NEW CLASSES OF UNIVALENT FUNCTIONS

Transcription

1 BABEŞ BOLYAI UNIVERSITY CLUJ NAPOCA FACULTY OF MATHEMATICS AND COMPUTER SCIENCE Aaaria-Geaia MACOVEI NEW CLASSES OF UNIVALENT FUNCTIONS PhD thesis-abstract Leader scietific: Acadeicia dr Petru T MOCANU Cluj-Napoca

2 CONTENTS PREFACE I GENERALS ON THE CONCEPT OF UNIVALENT FUNCTION 4 I Geeral issues coceri the theory of uivalet fuctios 4 I Special failies of uivalet fuctios i U 4 I3 Aalytic fuctios with positive real part 5 I4 Subordiatio 5 II SPECIAL CLASSES OF UNIVALENT FUNCTIONS 6 II Starlie fuctios 6 II Covex fuctios 6 II3 Close-to-covex fuctios 8 II4 Alpha-covex fuctios 8 II5 p-fold syetric alpha-covex fuctios 9 II6 Starlie fuctios type 9 II7 Spirallie fuctios II8 Starlie ad covex fuctios of order II9 Aalytic fuctios with eative coefficiets III DIFFERENTIAL SUBORDINATIONS AND SUPERORDINATIONS III Differetial subordiatios III Briot-Bouquet differetial subordiatios 3 III3 Applicatios differetial subordiatios 5 III4 Applicatios of Briot-Bouquet differetial subordiatios 7 III5 Differetial superordiatios 9 III6 Briot-Bouquet differetial superordiatios III7 Applicatios of differetial superordiatios of type usi a iteral operator III8 Applicatios of differetial subordiatios ad superordiatios sadwich theores 4 III9 Differetial subordiatios ad superordiatios for aalytic fuctios defied by the Ruscheweyh liear operator 9 III Differetial subordiatios ad superordiatios for aalytic fuctios defied by a class of ultiplier trasforatios 35 IV SUBCLASSES OF UNIVALENT FUNCTIONS 39 IV Subclasses of uivalet fuctios defied by covolutio 39 IV Subclasses of oralied starlie ad covex fuctios 4 IV3 Subclasses of oralied uivalet fuctios defied by covolutio 4 IV4 Subclasses of oralied starlie ad covex fuctios of order 45 IV5 Subclasses of oralied alpha-covex fuctios 45 IV 6 Subclasses of uivalet fuctios with eative coefficiets 47 BIBLIOGRAPHY 5

3 Keywords: uivalet fuctio aalytic fuctios with positive real part starlie fuctios covex fuctios close-to-covex fuctios alpha-covex fuctios spirallie fuctio aalytic fuctios with eative coefficiets differetial subordiatio Briot-Bouquet differetial subordiatio differetial superordiatio Briot-Bouquet differetial superordiatio Sălăea differetial operator Ruscheweyh differetial operator PREFACE Geoetrical theory of uivalet fuctios is oe of the areas of coplex aalysis the aalytical rior of reasoi is closely itertwied with the eoetric ituitio ad the first cocepts were itroduced i the early twetieth cetury whe they appeared first ajor wors such as writte by P Koebe T H Growall I W Alexader L Bieberbach Is oteworthy that i developi this area of atheatics Roaia atheaticias had distiuished erit Creator of Roaia school uivalet fuctios theory is G Căluareau who obtaied ecessary ad sufficiet coditios for uivaletio Cotiuer it is Acadeicia PT Mocau who has iposed worldwide with outstadi results ad we etio a few has itroduced ew classes of uivalet fuctios alpha-covex fuctios or fuctios Mocau has addressed the issue of oaalytic fuctio ijectivity ad has created toether with SS Miller a ew ethod of study of specific classes of uivalet fuctios ad aely "adissible fuctios ethod" or "ethod of differetial subordiatio" Later Acadeicia PT Mocau ad SS Miller itroduced the dual cocept of differetial subordiatio called differetial superordotio Uder the leadership of Acadeicia Mr PT Mocau has bee fored a stro school of eoetric fuctios theory i Cluj Ao his uerous collaborators at the atioal level we etio: NN Pascu GŞ Sălăea T Bulboacă G Kohr P Curt Gheorhe Oros M Acu ad others ad iteratioally SS Miller MO Reade S Ruscheweyh S Owa R Fourier MK Aouf ad others I this PhD thesis have bee obtaied ew results reardi the differetial subordiatios ad superordiatios ad soe subclasses of uivalet fuctios The paper cotais four chapters a itroductio ad a biblioraphy cotaii 38 titles ao which are sied by the author I the first chapter etitled "Geeral o the cocept of uivalet fuctio" ad structured i four pararaphs are preseted eeral probles coceri the theory of uivalet fuctios their special faily aalytic fuctios with positive real part ad the cocept of subordiatio The secod chapter etitled "Special classes of uivalet fuctios" is divided ito ie sectios Here are the iportat results o the class of starlie fuctios covex close-to-covex alpha-covex alpha-covex p-syetric type starlie of type spirallie stellate ad covex of order ad aalytical with eative coefficiets bei exposed defiitios leas ad fudaetal theores These otios ad results are ecessary to cofir the oriial results cotaied i Chapter IV The ext two chapters cotai oriial results already published or uder publicatio "Differetial subordiatio ad superordiatio" is the third chapter ad is divided ito te pararaphs I the first two pararaphs are defiitios leas ad fudaetal theores for differetial subordiatio ad Briot-Bouque differetial subordiatio These otios ad results are ecessary to cofir the oriial results cotaied i the followi pararaphs of this chapter I pararaphs III3 ad III4 were deteried applicatios of differetial subordiatio ad Briot-

4 Bouque differetial subordiatio ad are cotaied i [57] [6] [64] I the followi two pararaphs are the basic defiitios ad theores for superordiatio differetial ad Briot- Bouque differetial superordiatio ecessary to cofir the oriial results cotaied i the followi pararaphs of this chapter Soe of these results are cotaied i [56] I pararaph III7 were deteried applicatios of Briot-Bouque differetial superordiatio obtaied by usi iteral operators ad [54] [55] [58] cotaii the results III8 pararaph cotais applicatios of differetial subordiatio ad superordiatio ad theores sadwich These results are cotaied i [63] [65] [66] Usi the Ruscheweyh operator i pararaph III9 were deteried ad subordiatio ad differetial superordiatio for aalytical fuctios The results of this pararaph are cotaied i [67] I the last pararaph of this chapter are preseted usi subordiatio ad superordiatio differetial operator I ( r λ ) f( ) are cotaied i [6] [68] The last chapter etitled "Subclasses of uivalet fuctios" is structured i six sectios All results of this chapter are oriial I pararaph IV are show subclasses of uivalet fuctios defied by covolutio deoted K S C ad are cotaied i [6] Pararaph IV etitled "Subclasses of oralied starlie ad covex fuctios" show the subclasses deoted S ( ζ ) K( ζ ) cotaied i [69] The followi pararaph is a iterili of the previous pararaphs withi the eai that we deteried subclasses of oralied uivalet fuctios defied by covolutio deoted S ( ζ ) K ( ζ ) C ( ζ ) M ( ζ ) S γ ( ζ ) These results are cotaied i [7] Subclasses of oralied starlie ad covex fuctios of order are preseted i pararaph IV4 are deoted S ( ; ζ ) K( ζ ; ) ad are cotaied i [7] Results of pararaph IV5 subclasses of oralied alpha-covex fuctios deoted M ( ζ ) are cotaied i [7] The last pararaph of this chapter is etitled "Subclasses of uivalet fuctios with eative coefficiets" presets a ew class of fuctios deoted TS λ ( ) ad these results are cotaied i [73] O this way I wat to bri sicere thas ad to express y feelis of estee ad respect for scietific leader of the wor acadeicia Mr Petru T Mocau for the way he direct the developet of this wor for the support ad cofidece that ispired e ad peraet ecourai Also y thas oes to prof dr Griore Şt Sălăea whose results i field of fuctios with eative coefficiets have bee very helpful Mrs prof dr Gabriela Kohr ad other professor of the Departet of Theory of Fuctios Ad last but ot least I would lie to thas y childre parets ad husbad for support uderstadi ecouraeet ad support I what follows I have selected the ost iportat results of each chapter 3

5 I GENERAL ON THE CONCEPT OF UNIVALENT FUNCTION I this chapter are preseted the basic cocepts ad results o theory uivalet fuctios their special faily aalytic fuctios with positive real part ad the otio of subordiatio I Geeral issues coceri the theory of uivalet fuctios Defiitio I [38]: Let the doai D ad let the fuctio f : D We say that fuctio f is uivalet fuctio if f H (D) ad f is ijective o D Defiitio I [38]: We deote with: H u (D) = { f H (D): f is uivalet o D } H u (D) deote the classes of uivalet fuctios Theore I [38]: If the fuctio f H u (D) the f ( ) for ay D Corollary I [38]: Let D be a covex doai i the plae f H (D) such that Re f ( ) > for ay D the f H u (D) I Special failies of uivalet fuctios i U We deote the ope uit disc i coplex pla: U = { : < } The set of fuctios f : U holoorphic i the uit disc is deoted by H (U) For a ad we deote H [ a ] = { f H (U) : f( ) = a+ a + a } + H (U) : + A ( ) = { f f = + a+ + a+ + } ad for = A = A A iportat place they occupy i the theory of uivalet fuctios of class S of fuctios of the for: 3 f( ) = + a + a3 + + a + U holoorphic ad uivalet i the uit disc U We deote: S = { f H u (D): f() = f () = } Theore I4 [7]: If f S 3 f( ) = + a + a3 + + a + U the a satisfie the relatio a Equality holds a = if ad oly if f is a Koebe fuctio if f( ) = Kτ ( ) = U τ iτ + e ( ) Cojectura I [9]: If f S f( ) = + a + a + + a + U the a for ay 3 3 4

6 Corollary I3 [95]: Class S is copact I3 Aalytic fuctios with positive real part Defiitio I3 [95]: We itroduce a class of fuctios P = { p H (U) : p () = Re p ( ) > U } called class of Caratheodory fuctios Defiitio I3 [95]: We itroduce a class of fuctios B = { ϕ H (U) : ϕ () = ϕ ( ) < U } called class of Schwar fuctios Theore I34 [8]: If p P p ( ) = + p+ p + U the p for ay Equality holds if ad oly if: + λ p ( ) = U λ λ = λ I4 Subordiatio Defiitio I4 [95]: Let the fuctios f F H (U) We say that fuctio f is subordiated to the fuctio F ad we ote f F or f ( ) F( ) U if there is a fuctio w H (U) with ( ) w = ad w( ) < U such that = U Theore I4 [95]: Let the fuctios f F H (U) ad suppose that F is uivalet i U The f ( ) F w( ) ( ) F( ) if ad oly if f ( ) = F( ) ad f ( U) F( U) f U Corollary I4 [49]: Let the fuctios f F H (U) such that F is uivalet i U a) If f ( ) = F( ) ad f ( U) F( U) the f ( Ur) F( Ur) b) The equality f ( Ur ) = F( U r ) there is a λ = < r < r < if ad oly if f ( U) = F( U) or f ( ) F( λ ) = II SPECIAL CLASSES OF UNIVALENT FUNCTIONS I this chapter are preseted the iportat results o starlie fuctios covex fuctios close-to-covex fuctios alpha-covex fuctios p-fold syetric alpha-covex fuctios starlie fuctios type spirallie fuctios starlie ad covex fuctios of order aalytic fuctios with eative coefficiets bei exposed to defiitios leas ad fudaetal theores II Starlie fuctios The starlie fuctios are first itroduced i 9 by J W Alexader Theore II [95]: Let the fucti f H (U) f () = The fuctio f is starlie i U if ad oly if f () ad 5

7 f ( ) Re > U f( ) Defiitio II6 [95]: Let S f ( ) = { f A : Re > U } S deote the class of starlie f( ) fuctio Theore II3 [95]: Class S is copact 3 Theore II5 [5]: If f S where f( ) = + a + a + + a + U the 3 a for ay Equality holds if ad oly if f ( ) = Kτ ( ) U τ II Covex fuctios The covex fuctios were itroduced i 93 by E Study [38] ad their study was cotiued T H Growall [35] ad K Lower [5] Lea II [3]: Let the fuctio p H (U) such that Re p () > ad let The: p ( ) Re p ( ) + > p ( ) U Re p ( ) > U Theore II [95]: Let the fuctio f H (U) The fuctio f is covex i U if ad oly if f () ad f ( ) Re + > U f ( ) f ( ) Defiitio II4 [95]: Let K = { f A : Re + > U } K deote the class of covex f ( ) fuctio Theore II [95]: The fuctio f K if ad oly if S where ( ) = f ( ) U or f K f ( ) S Theore II5 [95]: Class K is copact Theore II6 [5]: If f K f( ) = + a + + a + U the a for ay Equality holds if ad oly if: f( ) = U τ iτ + e G S Sălăea ad S Ruscheweyh itroduce two differetial operators which allow i certai situatios study the stars ad covex fuctios siultaeously ad of their subclasses Defiitio II8 [5]: Let D be the Sălăea differetial operator D : A A defied as: D f( ) = f( ) Df( ) = Df( ) = f ( ) ( ) D f( ) = D D f( ) 6

8 Observatio II: If fuctio f A j f ( ) = + a U the: j= j= D f( ) j a = + j U j j Defiitio II9 [5]: We say that fuctio f A is -starlie if verify iequality: + D f( ) Re > U D f( ) We ote S class of these fuctios Theore II [5] Let φ A is covex S ad F H (U) such that Re F( ) > U φ F The is covex φ Defiitio II []: Let R be the Ruscheweyh differetial operator R : A A defied as: R f( ) = f( ) = ( ( )) ( f ) + ( )! Observaţia II []: If the fuctio f A Defiitio II3 [95]: A fuctio f : U R f( ) C a j= U j f ( ) = + a U the = + + j j j U j= j II3 Close-to-covex fuctios f H (U) is called close-to-covex if there a covex fuctio ϕ i U such that: f ( ) Re > U ϕ ( ) We say that fuctio f is close-to-covex aaist with the fuctio ϕ f ( ) Defiitio II3 [95]: We deote C = { f A : ( ) ϕ K Re > U } C deote the ϕ ( ) class of close-to-covex fuctio Theore II3 [436]: Let the doai D ad let the fuctio f H (D) Suppose that f ( ) there a fuctio ϕ H u (D) such that ϕ ( D) = Δ is a covex doai The Re > D (ie ϕ ( ) fuctio f is close-to-covex aaist with ϕ ) ivolvi the fuctio f is uivalet i D Theore II33 [67]: If f Equality holds if ad oly if C f = + a + + a + U the a for ay ( ) f( ) = K ( ) = τ iτ ( + e ) U τ 7

9 II4 Alpha-covex fuctios (Mocau fuctios) I order to establish a li betwee the otios of covexity ad the star i 969 P T Mocau [9] itroduces the cocept of alpha-covexity Later their various properties were obtaied by P T Mocau S S Miller ad M O Reade [8889] : = r if ad oly if Theore II4 [9]: Let the fuctio f alpha-covex o the circle { } Re J( f; ) > for = r where: f ( ) f ( ) J( f; ) = ( ) + + f ( ) f ( ) Defiitio II43 [9]: We deote M = { f H (U) : f() = f () = f( ) f ( ) Re J( f; ) > U } M deote the class of alpha-covex fuctios or Mocau fuctios II5 p-fold syetric alpha-covex fuctios Defiitio II5 [6]: Let ad p p We deote M { p p p = f M : f( ) = + ap + + ap U} M p deote the class of p-fold syetric alpha-covex fuctios Theore II5 [6]: If the fuctio f where > ad U is a fixed poit the: where M( r ) = r ρ p p f ( ) = K ( p) τ ( ρ) M p p p p p M( r p) f( ) M( r p) dρ Defiitio II6 : Let the fuctio Equality is achieved (both sides) if the fuctio f is for II6 Starlie fuctios type f S We say that fuctio f is starlie type ad ote f S [ ] If = ( f ) = sup{ β : f M β } Defiitio II6: Let ad p p We deote p+ p+ { ap+ + p + U} S [ ] = f S[ ]: f( ) = + a + p starlie fuctio type p- syetric ] deote the class of S p [ 8

10 Theore II6 [6]: The fuctio f S [ ] if ad oly if p S [ p] where f( ) = ( p ) p p = 3 II7 Spirallie fuctios I 93 L Space [36] presets a eeraliatio of specific fuctios ad class spirallie fuctios Theore II7 [8]: Let the fuctio f H (U) with f () = f () ad f( ) U π π ad let γ The fuctio f is spirallie type γ if ad oly if: iγ f ( ) Re e > f( ) U Defiitio II77 [8]: We deote S iγ f ( ) γ = f A : Re e > f( ) U where π π γ S γ deote the class of type γ spirallie fuctios II8 Starlie ad covex fuctios of order Defiitio II8 [95]: Let < We deote deote the class starlie fuctios of order Defiitio II8 [95]: Let < We deote: K( ) K( ) deote the class covex fuctios of order f ( ) S ( ) = f A :Re > U f( ) S ( ) f ( ) = f A :Re + > U f ( ) II9 Aalytic fuctios with eative coefficiets Defiitio II9: We deote: T = f S: f( ) = a a \{} U ad = T = T S T deote the class starlie fuctios with eative coefficiets T ( ) = T S ( ) T ( ) deote the class starlie fuctios of order with eative coefficiets c T = T K c c T deote the class covex fuctios with eative coefficiets ad T ( ) = T K( ) with < deote the class covex fuctios of order with eative coefficiets f ( ) Defiitio II9: We deote: Td = f T : < U f( ) 9

11 Theore II9 [34]: Let the fuctio f defied by the relatio (II9) The oly if: c a ad f T ( ) if ad oly if: = ( ) a III DIFFERENTIAL SUBORDINATIONS AND SUPERORDINATIONS = f T ( ) if ad I this are preseted chapter the defiitios leas ad the fudaetal theore o differetial subordiatios ad superordiatios Briot-Bouque differetial subordiatios ad superordiatios ad their applicatios ad usi various fuctios ad liear operator III Differetial subordiatios The ethod of differetial subordiatio ow as the ethod adissible fuctios is oe of the latest ethods used i the eoetric theory of aalytic fuctios ad was itroduced by S S Miller ad P T Mocau i [7677] ad the developed i ay other wor 3 Defiitio III: Let ψ : U ad let the uivalet fuctio h i uit disc U If the fuctio p H [ a ] satiesfies the differetial subordiatio: ( ) ψ p( ) p'( ) p"( ); h( ) U (III4) the fuctio p is called a (a ) solutio of the differetial subordiatio (III4) Defiitio III: Subordiatio of the relatio (III4) is called a secod-order differetial subordiatio ad the uivalet fuctio q i U is called (a ) doiat solutio of the differetial subordiatio (III4) Defiitio III3: A doiat q ~ that satisfies q ~ ( ) q( ) for all doiats q of relaţia (III4) is said to be the best (a ) doiat 3 Defiitio III4: Let ψ : U ad let the uivalet fuctio h i uit disc U If p is a fuctio aalytic i uit disc U ad satiesfies the differetial subordiatio of relatio (III 4) the fuctio p is called solutio of the differetial subordiatio Defiitio III5: The uivalet fuctio q is called a doiat the differetial subordiatio of relatio (III4) If p q for ay p care satisfies the relatio (III4) i Lea III [76]: Let = re θ with < r < ad let f ( ) = a + a a fuctio U ; r with f ad If be cotiuous o U (; r ) ad aalytic o ( ) { } ( ) f ( ) ax f ( ) : U ( r ) { ; = } the there a real uber such that: f '( ) f ''( ) a) = ; b) Re + f( ) f '( ) Defiitio III6: We deote by Q the set of fuctios q that are aalytic ad ijective o U \ E( q) where E { = } ( q) = ζ U liq( ) : ζ

12 ad are such that q '( ζ ) for ζ U \ E( q) the subclass Q for which q() = a Let E(q) is called exceptio set Deote by Q (a) Defiitio III6 [76 77]: Let the set of Ω let the fuctio q Q ad with Ψ [ Ω q] 3 class fuctios ψ : U that satisfy the coditio ψ ( rst; ) Ω We ote wheever: t ζ q"( ζ) r = q( ζ) s= ζ q'( ζ) Re + Re s + q'( ζ ) (III5) ζ U \ E q ad The set of Ψ [ Ωq] is called class of adissibile ψ r s t; is called the adissibility coditio where U ( ) fuctios ad the coditio ( ) Ω Theore III3 [8]: Let [ ] p( ) = a+ p + p H [ ] have: ψ Ψ where q() = a ad ψ ( a ;) = h () If the fuctio a ad the fuctio ψ ( p( ) p'( ) p"( ); ) ( p( ) p'( ) p"( ); ) h( ) ψ ( ) q( ) p H (U) the we Theore III4 [7677]: Let the uivalet fuctios hq H u (U) with q() = a ad we ote 3 hρ ( ) = h( ρ ) qρ ( ) = q( ρ ) Let the fuctio ψ : U with ψ ( a;) = h() satisfy oe of the followi coditios: a) ψ Ψ [ Ω q ρ ] for soe ρ ( ) or ψ Ψ h q ρ ρ ρ such that [ ] b) there exists ( ) ρ ρ for ay ( ) p( ) = a+ p + p H [ ] If the fuctio ψ ( p( ) p'( ) p"( ); ) H (U) the we have: ( p( ) p'( ) p"( ); ) h( ) ψ ( ) q( ) Theore III7 [53]: Let the fuctio p( ) = a+ p + p H [ ] a) If ψ Ψ [ Ω a] the we have: b) If ψ Ψ [ a ] ( p ( ) p'( ) p "( ); ) a ad the fuctio p a ψ Ω U p ( ) < U the we have: ( p p p ) ψ ( ) '( ) "( ); < U p ( ) < U Theore III9 [95]: Let the fuctio p( ) = a+ p + p H [ ] a) If ψ Ψ { Ω a} the we have: ( p ( ) p'( ) p "( ); ) a ψ Ω U Re p( ) > U b) If ψ Ψ {} a the we have: ( p p p ) ψ ( ) '( ) "( ); > U Re p( ) > U Theore III [53]: Let the fuctio p( ) = a+ p + p H [ ] the fuctio P: U with P ( ) < U The we have: p ( ) + Pp ( ) '( ) < U p ( ) < U a where a < ad let

13 Theore III [53]: Let the fuctio p( ) = a+ p + p H [ ] M > ad let the fuctio P: U with P ( ) p( ) + P( ) p'( ) < M U p( ) Defiitio III7: Let c with Re c > let ad let < M U The we have: < M U a where a < M Rec C = C( c) = c + + Ic Rec C If the uivalet fuctio R is defied i U by R( ) = the we ote with R c the Ope Door fuctio defied by the relatio: + b ( + b)( + b ) Rc ( ) = R = C where b = R ( c) + b ( + b ) ( + b) Lea III7 ( lea Ope Door ) [85]: Let c with Re c > let ad R c the Ope Door fuctio ad let the fuctio P( ) R ( ) If the fuctio c p H p' ( ) + P( ) p( ) = the Re p( ) > U P H [ c ] satisfy the differetial subordiatio c satisfies the differetial equtio Theore III4 [8384]: Let the fuctios φ ϕ H [ ] with φ ( ) ϕ( ) U Let βγδ with β + δ = β + γ ad Re( +δ ) > Let the fuctio f A ad suppose that f '( ) ϕ'( ) P( ) + + δ R + δ ( ) f ( ) ϕ( ) where R c is the Ope Door fuctio If F = I ( ) is defied by β γ f δ ϕ β + + β + γ F( ) = f ( t) t ( t) dt γ φ( ) = + A + F( ) the F A U ad F'( ) φ'( ) Re β + + γ > ( ) ( ) U F φ III Briot-Bouquet differetial subordiatios Defiitio III: By Briot Bouquet differetial operator eas a operator of the for: s Φ ( p( ) p'( ) ) where Φ( r s) = r + β r + γ Defiitio III: Let β γ let the fuctio h H(U) ad let the fuctio p H(U) p ( ) = h() + p + with p () = h() By Briot Bouquet differetial subordiatio uderstad for:

14 Lea III [95]: The fuctio ad p'( ) p( ) + h( ) β p ( ) + γ L( t) = a ( t) + a ( t) + a ( t) + with a ( t) for t 3 3 li a ( t) = is a subordiatio chai if ad oly if there exist costats r (] ad M > t such that: a) L(t) is aalytic i < r for each t is easurable i [ ) for each < r ad satisfies L ( t ) M a ( t ) for < r ad t b) There a fuctio p(t) aalytic i U for ay t [ ) ad easurable i [ ) for each L( t) L( t) U such that Re p ( t) > U t ad = p( t) for < r ad for t alost all t [ ) Theore III [7879]: Let β γ with β ad let be the covex fuctio h which satisfies: Re β h ( ) + γ > U If the fuctio p H [ h() ] the we have: [ ] p'( ) p( ) + h( ) p( ) ( ) β p ( ) + γ h Theore III4 [85]: Let β γ with β ad let be the uivalet fuctio q H u (U) with q() = a such that β q ( ) + γ U ad Re [ β q () + γ ] > We deote: q'( ) q'( ) Q ( ) = ad h ( ) = q( ) + Q( ) = q( ) + β q( ) + γ β q( ) + γ Suppose that: h'( ) Q'( ) a) Re = Re ( ) > ( ) β q + γ + Q ( ) U β Q ad b) h is covex or b ) lo[ β q + γ ] is covex (or Q is starlie) If p H [ a ] the satiesfies the Briot Bouquet differetial subordiatio p'( ) p( ) + h( ) (III) β p( ) + γ p( ) q( ) ad the fuctio q is (a ) doiat solutio of the differetial subordiatio (III) The extreal fuctio is p ( ) = q( ) ad the fuctio q is solutio of Briot Bouquet differetial equatio Theore III3 [78]: Let be the uivalet fuctio q H u (U) ad let θ φ H(D) where D q(u ) such that φ ( w) w q(u ) Să ote Q( ) = q'( ) φ( q( ) ) θ ( ) h ( ) = q ( ) + Q ( ) ad suppose that: a) h is covex or Q is starlie iu 3

15 ( ) h'( ) θ ' q( ) Q'( ) b) Re = Re > ( ) + ( ( )) ( ) U Q φ q Q If fuctio p H(U) with p() = q() ad p( U ) D the we have: ( p ( ) ) + p'( ) φ( p( ) ) θ ( q( ) ) + q'( ) φ( q( ) ) = h( ) θ (III3) iplică p( ) q( ) ad the fuctio q is the best doiat of the subordiatio (III3) III3 Applicatios of differetial subordiatios I pararaph were deteried applicatios of differetial subordiatios usi + A fuctio q ( ) = The results are oriial ad are cotaied i [57] [6] A + A Theore III3 [6]: Let q H u (U) q ( ) = with A ( ) () ad let (] A such that + A + > If p H(U) with p() = q() = ad A + A A ( ) p ( ) + p'( ) ( ) + ( A A) the p( ) q( ) Theore III3 [6]: Let A ( ) () ad let (] such that + A + > If A p H(U) with p() = ad + A A ( ) p ( ) + p'( ) ( ) + ( A A) the Re p ( ) > + A Theore III33 [6]: Let q H u (U) q ( ) = with A ( ) () ad let β > A γ (] such that + A β + A + + > (III39) γ A γ A If p H(U) with p() = q() = ad + A + A A p ( ) + βp( ) + γ p'( ) + β + γ the p( ) q( ) A A ( A) Theore III34 [6]: Let A ( ) () ad let β > γ (] suppose that satisfies the relatio (III39) If p H(U) with p() = ad + A + A A p ( ) + β p( ) + γ p'( ) + β + γ the Re p ( ) > A A ( A) + A Theore III35 [6]: Let q H u (U) q ( ) = with A ( ) () ad let > A ad β (] such that: 4

16 A A A t ( β ) 4 A A A A > (III36) ( β ) + A A + β ( β ) A A > (III37) If p H(U) with p() = q() = ad β β + A + A A + A ( p ( )) [( ) + p ( )] + p'( ) ( p ( )) ( ) A + + A ( A) A the p( ) q( ) Theore III36 [6]: Let A ( ) () let > ad β (] suppose that satisfies the relatios (III36) ad (III37) If p H(U) with p() = ad β β + A + A A + A ( p ( )) [( ) + p ( )] + p'( ) ( p ( )) ( ) A + + A ( A) A the Re p ( ) > Theore III37 [57]: If the fuctio f A the: ( a) ( b) f''( ) + f'( ) f( ) + f '( ) f( ) Theore III38 [57]: If the fuctio f A ad < A the: ( a) ( b) f''( ) + A f'( ) f( ) + f '( ) A f( ) A A Theore III39 [57]: If the fuctio f A the: ( a) ( b) f''( ) + f( ) + f '( ) f '( ) ( ) Theore III3 [57]: If the fuctio f A ad < A the: f + A f f A A ( a) ( b) ''( ) ( ) + '( ) '( ) f ( ) β β A β β III4 Applicatios of Briot-Bouquet differetial subordiatios I pararaph were deteried applicatios of Briot-Bouquet differetial subordiatio + A usi the fuctio q ( ) = The results are oriial ad are cotaied i [64] A + A Theore III4 [64]: Let q H u (U) q ( ) = where A ( ) () If p H(U) A with p() = q() = ad p'( ) + A A p ( ) + p( ) + A A the p( ) q( ) 5

17 + A Theore III4 [64]: Let q H u (U) q ( ) = where A ( ) () If p H(U) A with p() = q() = ad p'( ) + A A p ( ) + p( ) + A A the Re p ( ) > Theore III43 [64]: Let A ( ) () ad let be the covex fuctio h i U with h () = Suppose that satisfies differetial equtio: q ( ) h ( ) = q ( ) + U (III49) q ( ) + A has the uivalet solutio q ( ) = satisfies q () = ad h ( ) q ( ) A f ( ) F ( ) If f A ad is uivalet H[] Q ad f ( ) F( ) f ( ) F ( ) h ( ) U the q ( ) U f( ) F( ) where f () t F( ) = dt (III4) t Theore III44 [64]: Let A ( ) () ad let the fuctio h defied by (III49) If f A f ( ) F ( ) ad is uivalet H [] Q ad f ( ) F( ) f ( ) F ( ) h ( ) U the Re > U f( ) F( ) where the fuctio F is defied by (III4) + A Theore III45 [64]: Let q H u (U) q ( ) = with A ( ) () such that: A ( ) A A γ + > (III49) A + γ + A Aγ > (III4) ( A) ( + γ + A Aγ ) If p H(U) with p() = q() = ad > 3 4 ( γ) Aγ( γ) A γ( γ ) A ( γ ) (III4) p'( ) + A A p ( ) + + the p( ) q( ) p( ) + γ A ( A)( + γ + A Aγ ) + A Theore III46 [64]: Let q H u (U) q ( ) = with A ( ) () ad suppose that A satisfies the relatios (III49) (III4) ad (III4) If p H(U) with p() = q() = ad p'( ) + A A p ( ) + + the Re p ( ) > p( ) + A ( A)( + γ + A Aγ ) 6

18 Theore III47 [64]: Let A ( ) () ad let be the covex fuctio i U with h () = Suppose that satisfies differetial equtio: q'( ) h ( ) = q ( ) + q ( ) + γ U (III49) + A f ( ) has the uivalet solutio q ( ) = satisfies q () = ad h ( ) q ( ) If f A ad A f ( ) F ( ) is uivalet H [] Q ad F( ) f ( ) F ( ) h ( ) U the q ( ) U f( ) F( ) where γ + γ F( ) = f( t) t dt γ (III43) Theore III48 [64]: Let A ( ) () ad let the fuctio h defied by (III49) If f A f ( ) F ( ) ad is uivalet H [] Q ad f ( ) F( ) f ( ) F ( ) h ( ) U the Re > U f( ) F( ) Where the fuctio F is defied by (III43) III5 Differetial superordiatio Differetial superordiatio ethod was itroduced by S S Miller ad P T Mocau i article Subordiats of Differetial Superordiatios [86] Usi these ethods allowed to obtai ew results i the eoetric theory of aalytic fuctios Defiitio III5: Let f ad F be ebers of H (U) The fuctio f is said to be subordiate to F or F is said to be superordiate to f if there exists a fuctio w aalytic i U with w () = ad w ( ) < such that f ( ) = F( w( )) I such a case we write f F or f ( ) F( ) If the fuctio F is uivalet the f F if ad oly if f() = F() ad f ( U) F( U) 3 Defiitio III5: Let ϕ : U ad let be the uivalet fuctio h i uit disc U If the fuctio p H [ a ] satiesfies the differetial subordiatio: ( ) h ( ) ϕ p ( ) p'( ) p"( ); U (III54) the fuctio p is called (a ) solutio of the differetial superordiatio (III54) Defiitio III53: The superordiatio (III54) is called the secod-order differetial superordiatio ad the fuctio q uivalet i U is called (a ) subordiat solutio of the differetial superordiatio (III54) 3 Defiitio III54: Let ϕ : U ad let be the uivalet fuctio h i uit disc U If p is a fuctio aalytic i uit disc U ad satisfy the differetial superordiatio (III54) the fuctio p is called solutio of the differetial superordiatio 7

19 Defiitio III55: The uivalet fuctio q is called a subordiat of the differetial superordiatio (III54) If q p for ay p which satisfies the relatio (III54) Defiitio III56: A subordiat q ~ such that q( ) q ( ) for all subordiats q of (III54) is said to be the best subordiat Defiitio III57 [77 8]: Let the set of Ω let the fuctio q H [ a ] ad Deote with [ q] ϕ( rst ; ) 3 Φ Ω the class fuctios ϕ : U that satisfy the coditio ζ Ω wheever: q'( ) t q"( ) r = q( ) s = Re + Re s + q'( ) where U ζ U ad The set of Φ [ Ω q] is called class of adissible fuctios iar the coditio ϕ ( rst ; ) Ω is called the adissible coditio Ω H [ a ] ad let [ q] Theore III5 [85]: Let q ϕ Φ Ω where q() = a If the fuctio p Q (a) ad ϕ ( p( ) p' ( ) p"( ); ) is is uivalet fuctio i uit disc U the Theore III5 [56]: Let soe ρ where q () q( ρ ) uivalet fuctio i uit disc U the { ϕ ( p ( ) p '( ) p "( ); ) U } Ω q( ) p( ) Ω let q H [ a ] with q() = a ad let ϕ Φ Ω ρ = If the fuctio p Q (a) ad ( p( ) p'( ) "( ); ) { ϕ ( p ( ) p '( ) p "( ); ) ; U} Ω q( ) p( ) q ρ for ϕ p is is Theore III53 [86]: Let q H [ ] ad let be h aalytic i U ad let a ϕ Φ [ hq ] fuctio p Q (a) ad the fuctio ( p( ) p'( ) p"( ); ) disc U the If the ϕ is is uivalet fuctio i uit ( ) ϕ ( ( ) '( ) "( ); ) h p p p q( ) p( ) Theore III54 [56]: Let Ω let the fuctios hq H [ a ] with q() = a ad ote 3 hρ ( ) = h( ρ ) qρ ( ) = q( ρ ) Let the fuctio ϕ : U with ϕ ( a;) = h() satisfy oe of the followi coditios: a) ϕ Φ Ω q ρ for soe ρ or b) there u ρ such that ϕ Φ hρ q ρ for ay ρ ( ρ ) If the fuctio p Q (a) ad the fuctio ϕ ( p( ) p'( ) p"( ); ) is uivalet i uit disc U the h ϕ p( ) p'( ) p"( ); q p ( ) ( ) ( ) ( ) 3 Theore III55 [86]: Let be h a aalytic fuctio i U ad let ϕ : U Suppose that the differetial equatio: ϕ q ( ) q'( ) q "( ); = h ( ) ( ) 8

20 has solutio q Q (a) If [ hq ] U the: ϕ Φ p Q (a) ad ( p( ) p'( ) p"( ); ) ( ) ϕ ( ( ) '( ) "( ); ) h p p p q( ) p( ) ϕ is uivalet i ad the fuctio q is the best subordiat 3 Theore III56 [56]: Let be the uivalet fuctio h H u (U) ad let ϕ : U that differetial equatio: ϕ q ( ) q'( ) q "( ); = h ( ) ( ) has solutio q with q() = a ad oe of the followi coditios is verified: a) Q ad ϕ Φ hq q [ ] b) q is uivalet i U ad ϕ Φ hq ρ for soe ρ Suppose c) q is uivalet i U ad there u ρ such that ϕ Φ hρ q ρ for ay ( ρ ) ρ If the fuctio p Q (a) ad the fuctio ( p( ) p'( ) p"( ); ) the: ( ) ϕ ( ( ) '( ) "( ); ) h p p p ϕ is uivalet i uit disc U q( ) p( ) ad the fuctio q is the best subordiat 3 Theore III57 [56]: Let be the uivalet fuctio h H u (U) ad let ϕ : U that differetial equatio: ϕ q ( ) q'( ) q'( ) + q "( ); = h ( ( ) ) ( ) has solutio q with q() = a ad oe of the followi coditios is verified: a) Q ad ϕ Φ hq q [ ] b) q is uivalet i U ad ϕ Φ hq ρ for soe ρ Suppose c) q is uivalet i U ad there ρ such that ϕ Φ hρ qρ for ay ( ρ ) ρ If the fuctio p Q (a) ad the fuctio ( p( ) p'( ) p"( ); ) U the ( ) ϕ ( ( ) '( ) "( ); ) h p p p ϕ is uivalet i the uit disc q( ) p( ) ad the fuctio q is the best subordiat Theore III55 [35]: Let be q a covex ( uivalet) fuctio i uit disc U let be the fuctios ϑ ad ϕ aalytic i a doai D q( U ) ad let μ H ( ) Suppose that: ( q) + ( q) ( tq ) U t ϕ( q ( )) μ ( tq ( ) ) ( ϑ( p( ) ) μ( p ( ) ) ϕ( p( )) ϑ' ( ) ϕ ( ) μ ( ) Re > If p H [ q( )] Q with p() = q() p U ) D ad uivalet i U ad ϑ q ( ) + μ tq ( ) ϕ q ( ) ϑ p ( ) + μ p ( ) ϕ p ( ) ( ) ( ) ( ) ( ) ( ) ( ) + is 9

21 the q ( ) p ( ) The fuctio q is the best subordiat Corolarul III56 [3]: Let be q a covex ( uivalet) fuctio i uit disc U let be the fuctio ϕ aalytic i a doai D qu ( ) ad let ϕ H ( ) Suppose that: a) ξ( ) = q'( ) ϕ( q( ) ) is starlie iu b) '( q ( )) ( q ( )) ϑ Re > U ϕ If p H [ q( )] Q with p() = q() p( U ) D ad ϑ( p( ) ) + p'( ) ϕ( p( )) is uivalet i U ad the ( q ( )) + ( ) ( q ( ) ) ( p ( )) + p'( ) ( p ( )) ϑ ϕ ϑ ϕ q ( ) p ( ) The fuctio q is the best subordiat Teorea III56: Let q H u (U) ad let be the fuctios D q( U ) with ϕ( w) where q(u ) Let ξ( ) q'( ) ϕ q( ) suppose that: a) ξ is starlie ϑ ad ϕ aalytic i a doai w = ( ) ( q) '( q ( )) ( ) l ( ) = ϑ ( ) + ξ ( ) ad l'( ) ϑ ξ '( ) b) Re = Re + > U ξ( ) ϕ q( ) ξ( ) If p H [ q( )] Q with p() = q() p( U ) D ad ϑ( p( ) ) + p'( ) ϕ( p( )) is uivalet i U ad the ( q ( )) + q'( ) ( q ( )) ( p ( )) + p'( ) ( p ( )) ϑ ϕ ϑ ϕ q ( ) p ( ) The fuctio q is is the best subordiat III6 Briot-Bouquet differetial superordiatio Defiitio III6: Let β γ let h H(U) ad let p H(U) p ( ) = h() + p + with property p ( ) = h() By Briot Bouquet differetial superordiatio uderstad for: p'( ) h ( ) p ( ) + β p ( ) + γ Theore III6 [87]: Let be h a covex fuctio i U with h() = a ad let be the fuctios Θ ad Φ aalytic i a doai D Let p H [ a] Q ad suppose that Θ ( p( ) ) + p ( ) Φ( p( )) is uivalet i U If differetial equatio: ( ) ( ) Θ q ( ) + q( ) Φ q ( ) = h ( ) it has uivalet solutio q q() = a qu ( ) D ad: The: The fuctio q is the best subordiat ( q) Θ ( ) h ( ) ( ) ( ) h ( ) Θ p ( ) + p( ) Φ p ( ) q ( ) p ( )

22 Theore III63 [87]: Let be the fuctios Θ ad Φ aalytic i a doai D ad let be q where q is uivalet fuctio i U with q() = a qu ( ) D Let Q ( ) = q ( ) Φ ( q ( )) h ( ) = Q ( ) +Θ ( q ( )) ad suppose that: a) Q( ) is starlie Θ ( q ( )) b) Re > Φ q ( ) If p H [ a] Q ( ) the The fuctio q is the best subordiat ( ) p U D ad suppose that ( p( ) ) p ( ) ( p( ) ) ( ) ( ) h ( ) Θ p ( ) + p( ) Φ p ( ) q ( ) p ( ) Θ + Φ is uivalet i U III7 Applicatios of Briot-Bouquet differetial superordiatio usi a iteral operator I pararaph were deteried applicatios of Briot-Bouquet differetial superordiatio with the help of iteral operators The results are oriial ad are cotaied i [54] [55] [58] + A A Theore III7 [54]: Let A ( ) () The fuctio h ( ) = + U is A A covex Theore III7 [55]: Let A ( ) () ad the fuctio h is covex i U with h () = Suppose that we have differetial equatio: q'( ) h ( ) = q ( ) + q ( ) + + A f ( ) with the uivalet solutio q ( ) = q () = ad q ( ) h ( ) If f A ad A f ( ) is F ( ) uivalet H [] Q ad: F( ) f ( ) F ( ) h ( ) U the q ( ) U f ( ) F( ) where: F( ) = f( t) dt (III7) Corollary III7 [55]: Let A ( ) () If f ( ) f ( ) f f A ad f( ) f( ) are uivalet F ( ) F ( ) H[ a ] Q ad: F( ) F( ) f ( ) A A f( ) f( ) + + U A A f( ) F ( ) + A F ( ) U F( ) A F( ) Where:

23 Fi( ) = fi( t) dt i = (III7) Theore III73 [58]: Let a A where A = ( 455) (757 + ) The fuctio: h ( ) = + a + + a + U is covex Theore III74 [58]: Let a A ad let h is covex fuctio i U with h () = a Suppose that differetial equatio: q'( ) h ( ) = q ( ) + q ( ) + U f ( ) has uivalet solutio q ( ) = + a q() = a ad q ( ) h ( ) If f A ad is uivalet f ( ) F ( ) H [ a] Q ad F( ) f ( ) F ( ) h ( ) U the q ( ) U f ( ) F( ) where the fuctio F is defied by the relatioship (III5) f ( ) f ( ) Corollary III7 [58]: Let a A If f f A ad are uivalet f( ) f( ) F ( ) F ( ) H [ a] Q ad F( ) F( ) f ( ) f ( ) F ( ) F ( ) + a+ U the + a U f ( ) + a + f ( ) F( ) F ( ) where Fi i = is defied by the relatioship (III5) III8 Applicatios of differetial subordiatios ad superordiatios sadwich theores I pararaph were deteried applicatios of differetial subordiatio ad superordiatio The results are oriial ad are cotaied i [63] [65] [66] Theore III8 [63]: Let be the covex fuctio q i U ad suppose that Re q ( ) > β Let f A ( ) ad > Suppose that the fuctio q satisfies the relatio: q ( ) Re + q ( ) β + > ( ) (III8) If f( ) f( ) f ( ) f( ) q ( ) ( ) q( ) ( ) + β β + + q the

24 f( ) q ( ) The fuctio q is the best doiat Theore III8 [63]: Let be the covex fuctio q i U ad suppose that Re q ( ) > β Let f A ( ) f( ) H [ q()] Q > ad let f( ) f( ) f ( ) f( ) ( β) + + is uivalet fuctio i U Suppose that the fuctio q satisfies the relatio: Re q( ) q ( ) βq ( ) > (III84) If the [ ] q ( ) f( ) f( ) f ( ) f( ) βq ( ) + q ( ) ( β) + + f ( ) q ( ) ad q is the best subordiat Theore III83 [63]: Let the fuctios is covex ad q is uivalet i U ad suppose that Re q ( ) > β Let f A ( ) q f( ) H [ q( )] Q > ad let f( ) f( ) f ( ) f( ) ( β) + + is uivalet i U Suppose that the fuctio satisfies the relatio (III84) ad the fuctio satisfies the relatio (III8) If q ( ) f( ) f( ) f ( ) f( ) βq( ) + q ( ) ( β) + + the q q q ( ) βq( ) + q ( ) f( ) q( ) q( ) ad is the best subordiat iar q is the best doiat q Theore III8 [65]: Let f A ( ) γ > ad > Let be the covex fuctio q i U ad suppose that the fuctio q satisfies the relatio: q ( ) Re + + > ( ) γ (III87) If: 3

25 the: f ( ) f ( ) f( ) γ q ( ) γ ( γ) q( ) + + f( ) q ( ) ad q is the best doiat Theore III85 [65]: Let f A ( ) let be the covex fuctio q i U ad let f( ) H [ q()] Q γ > ad > Let f ( ) f ( ) f( ) γ ( γ) + is uivalet i U Suppose that the fuctio q satisfies the relatio γ q ( ) Re > (III8) If : the: ad q is the best subordiat Theore III86 [65]: Let f A ( ) Let γ q ( ) f( ) f ( ) f( ) q ( ) + γ ( γ) + f ( ) q ( ) f( ) H [ q()] Q γ > ad > f ( ) f ( ) f( ) γ ( γ) + is uivalet i U Let the fuctios is covex ad q is uivalet i U Suppose that the q fuctio satisfies the relatio (III8) ad q satisfies the relatio (III87) If the q γ q ( ) f( ) f ( ) f( ) γ q ( ) q ( ) + γ ( γ) q ( ) + + f( ) q( ) q( ) ad is the best subordiat iar q is the best doiat q Theore III87 [66]: Let f A ( ) γ > ad > i U ad suppose that satisfies the relatio: Let be the uivalet fuctio q 4

26 If : the: ad q is the best doiat q ( ) q ( ) Re + > ( ) q( ) (III83) f ( ) γ q ( ) + γ γ + f ( ) q( ) f( ) q ( ) f( ) Theore III88 [66]: Let f A ( ) H[ q()] Q γ > ad > Let f ( ) + γ γ f( ) uivalet i U Let be q covex fuctio i U ad suppose that satisfies the relatio (III73) If γ q ( ) f ( ) + + γ γ q ( ) f( ) the: ad q is the best subordiat f ( ) q ( ) f( ) Theore III89 [66]: Let f A ( ) H[ q()] Q γ > ad > Let f ( ) + γ γ is uivalet i U Let the fuctios q is covex ad q is uivalet i U f( ) ad suppose that satisfies the relatio (III83) If γ q( ) ( ) ( ) f γ q + + γ γ + q( ) f( ) q( ) the : f( ) q( ) q( ) ad is the best subordiat iar q is the best doiat q Theore III8 [66]: Let f A ( ) ad > Let be the uivalet fuctio q i U ad suppose that satisfies the relatios: Re q ( ) > (III88) ad q ( ) q ( ) Re + > ( ) q( ) (III89) If 5

27 the ad q is the best doiat Theore III8 [66]: Let f A ( ) f ( ) f ( ) q ( ) q( ) + γ γ + f( ) q( ) f( ) q ( ) f( ) H [ q()] Q ad > Let f( ) f ( ) + γ γ f( ) is uivalet i U Let be the covex fuctio q i U Suppose that the fuctio q satisfies the relatios (III89) ad Re q ( ) ( ) > (III8) If: the: ad q is the best subordiat Theore III8 [66]: Let f A ( ) [ ] q ( ) f( ) f ( ) q ( ) + + q ( ) f( ) f ( ) q ( ) f( ) H[ q()] Q ad > Let f( ) f ( ) + f( ) is uivalet i U Let the fuctios is covex ad q is uivalet i U Suppose that the q q fuctio satisfies the relatios (III89) ad (III8) ad the fuctio q satisfies the relatios (III88) ad (III89) If the q( ) f( ) f ( ) q ( ) q ( ) + + γ γ q ( ) + q ( ) f( ) q ( ) f( ) q( ) q( ) ad is the best subordiat iar q is the best doiat q Theore III83 [65]: Let f A ( ) λ > ad > U If Let be the covex fuctio q i f( ) f ( ) f( ) λ ( λ λ) ( λ) q( ) λ ( ) + + q 6

28 the ad q is the best doiat Theore III84 [65]: Let f A ( ) Let: f( ) q ( ) f( ) H [ q()] Q λ > ad > f ( ) f ( ) f( ) λ ( λ λ) + is uivalet a fuctio i U Let be the covex fuctio q i U Suppose that q satisfies the relatio: ( λ) q ( ) Re > λ (III87) If : the: f ( ) f ( ) f( ) ( λ) q ( ) + λq ( ) λ ( λ λ) + f ( ) q ( ) ad q is the best subordiat Theore III85 [65]: Let f A ( ) H [ q( )] Q λ > ad > Let f ( ) f ( ) f( ) λ ( λ λ) + is uivalet i U Let the fuctios is covex ad q is uivalet i U ad suppose that the fuctio q satisfies the relatio (III87) If q f( ) f ( ) f( ) ( λ) q( ) + λq ( ) λ ( λ λ) ( λ) q ( ) λq( ) + + the: f( ) q( ) q( ) ad is the best subordiat iar q is the best doiat q III9 Differetial subordiatios ad superordiatios for aalytic fuctios defied by the Ruscheweyh liear operator I [59] ad [67] the author obtaied differetial subordiatio ad superordiatio usi Ruscheweyh liear operator These results are oriial We defie the operator Ruscheweyh R : A A {} 7

29 R f( ) = f( ) Rf( ) = f ( ) + ( + ) R f( ) = R f( ) + R f( ) U If f A the we have: j R f( ) = + C a j= + + j j Theore III9 [59]: Let f A {} ad > Let q is uivalet fuctio i U ad suppose that: Re q ( ) > (III9) ad q ( ) q ( ) Re + > ( ) q( ) (III9) If : + R f( ) ( + ) R f( ) q ( ) + ( + ) q( ) + R f( ) q( ) the: ad q is the best doiat R f( ) q ( ) R f( ) Theore III9 [59]: Let f A H [ q()] Q {} ad > Let + R f( ) ( + ) R f( ) + ( + ) R f( ) is uivalet i U Let be the covex fuctio q i U ad suppose that satisfies the relatios (III9) ad Re q ( ) ( ) > (III95) If: the: ad q is the best subordiat [ ] + ( ) ( ) ( ) ( ) q R f + R f q ( ) + + ( + ) q ( ) R f( ) R f( ) q ( ) 8

30 R f( ) Theore III93 [59]: Let f A H [ q()] Q {} ad > Let + R f( ) ( + ) R f( ) + ( + ) is uivalet i U Let the fuctios q is covex ad R f( ) is uivalet i U Suppose that the fuctio q satisfies the relatios (III9) ad (III95) ad q the fuctio q the satisfies the relatios (III9) ad (III9) If + R f( ) ( + ) R f( ) ( ) + + ( + ) ( ) q( ) q ( ) q q + q ( ) R f( ) q ( ) R f( ) q( ) q( ) ad is the best subordiat iar q is the best doiat q Theore III94 [59]: Let f A {} ad > Let q is uivalet fuctio i U ad suppose that satisface relatios (III9) ad (III9) If the: R f R f R f q ( ) ( ) q( ) ( ) ( ) ( ) ( ) R f( ) R f( ) R f( ) q( ) ad q is the best doiat Theore III95 [59]: Let f A > Let + R f( ) q ( ) R f( ) + R f( ) H [ q()] Q {} ad R f( ) R f( ) R f( ) R f( ) ( ) ( ) R f( ) + R f( ) + R f( ) is uivalet i U Let be the covex fuctio q i U ad suppose that satisfies the relatios (III9) ad (III95) If ( ) ( ) ( ) ( ) + ( ) ( ) ( ) ( ) q R f R f R f q ( ) + + ( + ) + ( + ) q R f R f R f The: ad q is the best subordiat q ( ) R f( ) + R f( ) 9

31 Theore III96 [59]: Let f A + R f( ) H [ q()] Q {} ad R f( ) R ( ) > Let f ( ) ( ) ( ) R f R ( ) f R f( ) + R f( ) + R f( ) is uivalet i U Let the fuctios is covex ad q is uivalet i U Suppose that the fuctio satisfies q the relatios (III9) ad (III95) ad the fuctio If the + + q( ) R f( ) R f( ) q ( ) R f( ) R f( ) q satisfies the relatios (III9) ad (III9) q ( ) ( ) R f( ) + q ( ) + ( ) q( ) R f( ) q( ) + R f( ) q ( ) q ( ) R f( ) ad is the best subordiat iar q is the best doiat q Theore III97 [59]: Let f A {} ad > Let q is uivalet fuctio i U ad suppose that satisface relatio (III9) ad (III9) If + + R f( ) R f( ) q ( ) ( + ) q( ) + + R f( ) R f( ) q( ) the + R f( ) q ( ) R f( ) ad q is the best doiat + R f( ) Theore III98 [59]: Let f A H [ q()] Q {} ad > Let R f( ) + + R f( ) R f( ) ( + ) is uivalet i U Let be the covex fuctio q i U ad + R f( ) R f( ) suppose that satisfies the relatios (III9) ad (III95) If + + q ( ) R f( ) R f( ) q ( ) + ( + ) + q ( ) R f( ) R f( ) the + R f( ) q ( ) R f( ) ad q is the best subordiat q 3

32 Theore III99 [59]: Let f A + R f( ) H [ q()] Q {} ad > Let R f( ) + + R f( ) R f( ) ( + ) is uivalet i U Let the fuctios q + is covex ad q is R f( ) R f( ) uivalet i U Suppose that the fuctio satisfies the relatios (III9) ad (III95) ad the fuctio q satisfies the relatios (III9) ad (III9) If + + q ( ) R f( ) R f( ) q ( ) q( ) + ( + ) q ( ) + + q( ) R f( ) R f( ) q( ) the + R f( ) q( ) q( ) R f( ) ad is the best subordiat iar q is the best doiat q q Theore III9 [67]: Let f A ( ) {} ad > Let be the uivalet fuctio q i U ad suppose that satisfies the relatios: Re q ( ) > (III93) ad q ( ) q ( ) Re + > ( ) q( ) (III94) If: the: ad q is the best doiat + ( ) ( ) ( ) ( ) R f + R f q ( ) ( ) + + q + R f( ) q( ) R f( ) Theore III9 [67]: Let f A ( ) ad > Let: q ( ) + ( ) ( ) ( ) R f( ) H [ q()] Q {} R f + R f + ( + ) R f( ) is uivalet i U Let be the covex fuctio q i U Suppose that the fuctio q satisfies the relatios (III94) ad Re qq ( ) ( ) > (III97) If : the: [ ] + ( ) ( ) ( ) ( ) q R f + R f q ( ) + + ( + ) q ( ) R f( ) 3

33 ad q is the best subordiat R f( ) q ( ) Theore III9 [67]: Let f A ( ) ad > Let + ( ) ( ) ( ) R f( ) H [ q()] Q {} R f + R f + ( + ) R f( ) is uivalet i U Let the fuctios is covex ad is uivalet i U Suppose that q satisfies the relatios (III94) ad (III97) ad the q q q satisfies the relatios (III93) ad (III94) If + R f( ) ( + ) R f( ) ( ) + ( ) ( ) + + q( ) q ( ) q q + q ( ) R f( ) q ( ) R f( ) q( ) q( ) ad is the best subordiat iar q is the best doiat q Theore III93 [67]: Let f A ( ) {} ad > Let be the uivalet fuctio q i U ad suppose that satisfies the relatios (III93) ad (III94) If the ad q is the best doiat + + R f( ) R f( ) + R f( ) R f( ) + ( + ) ( + + ) + ( + ) + R f( ) q ( ) ( + ) q( ) + R f( ) q( ) + R f( ) R f( ) Theore III94 [67]: Let f A ( ) {} ad > Let q ( ) + R f( ) H [ q()] Q R f( ) R f R f R f ( ) ( ) ( ) ( ) R f( ) R f( ) R f( ) ( ) ( ) ( ) is uivalet i U Let be the covex fuctio q i U Suppose that the fuctio q satisfies the relatios (III94) ad (III97) If 3

34 the + + ( ) ( ) ( ) + ( ) ( ) ( ) q R f R f q ( ) + + ( + ) ( + + ) q R f R f ad q is the best subordiat + R f( ) + ( + ) ( + ) R f( ) q ( ) R f( ) + R f( ) Theore III95 [67]: Let f A ( ) {} ad > Let + R f( ) H [ q()] Q R f( ) R f R f R f ( ) ( ) ( ) ( ) R f( ) R f( ) R f( ) ( ) ( ) ( ) is uivalet i U Let the fuctios is covex ad is uivalet i U Suppose that satisfies the relatios (III94) ad (III97) ad (III94) If : the: q q q q satisfies the relatios (III93) ad + + q ( ) R f( ) R f( ) q ( ) R f( ) R f( ) q ( ) ( ) ( + + ) R f( ) q ( ) ( ) ( ) q( ) R f( ) q( ) + R f( ) q ( ) q ( ) R f( ) ad is the best subordiat iar q is the best doiat q III Differetial subordiatios ad superordiatios for aalytic fuctios defied by a class of ultiplier trasforatios The results of this pararaph are oriial ad are cotaied i [6] [68] Defiitio III: Let f A ( ) Let the operator I ( r λ): A ( ) A ( ) defied by: j + λ j I( r λ) f() = + aj j= + + λ λ ( + λ) I ( r+ λ) f( ) = I ( r λ) f( ) + λi ( r λ) f( ) [ ] Theore III4 [6]: Let f A ( ) ad λ suppose that satisfies the relatios: r Let q is uivalet fuctio i U ad 33

35 Re q ( ) > (III4) ad q ( ) q ( ) Re + > ( ) q( ) (III5) If : I( r+ λ) f( ) I( r+ λ) f( ) q ( ) ( + λ) ( + λ ) q( ) + (III6) I( r+ λ) f( ) I( r λ) f( ) q( ) the: I ( r+ λ) f( ) q ( ) I ( r λ) f() ad q is the best doiat I ( r+ λ) f( ) Theore III5 [6]: Let f A ( ) H [ q()] Q ad λ I ( r λ) f() Let I( r+ λ) f( ) I( r+ λ) f( ) ( + λ) ( + λ ) I( r+ λ) f( ) I( r λ) f( ) is uivalet i U Let be the covex fuctio q i U Suppose that the fuctio q satisfies the relatios (III5) ad: Re q ( ) ( ) > (III8) [ ] If : q ( ) I( r+ λ) f( ) I( r+ λ) f( ) q ( ) + ( + λ) ( + λ ) (III9) q( ) I( r+ λ) f( ) I( r λ) f( ) the: I ( r+ λ) f( ) q ( ) I ( r λ) f() ad q is the best subordiat I ( r+ λ) f( ) Theore III6 [6]: Let f A ( ) H [ q()] Q ad λ I ( r λ) f() I( r+ λ) f( ) I( r+ λ) f( ) Let ( + λ) ( + λ ) is uivalet i U Let the fuctios q is I ( r+ λ) f( ) I ( r λ) f( ) q covex ad is uivalet i U Suppose that satisfies the relatios (III5) ad (III8) ad the q q satisfies the relatios (III4) ad (III5) If q ( ) I( r+ λ) f( ) I( r+ λ) f( ) q ( ) q ( ) + ( + λ) ( + λ ) q( ) + q ( ) I ( r+ λ) f( ) I ( r λ) f( ) q ( ) I ( r+ λ) f( ) q ( ) q ( ) I ( r λ) f() ad is the best subordiat iar q is the best doiat q 34

36 Theore III7 [6]: Let f A ( ) ad λ Let q be uivalet fuctio i U ad suppose that satisfies the relatios (III4) ad (III5) If the: I( r λ) f() I( r+ λ) f() q ( ) ( ) ( ) q( ) + + λ + λ + I ( r λ) f() q() ad q is the best doiat I ( r λ) f() q ( ) I ( r λ) f() Theore III8 [6]: Let f A ( ) H [ q()] Q ad λ ( ) ( ) ( Let I r λ f I r+ λ) f( ) ( ) ( ) + + λ + λ is uivalet i U Let be the covex I ( r λ) f() fuctio q i U Suppose that the fuctio q satisfies the relatios (III5) ad (III8) If the: q ( ) I( r λ) f( ) I( r+ λ) f( ) q ( ) + ( ) ( ) + + λ + λ q ( ) I( r λ) f( ) ad q is the best subordiat I ( r λ) f() q ( ) Theore III9 [6]: Let f A ( ) I ( r+ λ) f( ) H [ q()] Q ad λ I ( r λ) f() ( ) ( ) ( Let I r λ f I r+ λ) f( ) ( ) ( ) + + λ + λ is uivalet i U Let the fuctios q I ( r λ) f() be covex ad be uivalet i U Suppose that satisfies the relatios (III5) ad (III8) ad q q satisfies the relatios (III4) ad (III5) If q ( ) I( r λ) f() I( r+ λ) f() q ( ) q ( ) + ( ) ( ) ( ) + + λ + λ q + q ( ) I ( r λ) f( ) q( ) the: I ( r λ) f() q( ) q( ) ad is the best subordiat iar q is the best doiat q Theore III [68]: Let f A ad λ Let q is uivalet fuctio i U ad suppose that satisfies the relatios (III4) ad (III5) If I( r+ λ) f( ) I( r+ λ) f( ) q ( ) ( λ+ ) λ q ( ) + I( r+ λ) f( ) I( r λ) f( ) q( ) the: q 35

37 Ir ( + λ) f( ) q ( ) Ir ( λ) f() ad q is the best doiat I( r+ λ) f( ) Theore III [68]: Let f A H [ q()] Q ad λ Let Ir ( λ) f() I( r+ λ) f( ) I( r+ λ) f( ) ( λ+ ) λ I( r+ λ) f( ) I( r λ) f( ) be uivalet i U Let be the covex fuctio q i U Suppose that the fuctio q satisfies the relatios (III5) ad (III8) If: q ( ) I( r+ λ) f( ) I( r+ λ) f( ) q ( ) + ( λ+ ) λ q ( ) Ir ( + λ) f( ) Ir ( λ) f( ) the: I( r+ λ) f( ) q ( ) I( r λ) f() ad q is the best subordiat I( r+ λ) f( ) Theore III [68]: Let f A H [ q()] Q ad λ Let Ir ( λ) f() I( r+ λ) f( ) I( r+ λ) f( ) ( λ+ ) λ I( r+ λ) f( ) I( r λ) f( ) be uivalet i U Let the fuctios be covex ad is uivalet i U Suppose that q q q satisfies the relatios (III5) ad (III8) ad q satisfies the relatios (III4) ad (III5) If q ( ) Ir ( + λ) f( ) Ir ( + λ) f( ) q ( ) q ( ) + ( λ+ ) λ q( ) + q ( ) I( r+ λ) f( ) I( r λ) f( ) q( ) the Ir ( + λ) f( ) q( ) q( ) Ir ( λ) f() ad is the best subordiat iar q is the best doiat q Theore III3 [68]: Let f A ad λ Let q is uivalet fuctio i U ad suppose that satisfies the relatios (III4) ad (III5) If the: ad q is the best doiat I ( r λ) f ( ) ( ) ( ) ( ) ( ) I r + λ f ( ) q ( ) q + λ + λ + + I( r λ) f() q() Ir ( λ) f() q ( ) Ir ( λ) f() Theore III4 [68]: Let f A H [ q()] Q ad λ Let 36

38 I( r λ) f() I( r+ λ) f() + λ ( + ) λ ( + ) I( r λ) f() be uivalet i U Let be the covex fuctio q i U Suppose that the fuctio q satisfies the relatios (III5) ad (III8) If the ad q is the best subordiat q ( ) I( r λ) f( ) I( r+ λ) f( ) q ( ) + + λ ( + ) λ ( + ) q ( ) Ir ( λ) f( ) I( r λ) f() q ( ) Ir ( λ) f() Theore III5 [68]: Let f A H [ q()] Q ad λ Let I( r λ) f() I( r+ λ) f() + λ ( + ) λ ( + ) I( r λ) f() be uivalet i U Let the fuctios be covex ad be uivalet i U Suppose that satisfies the relatios (III5) ad (III8) ad (III5) If : the: q q q q satisfies the relatios (III4) ad q ( ) Ir ( λ) f() Ir ( + λ) f() q ( ) q ( ) + + λ ( + ) λ ( + ) q ( ) + q ( ) I( r ) f( ) q ( ) λ Ir ( λ) f() q( ) q( ) ad is the best subordiat iar q is the best doiat q IV SUBCLASSES OF UNIVALENT FUNCTIONS The chapter is preseti subclasses of uivalet fuctios defied with the covolutio subclasses of oralied starlie ad covex fuctios subclasses of oralied uivalet fuctios defied with the covolutio subclasses of oralied starlie ad covex fuctios of order ad subclass uivalet fuctios with eative coefficiets The results preseted i this chapter are oriial ad are cotaied i [6] [69] [7] [7] [7] IV Subclasses of uivalet fuctios defied by covolutio Usi the defiitio of covolutio have obtaied ew subclass of uivalet fuctios The results of this subpararaph are oriial ad are cotaied i [6] Defiitio IV: Let f A defied as with ( ) ad we deote: U f () = j j f ( ) = + a U ad ( ) b j= j = + j= j 37