Sufficient Conditions for Subordination of Meromorphic Functions

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1 Joural of Mathematics ad Statistics 5 (3): ISSN Sciece Publicatios Sufficiet Coditios for Subordiatio of Meromorphic Fuctios Rabha W. Ibrahim ad Maslia arus School of Mathematical Scieces Faculty of Sciece ad Techology iversity Kebagsaa Malaysia Bagi Selagor. Ehsa Malaysia Abstract: Problem statemet: The problem of givig sufficiet coditio for certai class of meromorphic fuctios defied as differetial operator was studied. Approach: The differetial operator of meromorphic fuctios cotaiig fractioal power was proposed ad defied. The prelimiary cocept of subordiatio was itroduced to give sharp proofs for certai sufficiet coditios of the differetial operator aforemetioed. Results: Havig ew operator subordiatio theorems established by usig stadard cocept of subordiatio ad reduced to well-ow results studied by various researchers. The operator was the applied for fractioal calculus ad obtaied ew subordiatio theorem. Coclusio: Therefore by havig ew operators ew criteria ad ew set of subordiatio theorems could be obtaied with some earlier results ad stadard methods. Key words: Fractioal calculus; Subordiatio; meromorphic fuctio INTROCTION The study of meromorphic fuctios has bee the major iterests for may authors i the field of uivalet fuctios. Recetly various differetial operators have bee itroduced for certai class of aalytic uivalet fuctios i the uit dis. I this article we follow the similar approach by itroducig a differetial operator of meromorphic fuctios i the puctured dis. We begi by givig some well-ow otatios ad prelimiary results o the class of meromorphic fuctios ad also the basic owledge of subordiatio. Later we derive the differetial operator aforemetioed. Oce the differetial operator beig derived we shall discuss o the coditios for subordiatio. Now let form: ε be the class of fuctios F() of the F() = a =0 Which are aalytic i the puctured uit dis :={ 0 > > } Ad let ε be the class of fuctios of the form: F() = a =0 Which are aalytic i the puctured uit dis. Let us recall the priciple of subordiatio betwee aalytic fuctios: let the fuctios f ad g be aalytic i :={ <} the we say that the fuctio f is subordiate to g if there exists a Schwar fuctio W() aalytic i such that: f () = g(w()) We deote this subordiatio by: f g or f() g() If the fuctio g is uivalet i the above subordiatio is equivalet to: f (0) = g(0) ad f ( ) g( ) 3 Let ϕ : ad let h be uivalet i Assume that pϕ are aalytic ad uivalet i if p satisfies the differetial superordiatio: h() 2 ϕ(p())p () p ();) () The p is called a solutio of the differetial superordiatio. (If f is subordiate to g the g is called to be superordiate to f.) A aalytic fuctio q is called a subordiat if q p for all p satisfyig (). A Correspodig Author: Maslia arus Faculty of Sciece ad Techology School of Mathematical Scieces iversity Kebagsaa Malaysia Bagi Selagor. Ehsa Malaysia 4

2 J. Math. & Stat. 5 (3): uivalet fuctio q such that p q for all subordiats p of () is said to be the best subordiat. Let ε be the class of aalytic fuctios i of the form f()= a =0 ad let ε - be the class of aalytic fuctios i of the form: f () = a =0 a 0 = 0... A fuctio f() ε is meromorphic starlie if f() 0 ad: f () R{ }>0 f() Similarly the fuctio covex if f() 0ad: f () R { } > 0 f() f() is meromorphic Ravichadra et al. [] studied sufficiet coditios for subordiatio for class ( ε ) of meromorphic fuctios: f () q() f() A fuctio F() ε( ε ) such that F() 0 is called meromorphic starlie if: F () R{ }>0 F() Ad the fuctio F() is meromorphic covex if F() 0 ad: F () R { } > 0 F() MATERIALS AN METHOS We defie a differetial operator as follows. Let F ε the: F()= F()= a 0 =0 λ F()=(2 )F() (2 λ)f () = [(2 λ )( ) ]a =0 =0 λ F() = ( F()) = [(2 λ )( ) ] a (2) We will establish some sufficiet coditios for fuctios F ε ad F ε to satisfy: [ F()] q() = 2... (3) F() where F() 0q() is a give uivalet fuctio i. Moreover we will give applicatios for this result i fractioal calculus. We shall eed the followig ow results. Lemma : Shamugam et al. [2] Let q() be covex uivalet i the uit dis ad ψ ad γ with q () ψ R { } > 0 If p() is aalytic i q() γ ad: ψ p() γp () ψ q() γq () the p() q() ad q is the best domiat. Lemma 2: Let q() be uivalet i the uit dis ad θ ad φ be aalytic i a domai cotaiig q( ) with ϕ(w) 0 whe w q( ) Set [3] : Q() := q () ϕ(q())h() := θ (q()) Q() Suppose that: Q() is starlie uivalet i ad h () R >0 for Q() If: θ (p()) p () ϕ(p()) θ (q()) q () ϕ(q()) The p() q() ad q() is the best domiat. 42

3 J. Math. & Stat. 5 (3): RESLTS Now we establish some sufficiet coditios for subordiatio of aalytic fuctios i the classes ε ad ε. Theorem : Let the fuctio q() be covex uivalet i such that q() 0 ad: q () ψ R { } > 0 γ 0 q() γ (4) [ F()] λ Suppose that is aalytic i If F ε λ F() satisfies the subordiatio: [ F()] [ F()] [ F()] λ λ λ { ψγ [ ]} F() [ F()] λ F() ψ q() γq () The: [ ad q() is the best domiat. F()] q() F() Proof: Let the fuctio p() be defied by: [ F()] p() := F() it ca easily be observed that: [ ψ p() γp () = F()] λ F() [ F()] [ F()] { ψγ [ ]} [ F()] F() ψ q() γq () The by the assumptio of the theorem we have that the assertio of the theorem follows by a applicatio of Lemma. Whe =0 we obtai the followig result: Corollary : Let the fuctio q() be uivalet i. If q satisfies: F () F () F () { ψγ [ ]} ψ q() γq() F() F () F() The: F () q() F() ad q() is the best domiat. Theorem 2: Let the fuctio q() be uivalet i q () such that q() 0 is starlie uivalet i q() ad: a q () q() R { q() [ ]}>0b 0q() 0 b q() q() If F ε satisfies the subordiatio: [ F()] [ F()] [ F()] λ λ λ a[ ] b[ ] F() [ F()] λ λ λ F() q () aq() b q() The: [ F()] q() F() ad q() is the best domiat. Proof: Let the fuctio p() be defied by: [ F()] p() := F() By settig: b θ( ω):=aωad ϕ( ω):= b 0 ω (5) It ca easily observed that θω ( ) is aalytic i ϕ( ω) is aalytic i \ {0} ad that ϕ( ω) 0 ω \ {0} Also we obtai: q () Q() = q () ϕ(q()) = b ad h() = θ (q()) Q() q() q () =aq() b q() 43

4 J. Math. & Stat. 5 (3): It is clear that Q() is starlie uivalet i : h () a q () q () R{ } = R { q() [ ]} > 0 Q() b q () q() Straightforward computatio we have: p () [ F()] [ F()] ap() = b = a[ ] b[ p() F() [ F()] [ F()] q () [ F() q() ] aq() b The by the assumptio of the theorem we have that the assertio of the theorem follows by a applicatio of Lemma 2. Whe =0 we obtai the followig result: Corollary 2: Let the fuctio q() be uivalet i. If q satisfies: F () F () q () (a b)[ ] b[ ] aq() b F() F () q() The: F () q() F() ad q() is the best domiat. ISCSSION Results obtaied i Theorem ad 2. ca be applied for the fractioal itegral operators. Assume that: f()= φ =0 Ad let us begi with the followig defiitios: efiitio : The fractioal itegral of order is defied for a fuctio f() by [4] : Γ ( ) If():= f()( ζ ζ) d;0 ζ < 0 requirig log( ζ ) to be real whe ( ζ ) > 0 Note that If()=f() for >0 ad 0 for 0 [5]. For Γ ( ) more properties [6]. From efiitio we have: If()=f() = = a Γ ( ) Γ ( ) φ =0 =0 ϕ where a := for all = Γ ( ) Thus: If() ε ( φ 0) If() ε ad The we have the followig results: Theorem 3: Let the assumptios of Corollary. hold the: ( I f ()) q() If() ad q() is the best domiat. Proof. Let the fuctio F() be defied by: F() := I f () Theorem 4: Let the assumptios of Corollary 2. hold the: ( I f ()) q() If() ad q() is the best domiat. Proof: Let the fuctio F() be defied by: F() := I f () where the fuctio f() is aalytic i simply-coected CONCLSION regio of the complex -plae ( ) cotaiig the The operator defied was motivated by various origi ad the multiplicity of ( ζ ) is removed by wor studied earlier by the researchers [7-9]. This 44

5 J. Math. & Stat. 5 (3): operator ca be geeralised further ad may other results such as the coefficiet estimates ad distortio theorem ca be obtaied. ACKNOWLEGEMENT The study here is fully supported by EScieceFud grat: SF0425 MOSTI Malaysia. REFERENCES. Ravichadra V. S.K. Sivaprasad ad M. arus O a subordiatio theorem for a class of meromorphic fuctios. J. Ieq. Pure Applied Math. 5: ml.sid?= Shamugam T.N. V. Ravichadra ad S. Sivasubramaia ifferetial sadwich theorems for some subclasses of aalytic fuctios Austral. J. Math. Aal. Applied 3: Miller S.S. ad P.T. Mocau ifferetial Subordiatios: Theory ad Applicatios. Pure ad Applied Mathematics No.225 eer New Yor ISBN: Srivastava H.M. ad S. Owa 989. ivalet Fuctios Fractioal Calculus ad Their Applicatios Halsted Press Joh Wiley ad Sos New Yor Chichester Brisbae ad Toroto ISBN: 0: Miller K.S. ad B. Ross 993. A Itroductio to The Fractioal Calculus ad Fractioal ifferetial Equatios. st Ed. Joh Wiley ad Sos Ic. ISBN: pp: Ibrahim R.W. ad M. arus Subordiatio ad superordiatio for uivalet solutios for fractioal differetial equatios. J. Math. Aal. Applied 345: OI: 0.06/J.JMAA Ibrahim R.W. ad M. arus O subordiatio theorems for ew classes of ormalie aalytic fuctios. Appllied Math. Sci. 2: hiari.com/ams/ams-password-2008/ams-password /darusams pdf 8. arus M. ad R.W. Ibrahim Coefficiet iequalities for a ew class of uivalet fuctios. J. Math. 29: Ibrahim R.W ad M. arus ifferetial subordiatio results for ew classes of the family ε(φ Ψ). J. Ieq. Pure Applied Math. 0: