Some Geometric Properties of a Class of Univalent. Functions with Negative Coefficients Defined by. Hadamard Product with Fractional Calculus I

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "Some Geometric Properties of a Class of Univalent. Functions with Negative Coefficients Defined by. Hadamard Product with Fractional Calculus I"

Μενέλαος Βικελίδης
5 χρόνια πριν
Προβολές:

1 Itrtol Mthtcl Foru Vol 6 0 o So otrc Proprts o Clss o Uvlt Fuctos wth Ntv Cocts Dd y Hdrd Product wth Frctol Clculus I Huss Jr Adul Huss Dprtt o Mthtcs d Coputr pplctos Coll o Sccs Uvrsty o Al-Muth Sw Irq Huss_lly77@yhooco Rd H But Dprtt o Mthtcs d Coputr pplctos Coll o Sccs Uvrsty o Al-Muth Sw Irq Rdh@yhooco Astrct I ths ppr w study suclss o uctos whch r uvlt d lytc uctos th ut ds U W ot coct stts dstorto ouds d so rsults Mthtcs Sujct Clsscto: 30C45 Kywords: Uvlt Fucto Hdrd Product Dstorto Bouds Itroducto Lt Ω dot th clss o th uctos dd y

2 380 H J A Huss d R H But U W dd suclss K o Ω cosst o uctos dd y Whch r uvlt d lytc th ut ds { C : < } 0 A ucto lo to th clss H stss R whr 0 < 0 0 U0 < < d Th Hdrd product or covoluto dd y xt dto Dto : I dd y d dd y 4 Th th Hdrd product or covoluto dd y th or : d U 0 5 Dto : [3] Frctol drvtv o ordr o lytc ucto s dd y d t D dt Γ d t 0 < < 6 0 whr s lytc ucto sply coctd ro o th pl cot th or d th ultplcty o t s rovd y rqur lo t to rl wh t s rtr th ro Clrly l D 0 d l D Dto 3: [6] Frctol trl o ordr o lytc ucto s dd y

3 otrc proprts o clss o uvlt uctos 38 D t dt Γ 7 t 0 whr s lytc ucto sply coctd ro o th pl cot th or d th ultplcty o t s rovd y rqur lo t to rl wh t > 0 Dto 4: [6] [Udr th codto o Dto 3] th rctol drvtv o ordr 0 s dd y d D D 8 d For th lytc ucto U w put M Γ D Ad! Γ Γ Γ D Th ro 0 w t whr Γ! Γ > < < 0! Γ Γ L : [] Lt w u v Th R w σ d oly w σ w σ L 3: [] Lt w u v d σ γ r rl urs Th R w > σ γ φ R w σ σ φ > γ So othr clss studd y W Atsh d S R Kulr [] SPousy [4]H M Srvstv [5] w d oly { }

4 38 H J A Huss d R H But Coct Estts I th xt thor w t th suct codto or th ucto th clss H Thor : Th ucto dd y s ' th clss H d oly [ ] 3 whr 0 < 0 00 < < Proo: By Dto 3 w t R Th y L w hv γ R π < γ π or quvltly γ R γ γ [ ] 4 γ A γ Lt [ ] [ ] γ d B By L 4 s quvlt to A B A B or 0 < But A B γ

5 otrc proprts o clss o uvlt uctos 383 γ γ Also B A γ γ γ d so B A B A [ ] 0 or

6 384 H J A Huss d R H But Ths s quvlt to Covrsly suppos tht holds Th w ust show R γ 0 γ Upo choos th vlus o o th postv rl xs whr r 0 < th ov qulty rducs to 0 R r r φ γ Sc R γ γ th ov qulty rducs to 0 R r r Ltt r w t dsrd cocluso Corollry : Lt H Th 3 Dstorto Thor I th xt thor w ot th dstorto thor or H Thor : I H Th

7 otrc proprts o clss o uvlt uctos [ ] [ ] 3 Proo : Sc ro 3 w t 3 [ ] 5 hc 3 [ ] Slrty w t 3 [ ] I th xt thor w shll prov tht th clss H s closd udr rthtc d covx lr cotos Now w dd th ucto y th or 0 IN 6 Thor 3: Lt th ucto dd y 6 th clss H or Th th ucto Φ c c 0 IN 7 Also th clss H whr c Proo : Lt th ucto H th ro thor w t Hc [ ] [ ] c

8 386 H J A Huss d R H But [ ] Hc Φ H Thor 4: Th clss H s closd udr lr cotos Proo : Lt th ucto dd y 6 th clss H Now w show tht th xt ucto E l l 0 l s lso th clss H Sc H th ro 3 w t [ ] Ad so H th ro 3 w t Th [ ] [ l ] l Thror y Thor w t E [ ] [ l ] Hc y Thor w hv E H l Thor 5: Lt th ucto dd y 6 th clss H whr 0 00 < < 0 For ch th th ucto S s lso th clss H whr { } { } d { }

9 otrc proprts o clss o uvlt uctos 387 Proo : Lt th uctos H th ro Thor w t hc [ ] [ ] Thror S H REFERENCES [] E S Aql So Prols Coctd wth otrc Fucto Thory Ph D Thss 004 Pu Uvrsty Pu [] W Atsh d S R Kulr So pplctos o rld Ruschwyh drvtvs volv rl rctol drvtv oprtor to clss o lytc uctos wth tv cocts I J Survys Mthtcs d ts Applctos [3] Ow S: O dstorto thors I Kyupoo Mth J [4] S Pousy Covoluto proprts o so clsss o roorphc uvlt uctos Proc Id Acd Sc Mth Sc [5] H M Srvstv Dstorto Iqults or lytc d uvlt uctos ssoctd wth crt rctol clculus d othr lr oprtor I Alytc d otrc Iqults d Applctos ds TM Rsss d H M Srvstv Kluwr Acdc Pulshrs [6] H M Srvstv d S Ow Eds Currt Topcs Alytc Fucto Thory World Sctc Pulsh Copy Spor 99

10 388 H J A Huss d R H But Rcvd: My 0

Σχετικά έγγραφα

PARTIAL SUMS OF CERTAIN CLASSES OF MEROMORPHIC FUNCTIONS

PARTIAL SUMS OF CERTAIN CLASSES OF MEROMORPHIC FUNCTIONS 5 Proc Pist Acd Sci Nilh 43: A 5-6 Al-ih 006 PATIAL SUMS OF CETAIN CLASSES OF MEOMOPHIC FUNCTIONS Nilh A Al-ih Girls Collg o Eductio iydh Sudi Arbi civd Jury 006 cctd Fbrury 006 Couictd by Pro r M Iqbl