ON CERTAIN CLASS OF ANALYTIC FUNCTIONS

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1 ON CERTAIN CLASS OF ANALYTIC FUNCTIONS Nailah Abdul Rahma Al Diha Mathematics Depatmet Gils College of Educatio PO Box 60 Riyadh 567 Saudi Aabia Received Febuay 005 accepted Septembe 005 Commuicated by Pof D Q K Ghoi Abstact: P [ A B] ad Q [ A B] deote classes of fuctios aalytic i the disc E { : < } defied by a bouded adius otatio fuctios I this pape we have obtaied the distotio theoems coefficiets estimate some adius poblems geometical popeties ad studied covolutio coditios Itoductio Keywods: Aalytic stalie covex positive eal pat fuctio bouded adius otatio covolutio Let A deote the class of aalytic fuctios f() i E { : < } give by + f a () ad let S S* ad C be classes of fuctios i A which ae espectively uivalet stalie ad covex i the uit disc E Jaowsi [] itoduced the class PAB [ ] as follows: Defiitio A aalytic fuctio i E give by the fom P()+C Z +C Z + belogs to PAB [ ] if it satisfies the coditio + Aw p ( ) B < A + Bw ( ) whee w(0)0 ad w() P[-]P (the class of aalytic fuctio with positive eal pat satisfyig Rep()>0) N_Al-Diha@Hotmailcom

2 Defiitio f f P A aalytic fuctio i E give by () belogs to S*[AB]- B<A if ad oly if [ A B] belogs to C[AB] if ad oly if Defiitio 3 ad S*[-] S* Also it is well ow that a aalytic fuctio give by () ( f ) f P [ A B] ad C[-]C A fuctio f A is close to covex deoted by K[ABCD] if a stalie fuctio f g() S*[CD] such that P[ A B] ad K[--]K (the well ow close to covex g class due to Kapla) Defiitio Let P ( ) ad 0< be the class of fuctios p aalytic i E ad have the epesetatio π it + ( ) e p ( ) dµ ( t) it e whee ( t ) π π µ is a fuctio with bouded vaiatio o [ ππ ] ad satisfies the coditios π dµ () t dµ () t We ote that ad P ( ) P [ ] P ( ) ae the class π π of aalytic fuctio with positive eal pat geate tha It ca easily be see [5] that p P ( ) if ad oly if thee exist two aalytic fuctios p p P( ) such that Let R ( ) deote a subclass of A of fuctios of bouded adius otatio of ode The f R ( ) if ad oly if + p p p f () P () E () f () I t is clea that R ( ) S*() g + Let f be give by () ad g give by ( ) a A The the covolutio f g is

3 defied by (f g) () + a Defiitio 5 b Let f A The f belogs to P [ A B] if it satisfies the coditio f + Aw g + Bw whee g R ( ) B < A w is egula w(0)0 ad w() ad 0< Defiitio 6 Let Q [ A B] deote the class of fuctios F + c0 + c + c + which ae egula i 0 < < ad satisfy the coditio F + Aw G + Bw whee B < A w is egula i 0 < < ad bouded adius otatio of ode ie G + d + d + d + is of 0 Distotio theoem fo the class P [ A B] Theoem If f P [ A B] the fo 0 < < G ( ) P ( ) 0 < < G ( )( )/ ( )( )/ A ( ) + A ( + ) f (3) ( + )( )/ ( + )( )/ B ( + ) + B ( ) This esult is shap Poof Sice f P [ A B] we have 3

4 f + Aw B < A g + Bw whee g R ( ) By Schwa's lemma we have w ( ) + Aw ( ) If p ( ) + Bw ( ) B< A the it is well ow [] that p P[AB] ad satisfies A + A p( ) () B + B Futhe if g is a fuctio of bouded adius otatio of ode the by [7] ( )( )/ ( )( )/ ( ) ( + ) g (5) ( + )( )/ ( + )( )/ ( + ) ( ) equatios ()(5) togethe imply the iequality (3) This esult is shap if we tae + A p ( ) ad + B ( + θ ) ( )( )/ ( ) θ ( )( )/ θ + ( + θ ) g Remas O taig we have a esult of Gaesa [] O taig B λβ ad A β with w ( ) eplaced by w we get the esult of Goel ad Sohi [3] Coefficiet estimates fo the class P [ A B] To fid the coefficiet estimates fo the class P [ A B] we eed the followig lemmas: Lemma [] Let p P[ A B] ad p ( ) + c The c A B Lemma If p P ( ) p + c the c ( )

5 Poof This ca be easily see usig Lemma ad the elatio + p p p with A - ad B- Usig Lemma we ca pove Lemma 3 Lemma 3 Poof 3 Let g R ( ) g + b + b + The 3 ( ) b ( ) ad b3 ( + ) Let g R ( ) The g P( ) g( ) P( ) ( ) If ad p ( ) + c+ c + the P g b ( ) b + 3 b + ( + b + b + )( + c + c + ) Equatig the coefficiet of b c+ b b c ( ) ad 3 o both sides ad usig Lemma ad Lemma we have ad 3b3 b3+ cb + c bc + c ( ) + ( ) ( ) b3 ( ( ) + ) Theoem Let f P [ A B] whee f + a The a ( ) + ( A B) ad ( ) a3 ( A B) + ( )( A B) + ( + ) 5

6 These bouds ae shap Poof Sice f P [ A B] thee exists a fuctio g R ( ) such that f g p p P[ A B] If ad g b ( ) + p + c+ c + the + a + a + ( + b + b + )( + c + c + ) Equatig the coefficiet of ad o both sides ad usig Lemma ad Lemma 3 we have a b + c a ( ) + ( A B) ad a3 c + bc+ b3 ( ) a3 ( A B) + ( )( A B) + ( + ) This esult is shap as ca be see by the fuctio f ( ( )( )/ ) ( ( + )( )/ ) + A + + B Remas i If this esult agees with the esult of Gaesa [] ad whe A β B λβ these esults coespod to the esult of Goel ad Sohi [3] f ii If B 0 we get + Aw ( ) ad if i E the iequality g a A( ) + with shap bouds as discussed i [3] is also obtaiable Agumet of f ( ) whe f P [ A B] To discuss the agumet of the class P [ A B] we eed the followig Lemma: 6

7 Lemma Let f R ( ) The f ag ( )si Poof It is well ow that if f ( ) the thee exist two fuctios s S ( ) + ( s ( )) ( s ) R f s such that Thus ag + s s f ag ag s + s ag + ag It is ow [8] that if s S*() the ag s ( ) si Hece ag f ( ) si Shapess is satisfied fo f Lemma 5 [] Let p P[ A B] The ( ) ( + θ ) + ( ) ( ) + θ ag ( A B) p si AB Usig Lemma ad Lemma 5 we ca pove 7

8 Theoem 3 Let f P [ A B] The f ( ) ( B ) AB ag ( )si + si Poof Sice f P [ A B] theefoe f g p p P[ A B] ad g R ( ) By Lemma we have g ag ( )si (6) ad by Lemma 5 we have ( A B) AB ag p ( ) si (7) Usig (6) ad (7) we have the esult Shapess follows by taig f + Aθ g B θ + θ (8) ad ( + θ ) ( )( ) / ( ) θ ( )( ) / + ( + θ ) g The ( ) ( ) ag f si A B g AB ad g ag ag( + θ ) + ag( + θ ) ( )( ) / ( )( + ) / 8

9 Usig Lemma we have g ag ( ) si (9) Usig (8) ad (9) we have that f ( ) ( B ) AB ag ( ) si + si Rema Fo agai this esult agees with the esult i [] ad whe A β > 0 B λβ ad eplacig w by w we have the esult of Goel ad Sohi [3] Some adius poblems fo P [ A B] Lemma 6 [] Let p P[ A B] The fo E [( A B) β + A] + A if R < R p ( ) ( A)( B) Re p ( ) + β > p ( ) A + B / β + {( LK ) β( AB )} if R < R A B ( A B)( ) whee / R L R A L A A ( )( + ) K B ad K B B ( )( + ) This esult is shap Lemma 7 [7] Let g ( ) R The g Re g( ) ( ) + ( ) Futhe sice g ( ) implies R g g f ( ) P we have fo all f P ( ) 9

10 ( ) + ( ) Re f () Theoem Let f P [ A B] The f ( ) M ( ) fo R R Re f M ( ) fo R R whee M ( ) + ( ) ( ) ( ) A B ( A)( B) ( ) + ( ) A + B M () + + A B ad R R LadK ae defied i Lemma 6 Poof ( )( ) ( ) ( ) L K AB A B Sice f P [ A B] thee exists a fuctio g R ( ) such that f P( ) P g( ) [ A B] Usig logaithmic diffeetiatio we obtai f ( ) g ( ) p ( ) + f g p ad f ( ) g ( ) p ( ) Re mi Re + mi Re f g p Usig Lemma 6 with 0 β ad Lemma 7 we have the esult Shapess of the bouds follow if we choose g i ( i ) of bouded adius otatio of ode such that 0

11 Case : If R R we tae ( + A g ( ) + ( ) + ( ) P ) ad + B g The P ( ) ( A B) P ( + A)( + B) P ( ) ( A B) Thus at - Re P ( A)( B) Case : If R R we tae p f + Aw ad g + Bw ( ) w + ( ) w g + g w with w ( c) ( c ) whee c defied by the coditio + Aw Re + Bw R at - ( A B) w ( + Aw )( + Bw ) p Now p AB I fact fom the iequalities R R c + p whee c B A + AT + A T w ( ) B + BT + B ( A B) p ad we have B Hece T ad T² ² which yields ( + c ) ( + c ) Thus c w Futhe w w fo w ( ) Hece R T BR A c ( c ) ( + c ) ( T ) ( ) ( q ) ( + ) T T ad q ( c ) ( c ) w c ad [ w w ] T ( )

12 Now p Re p ( A B) T T ( )( ) AT BT Usig T R BR A with R ( A)( + A ) ( B)( + B ) (see []) ad simplifyig we have p Re p A + B + L K AB ( ) ( )( ) ( ) ( ) A B A B whee L ( A)( + A ) K ( B)( + B ) (see[]) Thus the equality i ou theoem holds at - fo f + A g ifr R + B + Aw ad fo f g ifr R whee g g R ( ) + Bw Theoem 5 If f P [ A B] the f is stalie i fo R R < fo R R whee Rad R ae defied as i Lemma 6 ad ae espectively the positive oots of the followig two equatios ( ( ) + ( ) )( A)( B) ( A B) ( ) 0 / ( ( ) + ( ) )( A B) + ( )( A + B) + ( LK) ( AB ) 0 whee K ad Lae defied i Lemma 6 This esult is shap Poof It follows fom Theoem that if f P [ A B] the f Re f M () ifr R ad

13 f Re f M () ifr R The ( ( ) + ( ) )( A)( B) ( A B) ( ) ( )( A)( B) f Re f > 0 foall < If R R ad ( ( ) + ( ) )( A B) + ( )( A + B) + ( ) L K ( AB ) ( )( A B) f Re f all if R R Lemma 8 Fo special cases see [] ad [3] > 0 fo Let g ad g ( ) ( ) R The whee ( )( ρ+ γ) G ( g ) ( g ) belogs to R ( ) ρ γ ( ρ + γ ) Poof A logaithmic diffeetiatio yields G G g g ρ + γ + + g g ( ( ρ γ )) + γk ( ) + ( ( ρ γ )) ρ K + whee K adk ( ) G G P Fom the defiitio of ( ) P thee exists i 3 P( ) + + ρ h 3 + h i such that h + γ h h ( ) + ( ( ρ γ )) It is well ow that if h P() the h() ca be witte as h ( ) p + whee Re p ad >0 3

14 G + ρ G + γ 3 [( ) p + ] ρ [( ) p + ] [( ) p + ] γ [( ) p + ] + ( ρ + γ ) ( ) + (0) Sice the class P is a covex set the ρp + γp ρ + γ 3 H whee 0 ρp ad + γp ρ + γ H Re H I > i Hece(0) ca be witte as G G + [( )( ρ + γ ) H + [ ( )( ρ + γ ) ] + T + T T T P( ) ad ρ + γ [( )( ρ + γ ) H + [ ( )( ρ + γ ) ] ( )( ) This shows that G R ( ) Theoem 6 Let f f P [ A B] The belogs to P [ A B ] Poof F ( f ) ( f ) ρ γ ( ρ + γ ) whee ( )( ρ + γ ) ρ γ ( ρ+ γ) Let G () be give byg ( g ) ( g ) The F f f G g g ρ γ ( h ) ( h ) ( ρ + γ) ρ γ whee h h P[ A B]

15 Hece F P [ A B] ( )( ρ + γ) Some geometical popeties i I this pat we shall ivestigate the behavio of ag f at a poit w ( θ ) F( e θ ) to the image Γ of the cicle C { : } 0 < ad whee θ is ay umbe of the iteval (0 π ) ude the mappig by meas of fuctio f fom the class p [ A B] We have Theoem 7 If F P [ A B] ad 0 < the fo θ < θ θ θ [0 π] θ ( ) e f e ag f ( e ) ag f ( e ) Re ( e ) θ whee - B<A ad 0< π + + θ θ + { ( ) ( )}( ) acc os Poof f If f P [ A B] the ( ) p ( ) whee p P[ A B] g A B AB Thus f ( ) g ( ) p ( ) Re Re + Re () f ( ) g ( ) p ( ) Let e 0 < < θ [0 π] Itegatig () with espect to θ i the iteval [ θ θ ] θ < θ we have l θ e f ( e ) Re dθ ag f ( e ) ag f ( e ) f ( e ) θ θ θ e g ( e ) e p ( e ) Re dθ + Re dθ g( e ) p( e ) θ θ Sice f R ( ) it follows that θ e g ( e ) ( ) + ( ) mi Re dθ ( θ θ) g R ( ) θ See [7] g( e ) 5

16 Now i the secod itegal we obseve that e p ( e ) ag pe ( ) Re{ il pe ( )} Re θ θ pe ( ) Cosequetly θ e p ( e ) Re dθ ag p( e ) ag p( e ) pe ( ) θ ad θ e p ( e ) max Re dθ max ag p( e ) ag p( e ) p P[ A B] pe ( ) p P[ A B] θ Usig Lemma 5 we have max ag pe ( ) si p P[ A B] ( A B) AB θ e p ( e ) max Re dθ max ag p( e ) mi ag p( e ) pe ( ) p P[ A B] p P[ A B] p P[ A B] θ ( A B) si AB ( A B) π cos AB Hece ( A B) ( ) + ( ) AB ag f ( e ) ag f ( e ) π + Cos + ( θ θ) The value of the ight side is depedig o the value of ad it taes its smallest value at Theeby we obtai the equied esult A covolutio coditios fo p [ A B] I 973 Rushweyh ad Sheil-Small [9] poved the polya-schoebeg cojectue amely if f is covex o stalie o close to covex ad φ is covex the f * φ belogs to the same class I the followig we shall pove the aalogue of this cojectue fo the class p [ A B] ad give some of its applicatios We eed the followig lemma with simple modificatio 6

17 Lemma 9 [6] Theoem 8 Poof: Let f ( ) R The G f * φ R ( ) whee φ is covex i E Let F P [ A B] ad φ is covex The F* φ P [ A B] Let F P [ A B] The F ()P() g() whee g belogs to R ( ) ad P()P[AB] It F * φ follows fom the Lemma 9 that g*φ R ( ) The PAB [ ] g * φ Rema As a applicatio of Theoem 8 we have the followig () The family P [ A B] is ivaiat ude the followig opeatos f ( ξ ) F( f ) dξ ( f * φ ) ξ 0 F ( f ) f ( ξ) dξ ( f * φ ) 0 f ( ζ) f ( xζ) F3 ( f ) dζ x x ζ x ζ 0 ( f * φ ) 3 + c c ( ) ξ ( ξ) ξ Re 0 c > 0 F f f d c whee F( f ) ( f φ ) ad φ ( i 3 ) i i i ae covex uivalet fuctios which satisfy φ log( ) [ + log( )] φ + x x φ3 log x x ( x) x + c φ Rec > 0 + c 7

18 Now let D F ( λ ) F + λf ( ψ F ) ( ) λ λ [ ( λ)] whee λ>0 ad let ψ λ ( ) The ψ λ ( ) is covex if λ λ+ λ λ+ (3) Thus we have () Let F P [ A B] The DF ( ) ψ * Fbelogs to the same class fo < whee λ is give by (3) λ λ λ Now let µ ( F) F This diffeetial opeato ca be witte as µ ( F) φ* F whee φ () It ca be easily veified that the adius of covexity of φ is give by c ( φ ) 3 This fact togethe with Theoem 8 yields (3) If f P [ A B] the φ* f P [ A B] whee φ is give by () if c < 3 Radius of stalieess fo the class Q [ A B] Now we geealie the esult of Goel ad Sohi [3] ad Gaese [] fo the class Q [ A B] The followig lemma ca be easily deived Lemma 9 Let i s o + ad s ( ) i s + d o + d + d + ad let s i i satisfy Re s s i be give by + c + c + c + b + b + b + G o such that the i ( ) ( ) > If + ( ) ( s) (5) ( s) G 8

19 G P ( ) G Poof Diffeetiatig (5) logaithmically yields G ( ) + s ( ) s ( ) G s s This implies that G ( ) + s ( ) s ( ) G s s o G ( ) + p p G whee Re p > i ad i G ( ) P ( ) G Theoem 9 If F Q [ A B] the fo < F Re F { M M () () for for R R whee M () ( ) + ( ) ( A B) ( A)( B) ( ) + ( ) A + B M () + + A B ad R R LadK ae defied i Lemma 6 Poof Sice F Q [ A B] theefoe ( )( ) ( ) ( ) L K AB A B 9

20 F + Aw p ( ) G + Bw whee B < A (6) w is aalytic i E ad satisfies w (0) 0 w < Diffeetiatig (6) logaithmically we have p ( ) F ( ) G ( ) + p ( ) F ( ) G ( ) o F ( ) G ( ) p ( ) + F G p Usig Lemma 6 we have ( ) ( ) ( ) Re F Re G A B if R R F G ( A)( B) / G ( ) ( A + B ) ( LK ) ( AB ) Re + + G ( A B) ( A B)( ) ad sice G is of bouded adius otatio of ode usig Lemma 7 we have G ( ) ( ) + ( ) Re < G (7) Usig (7) we have the equied esult The bouds ae shap This ca be see by choosig G of bouded adius vaiatio of ode such that G ( ) ( ) + ( ) if R R G G ( ) ( ) w ( ) + ( ) w ( ) if R R G w ad tae F ( ) such that it satisfies F + A p if R R G + B 0

21 + Aw ( ) if R R + Bw ( ) ( c) whee w with c Poceedig i the same way as i povig the shapess of c Theoem we ca pove that this esult is shap Theoem 0 If F Q [ A B] the F is stalie fo i <i i 0 < < fo R R ii 0 < < fo R R whee ad ae the smallest positive oots of the followig equatios espectively ( ) + ( ) ( A)( B) ( A B) ( ) 0 / ( ) + ( ) ( A B) + ( )( A + B) + ( LK) ( AB ) 0 Poof Usig Theoem 9 we have Re F M () wher R ad F Re F F ( Z ) M () wher R Hece Re F F > 0 Fo < i i ad this gives a sufficiet coditio fo ay fuctio F to be stalie Poceedig i the same way as i Theoem 5 we obtai the equied esult Acowledgemets Refeeces The autho is gateful to the efeee fo his valuable commets ad suggestios Ah A ad Tua P 979 O β -covexity of cetai stalie fuctios Rev Roum Math Pues Appl 5:3- Gaesa M 98 O cetai classes of aalytic fuctios Idia J Pue Appl Math 3: Goel R ad Sohi N 980 O cetai aalytic fuctios Idia J Pue Appl Math :308-3

22 Jaowsi W 973 Some exteal poblems fo cetai families of aalytic fuctios I A Polo Math 8: Noo K 99 O subclasses of close to covex fuctios of highe ode Ite J Math Math Sci 6: Noo K 996 O some subclasses of fuctios with bouded adius ad bouded bouday otatio PaAme Math J 6: Padmaabha K ad Paavatham R 975 Popeties of a class of fuctios with bouded bouday otatio A Polo Math 3: Pichu B 968 O stalie ad covex fuctios of ode Due Math J 35:7-3 9 Rusheweyh S ad Sheil-Small T 973 Hadamad poducts of Schlicht fuctios ad the polya-schoebeg cojectue Commet Math Helv 8:9-35