MAJORIZATION PROBLEMS FOR SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING

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1 Iteratioal Joural of Civil Egieerig ad Techology (IJCIET) Volume 9, Issue, November 08, pp. 97 0, Article ID: IJCIET_09 6 Available olie at aeme.com/ijciet/issues.asp?jtypeijciet&vtype 9&IType ISSN Prit: ad ISSN Olie: IAEME Publicatio Scopus Idexed MAJORIZATION PROBLEMS FOR SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING Q-CALCULUS OPERATOR K.Thilagavathi, K.Vijaya ad K.Uma Departmet of Mathematics, School of Advaced Scieces, VITT Uiversity, Vellore , Idia ABSTRACT I the preset paper, we ivestigate majorizatio problems for certai classes of aalytic fuctios ivolvig Ruscheweyh - differetial operator. AMS Subject Classificatio: [00]Primary 30C45; Secodary 30C50. Key words: Aalytic fuctios, starlike fuctios, subordiatio, majorizatio, - calculus operator. Cite this Article: K.Thilagavathi, K.Vijaya ad K.Uma, Majorizatio Problems For Subclasses of Aalytic Fuctios Ivolvig Q-Calculus Operator, Iteratioal Joural of Civil Egieerig ad Techology (IJCIET) 9(), 08, pp IET/issues.asp?JTypeIJCIET&VType9&ITy ype. INTRODUCTION Let A deote the class of fuctios of the form f ( z + a z which are aalytic ad uivalet i the the ope uit disk () U { z : z C ad z < }. We ow give some basic defiitios ad related details of -calculus which are relevat for our study. The -shifted factorial is defied for α, C as a product of factors by ( α; ) ( α )( α ) L( α ad i terms of the basic aalogue of the gamma fuctio ) ; 0 ; N, () 97 editor@iaeme.com

2 K.Thilagavathi, K.Vijaya ad K.Uma Γ ( α + )( ) α ( ; ) ( > 0), Γ ( α) (3) the -gamma fuctio is defied by ([3], p. 6, e. (.0.)) ( ; ) ( ) Γ ( x) x ( ; ) x (0 < < ). (4) If <, the euatio () remais meaigful for as a coverget ifiite product: j0 ( α; ) ( α I view of the relatio j ). α ( ; ) Lim ( α), ( ) (5) we observe that the -shifted factorial () reduces to the familiar Pochhammer symbol α ), ( α) α( α + ) L ( α + ). The -derivative ad -itegral of a fuctio o a subset of C are respectively give by(see[[3],pp.9 ]) ( f ( f ( f ( ( z 0, 0) z( ) (6) ad 0 z f ( t) d( t; ) z ( ) k 0 Therefore,the -derivative of z z ( z( ) z k f ( z k ). f (7) ( z, is a positive iteger,is give by ( z 0, 0). For ay o-egative iteger, the -iteger umber, [] is defied by: (8) [ ] ,[0] 0. The -umber shifted factorial is defied by fixed umber betwee 0 ad. [0]! ad (9) [ ]! [][]...[ ]. Here to be a As we have [ ] editor@iaeme.com

3 Majorizatio Problems For Subclasses of Aalytic Fuctios Ivolvig Q-Calculus Operator Here we used the the followig relatios [ + m m + ] [ m] [ ] [ m ] [ m] [ ],[0] 0,[] Recetly, may authors have itroduced ew classes of aalytic fuctios usig - calculus operators ad related topics, we refer the reader to [, 6, ] ad the refereces cited therei. Usig the defiitio of Ruscheweyh differetial operator [] for f A, Kaas ad Raducau[4] defied ad discussed the Ruscheweyh -differetial as R [ + ] f ( f ( * F, + ( z + az. [ ]! Makig use of (0) ad properties of Hadamard product, Kaas ad R a ( ducau [4] obtai the followig euality (0) ( R [ ] f ( ) (+ ) R + It is easily verified from (0) if [ ] f ( R f (. the euality ()implies () ' + z( R f ( ) (+ ) R f ( R f (. () Recetly Selvakumara et.al.[3] proved the followig result aalogues to Nehari [0] result. Lemma [3]If f ( is aalytic ad bouded i U,the f ( f ( z D,( z U). For two fuctios f ( ad g ( are aalytic fuctios i U, we say that f is subordiate to g writte f ( p if there exists a schwarz fuctio ω ( which is aalytic i U with ω (0) 0 ad ω ( < for all z U, such that f ( ω ( ), z U. Let f ad g be two aalytic fuctios i U. We say that f ( is majorized by i U ad write f ( << ( z U) if there exists a fuctio φ (, aalytic i U, such that φ ( < adf ( φ( ( z U). It is oted that the otatio of majorizatio is closely related to the cocept of uasisubordiatio betwee aalytic fuctios. Majorizatio problem for the class of aalytic fuctios had bee ivestigated by MacGregor [5] ad Altitas et al. [] further by Murugusudaramoorthy et al., [7, 8] for certai classes of aalytic fuctios ivolvig liear operators. Motivated by the results give i Kaas ad R a ( ducau [4]ad Nehari [0] i this paper,we defie the followig ew class of starlike fuctios of complex order ivolvig Ruscheweyh - differetial operator ad obtai majorisatio result for f S (b). For fuctios f A we let S (b), the class of starlike fuctios of complex order if 99 editor@iaeme.com

4 K.Thilagavathi, K.Vijaya ad K.Uma ( ( )) R f z R + [] > 0 ( ) b R f z ( z U, b C \{0},0 < < ) It ca be see that, by specializig the parameters, the class S (b), kow subclasses of aalytic fuctios. For istace, if the reduces to may 0 ( i ) S ( b) S( ), the class of starlike fuctios of complex order b (see [9]) b ( ii) S ( α) S( α),(0 0 α <) the class of starlike fuctios of complex order α.. MAIN RESULT Theorem Let the fuctio f ( be i the class A ad suppose that S ( b) f R ( is majorized by R.If R i U, the. If + + f ( R,( z r ) (3) r r ( : b) K K 4( + [ ])( (b ) ((b ) ]) ] )) (4) K 3( + [ ]) + (b ) ], ( b C \ 0,0 < <, 0) Proof. Let ( R ) h ( ] b R + S ( b), Sice we have Reh ( > 0( z U) ad + ω( h(,( ω A) ω( (5) (6) ω ( c z + c z +... ad A deotes the well kow class of bouded aalytic fuctios i U ad satisfies the coditios ω (0) 0, ad ω( z, ( z U) (7) 00 editor@iaeme.com

5 Majorizatio Problems For Subclasses of Aalytic Fuctios Ivolvig Q-Calculus Operator It follows from (5) ad (6) that ( R ) + (b ) ω( R ω( I view of the idetity (5), we have the followig ieuality from (8) by makig some simple calculatios Sice R f ( R f ( φ( ( R ) (+ z )( + [ ]) + [ ] + ((b ) ]) z is majorized by R + i U, there exists a aalytic fuctio φ (, such that Applyig differetiatio with respect to z ad the multiplyig by z we get ( R f ( ) ( φ( )( R ) + zφ( ( R )) Notig that the φ ( is bouded i U ad usig lemma (),we obtai φ( ( φ( ) z ad usig (9),() i () we have (8) (9) (0) () () + R φ( f ( φ( + z ( [ ]) z [ ]) ((b ) [ ]) z + + ( + + (3) ( + [ ]) rρ + ( (( ( ( z r ad φ ( ρ (0 ρ ) ) + [ ]) + ((b ) ]) z ) ρ + ( ( + [ ]) + ((b ) ]) + [ ]) r + + f ( ( ψ ( ρ) ( ψ ( ρ) ( + [ ]) + ((b ) ]) + [ ]) rρ + ( (( + [ ]) + ((b ) + ]) ρ + ( + [ ]) r takes its maximum value at ρ with r r ( : ) give by (4) Furthermore, b (4) (5) if 0 σ r r ( : b),the fuctio ϕ (ρ) defied by ϕ( ρ) ( + [ ]) σρ + ( σ )(( + [ ]) + ((b ) is a icreasig fuctio o (0 ρ ) so that ]) σ ) ρ + ( +[ ]) σ 0 editor@iaeme.com

6 K.Thilagavathi, K.Vijaya ad K.Uma ϕ( ρ) ϕ() ( + [ ]) σ + ( σ )(( + [ ]) + ((b ) ]) σ ) + ( +[ ]) σ (6) (0 ρ );0 σ r r ( : b) the settig ρ i (4), we coclude that holds true for z r ( : b). This completes the proof of Theorem (). REFERENCES [] M.H.Abu-Risha, M.H.Aaby, M.E.H.Ismail ad Z.S.Masour, Liear -differece euatios. Z. Aal. Awed. 6 (007) [] O.Altitas, O.Ozka ad H.M.Srivastava, Majorizatio by starlike fuctios of complex order,complex variable Theory Appl.,46(00), [3] G.Gasper ad M.Rahma, Basic Hypergeometric Series, Cambridge Uiversity Press, Cambridge, 990. [4] S. Kaas ad D. Ra ( ducau,some subclass of aalytic fuctios related to coic domais, Math. Slovaca 64(04), o. 5, [5] T.H. MacGreogor, Majorizatio by uivalet fuctios, Duke Math. J.34(967), [6] Z.S.I. Masour, Liear seuetial -differece euatios of fractioal order, Fract. Calc. Appl. Aal., () (009) [7] G.Murugusudaramoorthy ad K.Thilagavathi, Majorizatio Of Starlike Ad Covex Fuctios Of Complex Order Ivolvig Liear Operators, Math. Slovaca, 66, No.,(06), [8] G.Murugusudaramoorthy ad K. Thilagavathi,Majorizatio for certai classes of aalytic fuctios of complex order associated with the Dziok-Srivastava ad the Srivastavawright covolutio operators,tamsui Oxford J. Iform. Math. Sci., 9 (03), [9] M.A.Nasrad ad M.K.Aouf, Starlike fuctio of complex order, The Joural of Natural Scieces ad Mathematics, 5,o.,(985). [0] Z.Nehari, Coformal mappig, MacGra-Hill Book Compay, New York; Toroto ad Lodo, (95). [] S.D. Purohit ad R. K.Raia, Certai subclasses of aalytic fuctios associated with fractioal -calculus operators, Math. scad., 09 (0) [] Ruscheweyh, S. (975). New criteria for Uivalet Fuctios, Proc. Amer. Math. Soc., 49, pp [3] K.A.Selvakumara, S.D. Purohit ad A.Secer, Majorizatio for a Class of Aalytic Fuctios Defied by -Differetiatio, Math. Problems i Egg.,(04), Volume 04, Article ID 65397, 5 pages. 0 editor@iaeme.com