SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR

Transcription

1 Joural of the Alied Matheatics Statistics ad Iforatics (JAMSI) 5 (9) No SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR SP GOYAL AND RAKESH KUMAR Abstract Here we itroduce a ew subclass of ultivalet aalytic fuctios of colex order by eas of a differetial oerator Certai iterestig roerties such as iclusio relatioshi ad radius robles are ivestigated for this fuctio class Matheatics Subject Classificatio : 3C45; 3C5 Key Words ad Phrases: Aalytic fuctios Covex set Multivalet fuctios Starlie fuctios INTRODUCTION Let A ( ) deote the class of fuctios of the for N { } () f = z + a z ( = 3 ) = + which are aalytic ad ultivalet i the oe uit disc U: = { z C : z < } Suose that Pβ ( ) be the class of fuctios hz ( ) of the for hz = + cz + c z + + () ( ) + which are aalytic i U ad satisfy Re{ hz ( )} > β β < z U We ote that P( ) P( ) P( ) that P is the class of fuctios with ositive real art P( β) P( β) 39 ad Let P ( β) β < be the class of fuctios aalytic i = if ad oly if U such P( β ) for z U It is easy to verify that P ( β) P( β) ad P ( ) P The class P ( β ) was itroduced ad studied (3)

2 SP GOYAL R KUMAR recetly by Noor [4] Agai P ( ) P the class itroduced by Pichu [5] where he has geeralized the cocet of fuctios of bouded boudary rotatio It is worth etioig that for > fuctios i P eed ot be with the ositive real art It is easy to see that P ( β ) if ad oly if there exists h P ( ) such that z ( ) = ( β) hz ( ) + β (4) Let ( q) f deotes the q th -order ordiary differetial oerator for the fuctio f A ( ) such that q q q = + ( ) f η( ; q) z η( ; q) a z = + (5) where η j! ( jq ; ) = ( j q j Nq ; N N {} z ) ( j q)! > = U (6) ( q Followig Frasi [] we defie the differetial oerator D f ) ( z ) by η q η q = + D f () z = ( q) ( ; q) z + ( q) (; q) a z ( N ; z U) Obviously ( ) = ( ) ( ) = ( ) ad () () D f z f z D f z f z (7) D f () = zf Also ( q) D f z z D f z ( ) = ( ( )) (8) Now for g A ( ) defied by g z b z = + (9) = + we itroduce a subclass of ultivalet aalytic fuctios i the oe uit disc U as follows: β DEFINITION Let > < > q N; q N z U ad f A ( ) The f Σ ( λ q β δ ) if it satisfies 4

3 SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR ( q) D f D f D f ( λ) ( ) + λ P ( q) β + D g D g D g where g A ( ) satisfies the coditio () ( q) D g ( q) D g P ( δ) ( δ < z ) We ote that if q = = = the g is starlie uivalet i U PRELIMINARY RESULTS U () To establish our ai results we shall require the followig ow results: LEMMA ([3]) Let r = r+ irad s = s+ is ad let ψ () rs be a colexvalued fuctio satisfyig the coditios: (i) () rs ψ is cotiuous i a doai (ii) ( ) D ad ψ ( ) > (iii) D C Re ψ ( ir s ) wheever ( ir s) D ad s ( + r ) If hz ( ) is aalytic i U with h () = such that ( hz ( ) zh ) D ad Re( h ( z )zh ' ) > for z U the Re h(z) > LEMMA Let h P( ) The for z U (i) r Re ( ) ( ) + r hz hz + r r r Re h zh' r (ii) For art (i) see [7] ad for art (ii) see [] LEMMA 3 ([6]) If hz ( ) is aalytic i U with h () = ad λ is a colex uber satisfyig Reλ ( λ ) the Re{ hz ( ) + λzh } > α ( α < ) ilies where γ is give by Re hz ( ) > α + ( α)(γ ) 4

4 SP GOYAL R KUMAR γ Reλ = ( + t ) dt which is a icreasig fuctio of Reλ ad γ < This estiate is shar i the sese that the boud caot be iroved 3 MAIN RESULTS THEOREM 3 The class P ( β ) is a covex set H H P ( β ) PROOF Let We shall show that for α α > H = [ αh α H ] P ( β ) α + α + By defiitio of P ( β ) we ca write H = α + α+ α 4 4 where P( β ) ( j = 34) j ( ) ( ) α + 3 z 4 z Now writig = ( β) h + β h P( ) j = 34 We have where j j j H β = + [ αh + αh3 ] β 4 α+ α [ αh + αh4 ] 4 α+ α = + r r = r 4 4 r r P( ) sice P( ) is a covex set Thus rz ( ) P( ) Now H = ( β) r + β ad rz ( ) P( ) 4

5 SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR Therefore by defiitio H P ( β ) colete ad roof of the THEOREM 3 is D f THEOREM 3 Let f Σ ( λ q β δ ) λ The D g P ( γ ) where ad g A ( ) PROOF Set ( q) β + λξ γ = ( q) + λξ satisfies the coditio () ad Re q ξ = q D f hz ( ) = γ γ D g h () = ad hz ( ) is aalytic i U ad we write q ( q) D g z = ( q) ( q) ( ) ( qd ) g () (3) By sile calculatios we get hz ( ) = + h h 4 4 (4) ( q) D f D f D f ( λ) + λ β ( q) D g D g D g λ( γ) zh = + ( γ) h + γ β + 4 ( q) q λ( γ) zh ( ) ( ) Now we for the fuctioal () rs = s + is Thus ( γ) h γ β q q z r = h = r + ir ad s = zh ψ by choosig j j 43

6 SP GOYAL R KUMAR λ( γ ) s ψ() rs = ( γ) r+ γ β + ( q) q The first two coditios of LEMMA are clearly satisfied We verify the coditio (iii) as follows λ( γ ) s Re ψ( ir s ) = γ β + ( q) q λ( γξ ) = γ β + ( q) s Re q (5) where Re q ξ = q (6) Now for Now s ( + r ) we have λ( γ) ( + r ) ξ Re ψ ( ir s) ( γ β) ( q) ( q)( γ β) λξ( γ) λ( γ) ξ r = ( q) A+ Br = C > C where A= ( q)( γ β) λ ξ( γ) Re ψ ( ir s ) if A B = λ( γ) ξ r which givesγ as defied by () Now we aly Lea to coclude that h P( ) z U ad thus h P ( ) defiitio P ( β ) j which gives the required result We ote that forδ = γ = β For = we thus get ad hece by P ( β ) THEOREM 33 For λ let f ( λ q β) Σ The ( q) D f z ( q) D g z ( ) ( ) 44

7 SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR PROOF We ca write λ ( q) ( q) D f D f D f λ = ( λ) λ ( q) ( q) D g + + D g D g This ilies that D f + ( λ ) (7) D g ( ) ( ) ( ) ( ) ( ) λ ( ) ( ) λ ( ) = H + ( ) H λ λ ( q) ( q) D f z D f z D f z D f z = ( λ) λ ( q) ( q) D g z + + D g z D g z + D g z H H P ( β ) by THEOREM 3 DEFINITION ad P ( β ) is Sice a covex set (by THEOREM 3) we get the required result Taig b = + N we get the result cotaied i the followig theore: THEOREM 34 Let > ad λ C such that Reλ > Let f A ( ) ad satisfy the coditio ( q) D f D f D f ( λ) λ q + q q ( q) η( ; q) z ( q) η( ; q) z ( q) η( ; q) z P ( β ) The ( q D f ) P ( σ ) z q U ( q) η( ; q) z where σ = β + ( β)(ρ ) ( q) ρ = F ; + ; Re{ λ} ad The value ofσ is best ossible ad caot be iroved 45

8 SP GOYAL R KUMAR PROOF We set D f = = + q hz ( ) h h ( q) η( ; q) z 4 4 where h () = ad h is aalytic i U The ( q) D f D f D f ( λ) λ q + q q ( q) η( ; q) z ( q) η( ; q) z ( q) η( ; q) z λzh = hz ( ) + P ( β ) z U ( q) Usig Lea 3 we ote that h P( σ ) j where σ = β + ( β) (ρ ) (8) ad Puttig Re{ λ} ( q) ρ = + t dt (9) Re{ λ} = λ > we have { } Re λ ( q) ( q) ( q) λ t dt = u ( u) du λ + ρ = + ( q) ( q) = F ; + ; λ λ ( q) = F; + ; λ O substitutig = ad λ R st λ i THEOREM 34 we get the followig result: COROLLARY 35 If f A ( ) satisfies () 46

9 SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR The D f ( q) D f q ( q) η( ; q) z q ( q) η( ; q) z ( λ ) + λ P ( β ) ( q) D f * P ( ) q σ z + ( q) η( ; q) z U where ad * * σ = β + ( β)(ρ )( λ ) * q ρ = F; + ; λ The value of f * σ is best ossible ad caot be iroved Taig q = i THEOREM 34 we get the followig result: COROLLARY 36 Let > ad λ C such that Reλ > A ad satisfy the coditio ( ) D f D f D f ( λ) P ( ) + λ β + z z z D f P ( σ ) z U z The σ = β + ( β)(ρ ) where ρ = ad F ; ; Re{ λ} + The value ofσ is best ossible ad caot be iroved Further taig = i COROLLARY 36 we get the followig result: the coditio COROLLARY 37 Let λ C such that Reλ > Let f A ( ) f z f f ( λ) P ( ) + λ β z f z Let ad satisfy 47

10 The SP GOYAL R KUMAR f P ( σ ) z U z where σ is defied with COROLLARY 36 Also the value of σ is best ossible ad caot be iroved λ < λ THEOREM 38 For Σ ( λ q β ) Σ ( λ q β ) PROOF If λ = the the roof is iediate fro THEOREM 3 So we let λ > ad f ( λ q β ) such that P ( β ) Σ The there exist two fuctios H H ad Now ( q) D f D f D f ( λ) ( ) ( ) ( ) ( ) q + λ H z q q = D g D g D g D f = D g H ( q) D f D f D f ( λ) + λ ( q) D g D g D g λ λ = H + H λ λ ad sice P ( β ) is a covex set it follows that RHS of () belogs to P ( β ) ad this coletes the roof We ow cosider the coverse case of THEOREM 3 as follows: () THEOREM 39 Let D f D g P ( β ) with ( q) D g ( ) ( ) qd g z 48

11 SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR P( δ ) for z U The f Σ ( λ q β δ ) for z < R( δ q λ) where R( δ q λ ) ( q) = [( δ ) ( q) + λ] + ( δ ( q)) + λ + λ( δ)( q) () PROOF Let H = D f D g The ad H P ( β ) Q P( δ ) Q ( q) D g z = Proceedig as i THEOREM 3 for β δ < ad we have ( ) ( qd ) g > q N; q N Reλ H = ( β) h+ β Q = ( δ) q + δ with h P ( ) ad q P( ) ( q) + D f D f D f ( λ) + λ β β D g z D g z D g z ( q) ( ) ( ) ( ) ( ) Usig LEMMA for h P( ) λ zh = hz ( ) + ( q) ( δ) q + δ λ zh ' = + h + 4 ( q) ( δ) q + δ λ zh ( ) ' z h + 4 ( q) ( δ) q + δ j r + r Re hj hj + r r 49

12 ad We have SP GOYAL R KUMAR r Re hj zhj ' r λ zhj ' Re hj + ( q) ( δ) q + δ λ r Re hj ( q) r ( δ) q + δ λ r + r = Re h q r δ r j ( ) ( ) λr = Re hj ( q)( r )[ ( δ) r ] ( q)[ r ( δ) r + ( δ) r ] λr = Re hj ( q)( r )[ ( δ) r ] ( q) ( δ) r [( δ ) ( q) + λ ] r + ( q) Re hj ( q)( r )[ ( δ) r ] RHS of (3) is ositive for r= z < R( δ q λ) where R( δ q λ ) is give by () f We ote that for = q = = = ad λ = δ = P ( β ) for g z U ilies f ' g ' P ( β ) for 4 ACKNOWLEDGMENT z * < R = ( + ) + + The secod author is thaful to CSIR Idia for rovidig Juior Research Fellowshi uder research schee No 9/49(498)/8-EMR-I (3) 5

13 SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR 5 REFERENCES [] BERNARDI SD 974 New distortio theores for fuctios of ositive real art ad alicatios to the artial sus of uivalet covex fuctios Proc Aer Mat Soc [] FRASIN BA 7 Neighborhood of certai ultivalet fuctios with egative coefficiets Al Math Cout 93-6 [3] MILLER SS 975 Differetial iequalities ad Caratheodory fuctios Bull Aer Math Soc [4] NOOR KI 8 O soe differetial oerators for certai classes of aalytic fuctios J Math Ieq 9-37 [5] PINCHUK B 97 Fuctios with bouded boudary rotatio Isr J Math 7-6 [6] PONNUSAMY S 995 Differetial subordiatio ad Bazilevic fuctios Proc Id Acad Sci [7] SHAH GM 97 O the uivalece of soe aalytic fuctios Pacific J Math SP Goyal Deartet of Matheatics Uiversity of Rajastha Jaiur-355 Eail: sorg@gailco Raesh Kuar Deartet of Matheatics Uiversity of Rajastha Jaiur-355 Eail: ryadav@gailco Received Jue 9 5