Dominant of Functions Satisfying a Differential Subordination and Applications

Transcription

1 Domiat of Fuctios Satisfyig a Differetial Subordiatio ad Applicatios R Chadrashekar a, Rosiha M Ali b ad K G Subramaia c a Departmet of Techology Maagemet, Faculty of Techology Maagemet ad Busiess, Uiversiti Tu Hussei O Malaysia, 864 Parit Raja, Batu Pahat, Johor, Malaysia b School of Mathematical Scieces, Uiversiti Sais Malaysia, 118 USM, Peag, Malaysia c School of Computer Scieces, Uiversiti Sais Malaysia, 118 USM, Peag, Malaysia Abstract Best domiat is obtaied for ormalied aalytic fuctios f satisfyig (1 ) f ( ) f( ) f( ) h( ) i the uit disk, h is a ormalied covex fuctio, ad, are appropriate real parameters This fudametal result is ext applied to ivestigate the covexity ad starlikeess of the image domais f ( ) for particular choices of h Keywords: Starlike ad covex fuctios, differetial subordiatio, domiat PACS: 3-f INTRODUCTION Let be the class of aalytic fuctios f defied i the ope uit disk : { : 1} For a, a positive iteger, ad, let k ( a) f : f( ) a a k ad k k f : f ( ) a, k k1 with The subclass of cosistig of starlike fuctios i satisfyig 1 f ( ) Re,, f( ) is deoted by, ad is the subclass of cosistig of covex fuctios i satisfyig f ( ) Re1, f ( ) For two aalytic fuctios f ad g, the fuctio f is subordiate to g, writte f ( ) g ( ) if there is a aalytic self-map w of with w() satisfyig f ( ) g ( w ( )) If g is uivalet, the f subordiate to g is equivalet to f () g () ad f ( ) g ( ) This paper cosiders a class of fuctios satisfyig a secod-order differetial subordiatio to a give covex fuctio Best domiat amogst the solutios to this differetial subordiatio is determied Further, sufficiet coditios are obtaied that esure these solutios are either starlike or covex fuctios i Such coditios i terms of differetial iequalities have bee ivestigated i several works, otably by [1,, 3, 4, 5, 6, 7] I particular, Kaas ad Owa [8] studied coectios betwee certai secod-order differetial subordiatio ivolvig expressios of the form f (), f () ad 1 f ( ) f ( ) The class studied i this paper presets a more geeral framework Proceedigs of the 1st Natioal Symposium o Mathematical Scieces (SKSM1) AIP Cof Proc 165, (14); doi: 1163/ AIP Publishig LLC /$3 58 This article is copyrighted as idicated i the article Reuse of AIP cotet is subject to the terms at: Dowloaded to IP: O: Mo, 1 Jul 14 8:1:49

2 The followig lemma will be eeded Lemma 1 [9, Theorem 1, p 19] Let h be covex i with h() a, ad Re If p ( a) ad p( ) p ( ) h ( ), the p( ) q( ) h( ), ( / ) 1 q ( ) htt ( ) dt / The fuctio q is covex ad is the best ( a, ) domiat SECOND ORDER DIFFERENTIAL SUBORDINATION I the followig sequel, we shall assume that h is a aalytic covex fuctio i with h() 1 For ad cosider the class of fuctios f satisfyig the secod-order differetial subordiatio Let ad satisfy ( ) (1 ) f f ( ) f( ) h( ) ad Note that Re ad Re The followig result gives the best domiat to solutios of the differetial subordiatio (1) Theorem 1 Let ad be give by (), ad, be real umbers such that ad If f satisfies ( ) (1 ) f f ( ) f( ) h( ), the f ( ) (1/ ) 1 (1/ ) 1 q( ): h( rs) r s drds, ad q is the best ( a, ) domiat (1) () Proof Let Evidetly f( ) 1 p ( ) 1 a a 1 f ( ) (1 ) f ( ) f( ) p( ) ( ) p( ) p( ), ad (1) ca be expressed as p( ) ( ) p( ) p ( ) h ( ) (3) 581 This article is copyrighted as idicated i the article Reuse of AIP cotet is subject to the terms at: Dowloaded to IP: O: Mo, 1 Jul 14 8:1:49

3 Writig F( ) p( ) p( ), it follows that F( ) F( ) p( ) ( ) p( ) p( ) h( ), ad are give by () Lemma 1 ow yields 1 (1/ ) 1 F( ) h( t) t dt, 1/ ad thus 1 1 (1/ ) 1 p( ) p( ) h( r) r dr A secod applicatio of Lemma 1 shows that (1/ ) 1 (1/ ) 1 p ( ) hrtr ( ) dr t dt, 1/ which i view of () implies that f ( ) (1/ ) 1 (1/ ) 1 q( ): h( rs) r s drds Sice q ( ) ( ) q( ) ( 1) q( ) q( ) h ( ), the fuctio Q ( ) q ( ) is a solutio of the differetial subordiatio f( ) (1 ) f ( ) f( ) h( ) This shows that q q for all ( a, ) domiats q, ad hece q is the best ( a, ) domiat The followig result is a immediate cosequece of Theorem 1 Corollary 1 Uder the assumptios of Theorem 1, if f( ) (1 ) f( ) f( ) 1+ M, (4) the f ( ) M 1, (5) 1 ( 1) ad the superordiate fuctio is the best domiat A applicatio of Corollary 1 gives the followig sufficiet coditio for starlikeess Theorem Let ad be real umbers with 1 ad Further let f ad M M(,, ), If f satisfies the differetial subordiatio the f ( )[1 ( 1)] M(,, ) ( 1) [ ( 1) ] f( ) (1 ) f( ) f( ) 1+ M, (6) 58 This article is copyrighted as idicated i the article Reuse of AIP cotet is subject to the terms at: Dowloaded to IP: O: Mo, 1 Jul 14 8:1:49

4 Proof Let A brief computatio shows that f ( ) w( ) ( ) ( ) Re f w Re 1 f ( ) w ( ) (7) (8) I view of the aalytical coditio for starlikeess, that is, f ( )/ f ( ) Re i, it remais to show that w( ) w( ) 1 (9) Usig (7), (4) ca be rewritte as Itegratig (1), evidetly w( ) ( ) w ( ) w ( ) 1 M (1) 1 (1 ) w ( ) w( ) w( ) M ( sds ), (11) is a aalytic self-map of with () It follows from (7) ad (11) that w( ) ( 1) w( ) 1 M ( s) ds w( ) w ( ) w( ) For (9) to hold true, it is sufficiet to prove ( 1) ( ) w 1 M ( s) ds 1 w ( ) w( ) (1) Now the subordiatio (5), implies 1 1 ( 1), w( ) 1 ( 1) M (13) M 1 ( 1), while w ( ) M 1 [1 ( 1)] (14) Sice ( ), a brief computatio shows that 1 M 1 M ( s) ds 1 (15) 583 This article is copyrighted as idicated i the article Reuse of AIP cotet is subject to the terms at: Dowloaded to IP: O: Mo, 1 Jul 14 8:1:49

5 Takig ito accout the iequalities (13), (14) ad (15), the coditio (1) is fulfilled wheever M M(,, ) with M (,, ) give by (6) This completes the proof The followig theorem which gives sufficiet coditios for covexity is also a cosequece of Corollary 1 Theorem 3 Let ad be real umbers with 1 M M(,, ), ad Further let f ad [1 ( 1)][( ) ] M(,, ) [ ( 1) ( 1)] ( )[ ( 1) ( 1)] (16) If f satisfies the differetial subordiatio the f f( ) (1 ) f( ) f( ) 1+ M, Proof I view of the fact f ( ) f ( ) Re1 1, f ( ) f( ) it is sufficiet to prove the iequality f ( ) f( ) 1 (17) Let f ( ) f ( ) 1 1 f( ) f ( ) (18) Proceedig similarly as i the proof of Theorem, with as a aalytic self-map of the uit disk, it follows from (18) ad (4) that f ( ) f( ) (1 ) f ( ) f( ) M( ) f ( ) (19) Subsequetly, f ( ) 1 1 ( 1) 1 ( ) f M, f( ) f( ) f( ) which leads to the coditio f 1 M( ) 1 f( ) f( ) () for (17) to hold true 584 This article is copyrighted as idicated i the article Reuse of AIP cotet is subject to the terms at: Dowloaded to IP: O: Mo, 1 Jul 14 8:1:49

6 Applyig (7) ad (11), as well as the iequalities (13), (14) ad (15), yield 1 [1 ( 1)] f( ) [1 ( 1)]( ) M[ ( 1) ( ( 1) )] (1) ad f ( ) [1 ( 1) M], f( ) [1 ( 1)]( ) M[ ( 1) ( ( 1) )] () ( )[1 ( 1)] M ( 1) [ ( 1) ] I view of (1), () ad the fact that 1 M( ) 1 M, () is fulfilled for M M(,, ), M (,, ) is give by (16) This completes the proof ACKNOWLEDGMENTS The work preseted here was supported i part by a research uiversity grat from Uiversiti Sais Malaysia REFERENCES 1 R M Ali, Rocky Moutai J Math 4, No, (1994) R M Ali ad V Sigh, Complex Variables Theory Appl 6, No 4, (1995) 3 R M Ali, S Pousamy ad V Sigh, A Polo Math 61, No, (1995) 4 R M Ali, S K Lee, K G Subramaia ad ASwamiatha, Abstr Appl Aal Art ID 9135, 1 pp (11) 5 R Fourier ad PT Mocau, Complex Variables Theory Appl 48, 83-9 (3) 6 Y C Kim ad H M Srivastava, It J Math Math Sci, No 3, (1999) 7 M Obradovic, Mat Vesik 49, (1997) 8 S Kaas ad S Owa, Surikaisekikekyusho Kokyuroku No 16, 5-33 (1998) 9 D J Hallebeck ad S Ruscheweyh, Proc Amer Math Soc 5, (1975) 585 This article is copyrighted as idicated i the article Reuse of AIP cotet is subject to the terms at: Dowloaded to IP: O: Mo, 1 Jul 14 8:1:49