz ,1,, ,;,mplgSpAB, then for some ,,arg ,,<2mmzIpl f zIplgz where  is the solutions for the following equation: Copyright © 2011 SciRes. APM O. S. KWON199 1111cos22=tan 111cos12btpA bbtB     b is given by (3.2), and 1,,