55 Characteristic Properties of the New Subclasses of Analytic Functions

Transcription

1 Doku Eylül Üiversitesi-Mühedislik Fakültesi Fe ve Mühedislik Dergisi Cilt 19 Sayı 55 No:1-Ocak/ Doku Eylul Uiversity-Faculty of Egieerig Joural of Sciece ad Egieerig Volume Volume 19 Issue 1955 Issue No:1-Jauary/ Characteristic Properties of the New Subclasses of Aalytic Fuctios Niami MUSTAFA DOI: 10105/deufmd Departmet of Mathematics Faculty of Sciece ad Letters Kafkas Uiversity Kars (Alıış / Received: Kabul / Accepted: Olie Yayılama / Published Olie: ) Keywords Aalytic fuctio Starlike fuctio Covex fuctio Abstract: I this study we itroduce ad ivestigate two ew subclasses of aalytic fuctios i the ope uit disk The object of the preset paper is to derive characteristic properties of the fuctios belogig to these classes Further several coefficiet iequalities for the fuctios belogig to these classes are also give Aalitik Foksiyoları Yei Alt Sııflarıı Karakteristik Öellikleri Aahtar kelimeler Aalitik foksiyo Yıldııl foksiyo Koveks foksiyo Öet: Bu çalışmada bi açık birim diskte aalitik foksiyoları iki yei altsııfıı taımladık ve araştırdık Mevcut çalışmaı amacı bu sııflara ait foksiyoları karakteristik öelliklerii elde etmektir Dahası bu sııflara ait ola foksiyolar içi çeşitli katsayı eşitsilikleri de verilmiştir Correspodig author: iamimustafa@gmailcom 47

2 N Mustafa / Characteristic Properties of the New Subclasses of Aalytic Fuctios 1 Itroductio ad Prelimiaries Let A be the class of aalytic fuctios f() i the ope uit disk U : 1 ormalied by f(0) 0 f(0) 1 of the form f ( ) a a a a (1) Further by S we will deote the family of all fuctios i A which are uivalet i U Let T deote the subclass of all fuctios f() with o-positive coefficiets i A of the form f ( ) a a a a 0 () May researchers have itroduced ad ivestigated several subclasses of aalytic fuctio class A (see for example [1-3]) Various subclasses of A were itroduced ad some geometric properties of these subclasses were ivestigated i several studies (see [4-7]) Recetly Prajapat [8] itroduced the subclasses R ( k ) V ( k ) T ( k ) a a a ad T ( k ) a of A ad several iclusio relatioships were established for these subclasses Also very soo Prajapat [9] itroduced a iterestig subclass t ( ) of aalytic ad close-tocovex fuctios i the ope uit disk U I [9] Prajapat derived several properties icludig coefficiets estimates distortio theorems coverig theorems ad radius of covexity for the fuctios belogig to the class t ( ) Soo after that Mustafa [10] itroduced ad ivestigated the subclass K( ) 01 which is the geeraliatio of the close-to-covex fuctios class amed close-to-covex with respect to a starlike fuctio g () of order ad type of aalytic fuctios i the ope uit disk U I [10] Mustafa foud sufficiet coditios for the parameters of the ormalied Wright fuctios to be i the class K( ) Very recetly Paigrahi ad Murugusudaramoorthy [11] have itroduced a ew subclass of the uivalet fuctios class S deoted by kt ad they have foud sharp M ( ) estimates for the differece of the coefficiets of the fuctios belogig to this class As it ca be see from the above metioed studies some of the importat ad well-ivestigated subclasses of S are the classes S ( ) ad C ( ) defied as follows Defiitio 11 (see also [1-14]) The class of starlike fuctios S ( ) of order 01 ad the class of covex fuctios C( ) of order 48

3 N Mustafa / Characteristic Properties of the New Subclasses of Aalytic Fuctios 01 are defied respectively by S ( ) f ( ) f A: Re f( ) U 01 ad C( ) f ( ) f A: Re 1 f( ) U 01 We will deote TS ( ) S ( ) T ad TC( ) C( ) T Iterestig geeraliatio of the fuctios classes S ( ) ad C( ) are deoted respectively by S ( ) ad C( ) ad defied by S ( ) f A: f ( ) Re f ( ) (1 ) f ( ) U ad C( ) f A: f ( ) f ( ) R e f ( ) f ( ) U 01 Moreover we will deote TS ( ) S ( ) T ad TC( ) C( ) T The classes TS ( ) ad TC( ) were extesively studied by Altıtaş ad Owa [15] ad certai coditios for hypergeometric fuctios ad geeralied Bessel fuctios for these classes were studied by Moustafa [16] ad by Porwal ad Dixit [17] Ispired by the above metioed studies we defie a uificatio of the fuctios classes S ( ) ad C( ) as follows Defiitio 1 A fuctio f A give by (1) is said to be i the class SC ( ; ) if the followig coditio is satisfied

4 N Mustafa / Characteristic Properties of the New Subclasses of Aalytic Fuctios f ( ) f ( ) f ( ) f ( ) Re (1 ) f ( ) (1 ) f ( ) U Further we will use TS C( ; ) S C( ; ) T I special case we have S C( ;0) S ( ); S C S C ( ;1) C( ); S C( 0;0) S ( ); TS C ( 0;1) C( ); TS C( ;0) TS ( ); ( ;1) TC( ); TS C( 0;0) TS ( ); TS C ( 0;1) TC( ) Suitably specialiig the parameters we ote that ( ; ) SC ad TS C( ; ) I this paper coefficiet bouds for the fuctios belogig i these classes are also determied Coeffıcıet Bouds for the Classes SC ( ; ) ad TS C( ; ) I this sectio we will examie some characteristic properties of the classes ( ; ) SC ad TS C( ; ) of aalytic fuctios i the ope uit disk U Here coefficiet bouds for the fuctios belogig to these classes are also give A sufficiet coditio for the fuctios i the class SC ( ; ) ) S C ( 0;0) S ( ) [18]; is give by the followig theorem ) S C( 0;1) C( ) [18]; 3) TS C( ;0) TS ( ) [19- ]; Theorem 1 Let f A The the fuctio f() belogs to the class SC ( ; ) ) TS C( 0;0) TS ( ) [18]; 5) TS C( ;1) TC( ) [15]; 6) TS C( 0;1) TC( ) [18] The object of the preset paper is to examie characteristic properties of the classes 50 if the followig coditio is satisfied 1 ( 1) a ( 1) The result is sharp for the fuctios 1 (3)

5 N Mustafa / Characteristic Properties of the New Subclasses of Aalytic Fuctios f ( ) (1 ) 1 ( 1) ( 1) U 3 Proof Let f S C ( ; ) 01 (4) 01 The accordig to Defiitio 1 we have f ( ) f ( ) f ( ) f ( ) Re (1 ) f ( ) (1 ) f ( ) U (5) f f f ( ) f ( ) ( ) ( ) (1 ) f ( ) (1 ) f ( ) 1 1 ( 1) a ( 1)(1 ) 1 ( 1) a 1 ( 1) 1 ( 1) a ( 1)(1 ) 1 ( 1) 1 a 1 ( 1) Also we ca easily show that the coditio (5) holds true if f f f ( ) f ( ) ( ) ( ) (1 ) f ( ) (1 ) f ( ) 1 1 (6) Therefore for the complete the proof of the theorem suffices to show that the coditio (6) is satisfied By simple computatio we obtai Last expressio of the above iequality is bouded by 1 if 1 ( 1) ( 1)(1 ) a (1 ) 1 ( 1) 1 a 1 ( 1) which is equivalet to (3) We ca easily see that the result of the theorem is sharp for the fuctios give by (4) Thus the proof of Theorem 1 is completed 51

6 N Mustafa / Characteristic Properties of the New Subclasses of Aalytic Fuctios By settig 0 ad 1 i Theorem 1 we ca readily deduce the followig results Corollary 1 The fuctio f() defied by (1) belogs to the class S ( ) 01 if the followig coditio is satisfied ( 1) a 1 The result is sharp for the fuctios 1 f( ) ( 1) U 3 Corollary The fuctio f() defied by (1) belogs to the class C( ) 01 if the followig coditio is satisfied ( 1) a 1 The result is sharp for the fuctios f ( ) 1 ( 1) U 3 By takig 0 i Corollary 1 ad respectively we have the followig results Corollary 3 (see [18 p 110 Theorem 1]) The fuctio f() defied by (1) belogs to the class S ( ) 01 if the followig coditio is satisfied a 1 The result is sharp for the fuctios 1 f( ) U 3 Corollary 4 (see [18 p 110 Corollary of Theorem 1]) The fuctio f() defied by (1) belogs to the class ( ) 01 C if the followig coditio is satisfied a 1 The result is sharp for the fuctios 1 f( ) U ( ) 3 Remark 1 Numerous cosequeces of the properties give by Corollary 3 ad 4 ca be obtaied for each of the classes studied by earlier researchers by specialiig the various parameters ivolved May of these cosequeces were proved by earlier researches o the subject (cf eg [18]) For the fuctio i the class ( ; ) Theorem 1 is also true TS C the coverse of Theorem Let f T The the fuctio f() belogs to the class 5

7 N Mustafa / Characteristic Properties of the New Subclasses of Aalytic Fuctios TS C( ; ) if ad oly if 1 ( 1) a ( 1) (7) 1 The result is sharp for the fuctios f ( ) (1 ) 1 ( 1) ( 1) U 3 (8) Proof The proof of the sufficiecy of the theorem ca be proved similarly to the proof of Theorem 1 We will prove oly the ecessity of the theorem Assume that f TS C( ; ) 01 That is 01 f ( ) f ( ) f ( ) f ( ) Re f ( ) (1 ) (1 ) f( ) U By simple computatio we obtai f ( ) f ( ) f ( ) f ( ) Re f ( ) (1 ) (1 ) f( ) 1 ( 1) a Re 1 ( 1) a 1 ( 1) The last expressio i the brackets of the above iequality is real if we choose as a real Hece from the previous iequality lettig 1 through real values we obtai 1 1 ( 1) a 1 ( 1) 1 a 1 ( 1) It follows that 1 ( 1) a ( 1) 1 which is the same as the coditio (7) Moreover it is clear that the equality i (7) is satisfied by the fuctios give by (8) Thus the proof of Theorem is completed 53

8 N Mustafa / Characteristic Properties of the New Subclasses of Aalytic Fuctios By takig 0 ad 1 i Theorem we ca readily deduce the followig results Corollary 5 The fuctio f() defied by () belogs to the class TS ( ) 01 if ad oly if ( 1) a 1 The result is sharp for the fuctios 1 f( ) ( 1) U 3 Corollary 6 The fuctio f() defied by () belogs to the class TC( ) 01 if ad oly if ( 1) a 1 The result is sharp for the fuctios f ( ) 1 ( 1) U 3 Remark The results obtaied by Corollary 5 ad Corollary 6 would reduce to kow results i [15] By takig 0 i Corollary 5 ad 6 respectively we have the followig results Corollary 7 (see [18 p 110 Theorem ]) The fuctio f() 54 defied by () belogs to the class TS ( ) 01 if ad oly if a 1 The result is sharp for the fuctios 1 f( ) U 3 Corollary 8 (see [18 p 111 Corollary ]) The fuctio f() defied by () belogs to the class TC( ) 01 if ad oly if a 1 The result is sharp for the fuctios 1 f( ) U 3 From Theorem we have the followig result Corollary 9 If f TS C( ; ) the 1 a 1 ( 1) ( 1) 3 Remark 3 Numerous cosequeces of Corollary 9 ca be deduced by specialiig the various parameters ivolved May of these cosequeces

9 N Mustafa / Characteristic Properties of the New Subclasses of Aalytic Fuctios were proved by earlier researchers o the subject (cf eg [18 19]) O the coefficiet bouds of the fuctios belogig i the class ( ; ) theorem TS C we give the followig Theorem 3 Let the fuctio f() defied by () belogs to the class TS C( ; ) The 1 (1 ) (1 ) a (9) ad a (1 ) (1 ) (1 ) Proof Usig Theorem we write (1 ) (1 ) 1 ( 1) a ( 1) 1 That is a 1 (1 ) (1 ) a Thus iequality (9) is provided Similarly we obtai (10) (1 ) (1 ) a 1 ( 1) a ( 1) 1 which is equivalet to (1 ) a 1 (1 ) (1 ) a Usig (9) i the last iequality we arrive at the followig (1 ) 1 (1 ) (1 ) a 1 (1 ) (1 ) (1 ) 1 This immediately yields the secod assertio (10) of Theorem 3 By settig 0 ad 1 i Theorem 3 we arrive at the followig results respectively Corollary 10 Let the fuctio f() defied by () belogs to the class TS ( ) 01 The 1 (1 ) a ad (1 ) (1 ) a 55

10 N Mustafa / Characteristic Properties of the New Subclasses of Aalytic Fuctios Corollary 11 Let the fuctio f() defied by () belogs to the class TC( ) 01 The 1 (1 ) a ad 1 (1 ) a Remark 4 Numerous cosequeces of the coefficiet iequalities (give by Corollary 10 ad Corollary 11) ca ideed be deduced by specialiig the various parameters ivolved For example by settig 0 we obtai the results for the classes TS ( ) ad TC( ) 01 respectively Moreover by settig 0 i Corollary 10 ad 11 we obtai iterestig results for the classes TS ad TC respectively Corollary 1 Let the fuctio f() defied by () belogs to the class TS The 1 a ad a 1 Corollary 13 Let the fuctio f() defied by () belogs to the class TC The 1 4 a ad 1 a 3 Cocludıg Remarks I this paper two ew subclasses SC ( ; ) ad ( ; ) TS C of aalytic fuctios i the ope uit disk are itroduced ad ivestigated I the preset paper the characteristic properties of the fuctios belogig to these classes are derived Further several coefficiet iequalities for fuctios belogig to these classes are obtaied Ackowledgmet The author is grateful to the aoymous referees for the valuable commets ad suggestios Refereces [1] Frasi B A 01 New subclasses of aalytic fuctios: Joural of Iequalities ad Applicatios Volume 4 pp 1-10 [] Cho N E ad Kim J A 006 Iclusio Properties of Certai Subclasses of Aalytic Fuctios Defied by a Multiplier Trasformatio: Computers ad Mathematics with Applicatios Volume 5 pp [3] Darus M ad Faisal I 01 Some subclasses of aalytic fuctios of complex order defied by ew differetial operator: Tamkag Joural of Mathematics Volume 43 pp 3-4 [4] Gao C Y Zhou S Q 005 O a class of aalytic fuctios related to the class to the starlike fuctios: Kyugpook Mathematical Joural Volume 45 pp [5] Kowalcyk J Les-Bomba E 010 O a subclass of close-to-covex fuctios: Applied Mathematics Letters Volume 3 pp

11 N Mustafa / Characteristic Properties of the New Subclasses of Aalytic Fuctios [6] Xu Q H Srivastava H M Li Z 011 A certai subclass of aalytic ad close-to-covex fuctios: Applied Mathematics Letters Volume 4 pp [7] Wag Z G Che D Z 009 O a certai subclass of close-to-coex fuctios: Hacettepe Joural of Mathematics ad Statistics Volume 38 pp [8] Prajapat J K 01 Iclusio properties for certai class of aalytic fuctios ivolvig multiplier trasformatio operator: Joural of Classical Aalysis Volume 1 pp 35-4 [9] Prajapat J K 016 A ew subclass of close-to-covex fuctios: Surveys i Mathematics ad Applicatios Volume 11 pp [10] Mustafa N 016 Close-tocovexity of ormalied Wright fuctios: Doku Eylul Uiversity- Faculty of Egieerig Joural of Sciece ad Egieerig Volume 18 pp [11] Paigrahi T Murugusudaramoorthy G 016 O Successive Coefficiet Estimate for Certai Subclass of Aalytic Fuctios: Applied Mathematics E- Notes Volume 16 pp [1] Dure P L 1983 Uivalet Fuctios Grudlehre der Mathematische Wisseshafte Bd 59 New York Spriger- Verlag 38p [13] Goodma A W 1983 Uivalet Fuctios Volume I Washigto Polygoal 46p [14] Srivastava H M ad Owa S (Ed) 199 Curret Topic i Aalytic Fuctio Theory World Scietific Publishig Compay New Jersey Lodo Hog Kog 456p [15] Altıtaş O ad Owa S 1988 O subclasses of uivalet fuctios with egative coefficiets: Pusa Kyogam Mathematical Joural Volume 4 pp [16] Moustafa A O 009 A study o starlike ad covex properties for hypergeometric fuctios: Joural of Iequalities i Pure ad Applied Mathematics Volume 10 pp 1-16 [17] Porwal S ad Dixit K K 013 A applicatio of geeralied Bessel fuctios o certai aalytic fuctios: Acta Uiversitatis Matthiae Belii series Mathematics pp [18] Siverma H 1975 Uivalet Fuctios with Negative Coefficiets: America Mathematical Society Volume 51 pp [19] Altıtaş O 1991 O a subclass of certai starlike fuctios with egative coefficiet: Mathematica Japoica Volume 36 pp [0] Altıtaş O Irmak H ad Srivastava H M 1995 Fractioal calculus ad certai starlike fuctios with egative coefficiets: Computers ad Mathematics with Applicatios Volume 30 pp 9-16 [1] Altıtaş O Öka Ö ad Srivistava H M 004 Neighbourhoods of a Certai Family of Multivalet Fuctios with Negative Coefficiets: Computers ad Mathematics with Applicatios Volume 47 pp [] Porwal S 014 A applicatio of a Poisso distributio series o certai aalytic fuctios: Joural of Complex Aalysis Article ID pp