On Certain Classes of Analytic and Univalent Functions Based on Al-Oboudi Operator

Transcription

1 Boig Itetiol Joul o t Miig, Vol, No, Jue 0 6 O Ceti Clsses o Alytic d Uivlet Fuctios Bsed o Al-Oboudi Opeto TV Sudhs d SP Viylkshmi Abstct--- Followig the woks o [, 4, 7, 9] o lytic d uivlet uctios i this ppe we itoduce two ew clsses (α,, ξ, γ, δ) d T V (α,, ξ, γ, δ), <,0, 0, / ξ, 0 α / ξ, / < γ, 0o mily o lytic uctios o the om N deied s: 0 () () () () () () [ ()] (), 0, o,3,, We hve obtied coeiciet estimtes, gowth & distotio theoems, exteml popeties o these two clsses The detemitio o exteme poits o mily o uivlet uctios leds to solve my exteml poits Keywods--- Al-Oboudi Opeto, Covex Fuctios, Stlike Fuctios, Uivlet Fuctios L I INTROUCTION E deote the clss o uctios o the om + () () tht e lytic d uivlet i the disk < Fo 0 α < let S * ( d K( deote the submilies o S cosistig o uctios stlike o ode α d covex o ode α espectively The submily T o S cosists o uctios o the om (), 0, o,3,, () Silvem [6] ivestigted uctio i the clsses T * ( T S * ( d C( T K( Let N d 0 eote by the Al-Oboudi opeto [3] deied by : A A, 0 () () () ( )() + () () () [ Note tht o () is give by (), ()] () + [+ ( )), whe, is the Sălăge dieetil opeto : A A, TV Sudhs, eptmet o Mthemtics, SIVET College, Chei , Idi E-mil: tvsudhs@edimilcom SP Viylkshmi, eptmet o Mthemtics, SIVET College, Chei , Idi eiitio : [8] Let, R, 0, 0 d + () we deote by the lie opeto deied by : A A () + (+ ( )) Remk : I T, (), 0,, 3,, the () (+ ( )) I this ppe usig the opeto we itoduce the clsses (α,, ξ, γ, δ) d T V (α,, ξ, γ, δ) d obti coeiciet estimtes o these clsses whe the uctios hve egtive coeiciets We lso obti gowth d distotio theoems, closue theoem o uctios i these clsses eiitio : We sy tht uctio () T is i the clss (α,, ξ, γ, δ) i d oly i + () () + + () () ξ α γ () () whee <, 0, / ξ, 0, 0 α / ξ, / < γ, 0 eiitio 3: A uctio () T is sid to belog to the clss (α,, ξ, γ, δ) i d oly i ξ () () () α γ () + + () () ISSN Boig

2 Boig Itetiol Joul o t Miig, Vol, No, Jue 0 7 whee <, 0, / ξ, 0, 0 α α / ξ, / < γ, 0 I we eplce 0, we obti the coespodig esults o SM Khi d Mee Moe [4] I we eplce 0, d γ we obti the esults o Aghly d Kulki [] d Silvem d Silvi [7] I we eplce 0, d ξ by, we obti the coespodig esults o [9] II MAIN RESULTS COEFFICIENT ESTIMATES Theoem : A uctio () ( 0) is i (α,, ξ, γ,δ) i d oly i Poo Suppose, ξδ( (+ ()) [(){ξδ + γδ} + δξ( ] ξ + () () + + () () ξ α γ () () (+ ( )) (+ ( )) (+ ( )) γ (+ ( )) + + α + (+ ( )) (+ ( )) we hve + (+ ( )) [(){ξδ + γδ} + δξ( ] () () δ ξ( + With the povisio, δ ξ γ (+ ()) (+ ()) (+ ()) ξδ( () α ()) γ( α + α + Fo < it is bouded bove by δ (+ ( )) ( ) δ ξ( () + () ()) < 0 (+ ()) δξ( (+ ()) (+ ()) < 0 (+ ( )) {ξα ξ ξ( ) + γ( )} p (α,, ξ, γ, δ) Now we pove the covese esult Let (+ ()) () ξ( + (+ ()) {ξ(α ) ξ()+ γ()} As Re () o ll, we hve (+ ()) () Re ξ( + (+ ()) {ξ( ξ()+ γ()} We choose vlues o o el xis such tht + is el d cleig the deomito o bove expessio d llowig though el vlues, we obti (+ ( )) {( )(ξδ + γδ) + δξ( } [ ξδ( (+ ( )) {( )(ξδ + γδ) + δξ( } ξδ( ] 0 Remk : I () (α,, ξ, γ, δ) the ISSN Boig

3 Boig Itetiol Joul o t Miig, Vol, No, Jue 0 8 ξδ(, (+ ( )) {( )(ξδ + γδ) + δξ( },3,4, d equlity holds o ξδ( () (+ ( )) {()(ξδ + γδ) + δξ( } Coolly : I () p (α,0, ξ, γ, δ), tht is, eplcig 0, we get ξδ(,,3,4, ( ) δ (ξα ξ + γ γ) d equlity holds o ξδ( () ( ) δ (ξα ξ + γ γ) This coolly is due to [4] Coolly : I () 0,, γ we get p (α,0, ξ,, δ), tht is, eplcig ξδ(,,3,4, ( ) δ (ξα ξ + ) d equlity holds o ξδ( () ( ) δ (ξα ξ + ) This coolly is due to [] d [7] Coolly 3: I () p (α,0,,, δ) we get δ(,,3,4, ( ) δ (α ) d equlity holds o δ( () ( ) δ (α ) This coolly is due to [9] Coolly 4: () p (α,0,,,) i d oly i ( ( Theoem : A uctio () ( 0) is i (α,, ξ, γ,δ) i d oly i [+ ( )] + {( ){δξ + γδ} + δξ( } δξ( Poo: The poo o this theoem is logous to tht o Theoem [], becuse uctio () T V (α,, ξ, γ, δ) i d oly i () (α,, ξ, γ, δ) So it is eough tht i Theoem is eplced with + Remk : I () T V (α,, ξ, γ, δ) the + () (+ ( )) d equlity holds o + (+ ()) ξδ(, {( )(ξδ + γδ) + δξ( },3,4, ξδ( {()(ξδ + γδ) + δξ( } Coolly 5: I () T V (α,, ξ, γ, δ), tht is, eplcig 0, we get ξδ(,,3,4, {( ) δ(ξα ξ + γ γ)} d equlity holds o ξδ( () {( ) δ(ξα ξ + γ γ)} This coolly is due to [4] Coolly 6: I () T V (α, 0, ξ,, δ), tht is, eplcig 0,, γ we get ξδ(,,3,4, {( ) δ(ξα ξ + )} d equlity holds o ξδ( () {( ) δ(ξα ξ + )} This coolly is due to [] d [7] Coolly 7: I () T V (α, 0,,, δ), the δ(,,3,4, {( ) δ(α )} d equlity holds o δ( () {( ) δ(α )} Coolly 8: () T V (α, 0,,, ), i d oly i ( ( ISSN Boig

4 Boig Itetiol Joul o t Miig, Vol, No, Jue 0 9 III GROWTH AN ISTORTION THEOREM Theoem 3: I () (α,, ξ, γ, δ) the (+ ) + (+ ) equlity holds o ξδ( { + ξδ γδ + ξδ( } ξ δ( () + 4ξδ δ(γ + ξ () ξδ( { + ξδ γδ + ξδ( } t ± Poo: By Theoem, we hve () (α,, ξ, γ, δ) i d oly i (+ ()) {( ){ξδ + γ} + ξδ( } ξδ( ξδ( Let t + ξδ γδ () (α,, ξ, γ, δ) i d oly i Whe (+ ( )) ( t) t (3) (+ ) ( t) (+ ( )) ( t) t This lst iequlity ollows om (3) we obti t () (+ ) ( t) Similly t () (+ ) ( t) So, t () + ( + ) ( t) tht is, (+ ) + (+ ) Hece the esult t ( + ) ( t) ξδ( () { + ξδ γδ + ξδ( } ξδ( { + ξδ γδ + ξδ( } Coolly 3: I p (α,0, ξ, γ, δ), tht is, eplcig d 0, the p tht is, ξδ( () + ξδ γδ + ξδ + ξδα + d equlity o + ξδ( + ξδ γδ + ξδ + ξδα ξδ( () + 4ξδ δ( γ + ξα ) ξδ( + 4ξδ δ( γ + ξα ) ξ δ( () (+ 4ξδ) δ(γ + ξ This coolly is due to [4] Coolly 3: I (), 0 d γ, the t ± p (α,0, ξ,, δ), tht is, eplcig ξδ( () + 4ξδ δ(+ ξ ξδ( + + 4ξδ δ(+ ξ with equlity o, ξ δ( () + 4ξδ δ(+ ξ t ± This coolly is due to [] d [7] Coolly 33: I () p (α,0,,, δ), the δ( () + 4δ δ(+ δ( + + 4δ δ(+ with equlity o, δ( () (+ 4δ) δ(+ t ± Theoem 3: I () T V (α,, ξ, γ, δ) the ξδ( () + ( + ) {( + ξδ γδ) + ξδ( } + ( + ) + ξδ( {( + ξδ γδ) + ξδ( } Poo: The poo o this theoem is logous to tht o Theoem 3, becuse uctio () T V (α,, χ, γ, δ) i d oly i () (α,, ξ, γ, δ) So it is eough tht i Theoem is eplced with + ISSN Boig

5 Boig Itetiol Joul o t Miig, Vol, No, Jue 0 0 Coolly 34: I () T V (α, 0, ξ, γ, δ) the ξδ( () (+ 4ξδ δ(γ + ξ ξδ( + (+ 4ξδ δ(γ + ξ with equlity o ξδ( () (+ 4ξδ) δ(γ + ξ t ± This coolly is due to [4] Coolly 35: I () T V (α, 0, ξ,, δ) the ξδ( () (+ 4ξδ δ(+ ξ ξδ( + (+ 4ξδ δ(+ ξ with equlity o ξδ( () (+ 4ξδ) δ(+ ξ t ± This coolly is due to [] d [7] Coolly 36: I () T V (α, 0,,, δ) the δ( () + 3δ δα δ( + + 3δ δα with equlity o δ( () + 3δ δα t ± This coolly is due to [9] Theoem 33: I () p (α,, ξ, γ, δ) the 4ξδ( () (+ ) {( γδ) + ξδ( + } 4ξδ( + (+ ) {( γδ) + ξδ( + } Poo Sice p (α,, ξ, γ, δ) we hve (+ ( )) ( t) t ξδ( whee t + ξδ γδ I view o Theoem 3, we hve ( t) (+ ) ( t) ( t) + t (4) + + Similly So, ( t) + ( + ) ( t) ( t) ( + ) ( t) ( t) ( t) () + ( + ) ( t) ( + ) ( t) Substitutig t, we hve 4ξδ( () (+ ) {( γδ) + ξδ( + } 4ξδ( + (+ ) {( γδ) + ξδ( + } Coolly 37: I (α,0, ξ, γ, δ) ) the 4ξδ( () ( γδ) + ξδ( 4ξδ( + ( γδ) + ξδ( o This coolly is due to [4] Coolly 38: I p (α,0, ξ,, δ) the 4ξδ( () ( δ) + ξδ( 4ξδ( + ( δ) + ξδ( This coolly is due to [] d [7] Coolly 39: I p (α,0,,, δ) the 4δ( () ( δ) + δ( 4δ( + ( δ) + δ( tht is, 4δ( () + 3δ δα 4δ( + + 3δ δα This coolly is due to [9] Theoem 34: I T V (α,, ξ, γ, δ) the p ISSN Boig

6 Boig Itetiol Joul o t Miig, Vol, No, Jue 0 o (+ ) + + (+ ) 4ξδ( () {( γδ) + ξδ( + } + 4ξδ( {( γδ) + ξδ( + } Poo: The poo o this theoem is simil to tht o Theoem 33 becuse uctio () T V (α,, ξ, γ, δ) i d oly i () (α,, ξ, γ, δ) So it is eough tht i Theoem 33 is eplced with + Coolly 30: I T V (α, 0, ξ, γ, δ) the ξδ( () ( γδ) + ξδ( ξδ( + ( γδ) + ξδ( o This coolly is due to [4] Coolly 3: I T V (α, 0, ξ,, δ) the ξδ( () ( δ) + ξδ( ξδ( + ( δ) + ξδ( o This coolly is due to [] d [7] Coolly 3: I T V (α, 0,,, δ) the δ( () ( δ) + δ( δ( + ( δ) + δ( δ( () + 3δ αδ δ( + + 3δ αδ o This coolly is due to [9] IV Theoem 4: Let () d CLOSURE THEOREM ξδ( () [+ ( )] [( ){ξδ + γδ} + δξ( ] o, 3, 4, The () T V (α,, ξ, γ, δ) i d oly i () c be expessed i the oms () () whee 0 d Poo: Let () (), 0,,, with () we hve () + () () ξδ( [+ ()] [(){ξδ + γδ} + δξ( ] The ξδ( [ + ( )] [( ){ξδ + γδ} + δξ( ] [+ ( )] [( ){ξδ + γδ} + δξ( ] ξδ( () T V (α,, ξ, γ, δ) Covesely, suppose () T V (α,, ξ, γ, δ) the emk o Theoem gives us ξδ( [+ ( )] [( ){ξδ + γδ} + δξ( ] we tke [+ ()] [(){ξδ + γδ} + δξ( ], ξδ(,3,4, d The () () Coolly 4: I () d ξδ( (), ( ) δ(ξα ξ + γ γ) o,3, The () T V (α, 0, ξ, γ, δ) i d oly i () c be expessed i the om () () whee 0,,,, This coolly is due to [4] ISSN Boig

7 Boig Itetiol Joul o t Miig, Vol, No, Jue 0 Coolly 4: I () d ξδ( (), ( ) δ(ξα ξ + ) o,3, The () T V (α, 0, ξ,, δ) i d oly i () c be expessed i the om () () whee 0, [6] H Silvem, Uivlet uctios with egtive coeiciets [7] H Silvem d E Silvi, Subclsses o pestlike uctios, Mth Jpo, Vol 9, No 6, Pp 99935, 984 [8] TV Sudhs, R Thiumlismy, KG Submi, Mugu Acu, A clss o lytic uctios bsed o extesio o Al-Oboudi opeto, Act Uivesittis Apulesis, Vol, Pp 7988, 00 [9] S Ow d J Nishiwki, Coeiciet Estimtes o ceti clsses o lytic uctios, JIPAM, J Iequl Pue Appl Mth, Vol 35, Aticle 7, 5pp (electoic), 00,,, This coolly is due to [] d [7] Coolly 43: I () d δ( (), ( ) δ(α ) o,3, The () T V (α, 0,,, δ) i d oly i () c be expessed i the om () () whee 0,,,, This coolly is due to [9] Coolly 44: I () d () The () T V (0, 0,,, ) i d oly i () c be expessed i the om () () whee 0,,,, V CONCLUSION I this ppe mkig use o Al-oboudi opeto two ew subclsses o lytic d uivlet uctios e itoduced o the uctios with egtive coeiciets My subclsses which e ledy studied by vious eseches e obtied s specil cses o ou two ew subclsses We hve obtied vious popeties such s coeiciet estimtes, gowth distotio theoems Futhe ew subclsses my be possible om the two clsses itoduced i this ppe REFERENCES [] M Acu, S Ow, Note o clss o stlike uctios, Poceedig o the Itetiol shot wok o study o clculus opetos i uivlet uctio theoy, Kyoto, Pp 0, 006 [] R Aghly d S Kulki, Some theoems o uivlet uctios, J Idi Acd Mth, Vol 4, No, Pp 893, 00 [3] FM Al-Oboudi, O uivlet uctios deied by geelied Sălăge opeto, Id J Mth Sci, No 5-8, 49436, 004 [4] SM Khi d Mee Moe, Ceti mily o lytic d uivlet uctios, Act Mthemtic Acdemie Pedogicl, Vol 4, Pp , 008 [5] SR Kulki, Some poblems coected with uivlet uctios, Ph Thesis, Shii Uivesity, Kolhpu, 98 ISSN Boig