On Some Coefficient Estimates For Certain Subclass of Analytic And Multivalent Functions

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1 IOSR Journal of Mathematis (IOSR-JM) e-issn: , -ISSN: 9-765X. Volume, Issue 6 Ver. I (Nov. - De.06), PP On Some Coeffiient Estimates For Certain Sublass of nalyti nd Multivalent Funtions M. Elumalai Deartment of Mathematis, Presideny College, Chennai , Tamilnadu, India. bstrat: In this aer, motivated by the wors of Jenins [], Leung [] Panigrahi Murugusundaramoorthy [6] we defined a sublass of - valent analyti funtions using a generalized differential oerator omute oeffiient differenes. We also oint out, as artiular ases, the results obtained earlier by various authors. Keywords hrases: Multivalent funtions, -valent starlie funtions, -valent onvex funtions, Toelitz determinant, Hanel determinant, Differential oerator 000 Mathematis Subet Classifiation: 0C45. I. Introdution Definition Let denote the lass of analyti funtions in the oen unit dis U : { z : z } of the form let. n (.) n f ( z) z a z {,,,...} n Let S denote the sublass of onsisting of multivalent funtions. funtion f given by (.) is said to be valently starlie if it satisfies the inequality () Re zf z 0, ( z U ). f () z We denote this lass of funtions by * * S. Note that the lass S redues to S * : * S, the lass of starlie funtions in U, introdued by Robertson [7]. funtion f is said to be -valently onvex if it satisfies the ondition zf ( z) Re 0, ( z U ). f () z We denote by C the familiar sublass of. In artiular, C : Cthe lass of onvex funtions in U, introdued by Robertson [7] (also see [4]). For n, Hayman [9] showed the differene of suessive oeffiients is bounded by an absolute onstant i.e. an an. Using different tehnique, Milin [5] showed that 9. Ilina [0] imroved this to 4.6. Further, Grisan [8] restrited to.6. For starlie funtion S *, Leung [] roved that the best ossible bound is. On the other h, it is nown that for the lass S, annot be redued to. When n, Golusin [5,6], Jenins [] Duren [4] showed that for f S, a a that both uer lower bounds in (.) are shar. When n n, Panigrahi [6] showed that for f C, a a 0.5 a4 a 0.5. lso for f S *, a a.5 a4 a both the inequalities are shar. by We now define the following differential oerator D,,, :,,, ( ) [( ) ( )( ) ] (,, ) n f z z n n n C n an z n D (.) DOI: / Page

2 On Some Coeffiient Estimates for Certain Sublass of nalyti Multivalent Funtions, 0 : {0},,, 0. ( n ) C(, n, ). ( n ) ( ) By seializing the arameters,,, we obtain the following oerators studied earlier by various researhers: Namely,, If, 0, 0 or 0,, the oerator D,0 D D is the oular Salagean 0,0,,0, oerator [9]; 0, When 0,, then D whih is the Rusheweyh differential oerator (see [8]);,,0 For 0, 0,, then D D whih is the differential oerator studied by l-oboudi (see []);,0,,0 If 0 then D D has been studied by Darus Ibrahim (see []); When, then,,, D D whih is the generalized differential oerator studied by Panigrahi,,,,, Murugusundaramoorthy (see [6]). Motivated by the above onet, in this aer, maing use of the differential oerator investigate a new sublass of multivalent funtions, as in D we introdue,,, Definition.. funtion f is said to be in the lass t, M,, ( ) if it satisfies the inequality,, ( t) z( D,, f ( z)) tz( D,, f ( z)) 0, ( z ),, U ( t),, f ( z) t,, f ( z) D D 0 t,, 0,, 0. (.) t, Note that by taing t 0 t, 0 the lass M,, ( ), redues the lasses C, resetively. * S Remar.. If t 0, then in U. Similarly, if we let t, 0 then onvex funtions in U. t, M,, ( ) redues to the well-nown lass of starlie funtions t, M,, ( ) redues to the well-nown lass of The urose of the resent study is to estimate the oeffiient differenes for the funtion lass when n n. II. Preliminary Results t, M,, ( ), In order to derive our main results, we have to reall the following reliminary lemmas: Let P be the family of all funtions h analyti in U, for whih n Lemma.. [4] If h P, then, for eah. Reh z 0 n h( z) n z, z U. (.) Lemma.. [7] The ower series for h given in (.) onverges in the unit dis U to a funtion in P if only if the Toelitz determinants. D,,,, DOI: / Page

3 On Some Coeffiient Estimates for Certain Sublass of nalyti Multivalent Funtions it z, are all non-negative. These are stritly ositive exet for h( z) h0e, 0, t t, for, in this ase D 0 for ( m) D 0 for m. This neessary suffiient ondition due to Caratheodory Toelitz an be found in [7]. m t real We may assume without restrition that 0 on using [Lemma.], for resetively, we get whih is equivalent to Then D 0 is equivalent to D 8 Re{ } 4 0, x(4 ), for some x, x. (.) D. (4 4 )(4 ) ( ) (4 ). (.) From the relations (.) (.), after simlifying, we get 4 (4 ) x (4 ) x (4 )( x ) z for some real value of z, with z. (.4) III. Main Results In this setion, we rove to estimate the oeffiient differenes for the funtion lass M,, ( ). t, ( ) ( ) Theorem.. Let f given by (.) be in the lass M,, ( ). If, then ( ) ( ) 8 ( ) a a, (.) 4 ( ) ( ) 8 ( ) a a, (.) 4 ( ) ( ) ( ) ( )[ (( ) ( ) ) t], ( )( ) ( ) ( ( ) ) [ (( ) ( ( ) ) ) t], ( )( )( ) ( ) ( ( ) ) [ (( ) ( ( ) ) ) t]. 6 t, Proof: Let the funtion f() z reresented by (.) be in the lass M,, ( ). By geometri interretation, there exists a funtion hp given by (.) suh that,, ( t) z( D,, f ( z)) tz( D,, f ( z)) hz ( ). (.),, ( t) D f ( z) td f ( z),,,, t, DOI: / Page

4 Relaing On Some Coeffiient Estimates for Certain Sublass of nalyti Multivalent Funtions D, f ( z), D f ( z), ( D f,, ( z )),,,,,, equivalent exression for hz () in series (.), we have D, f,, ( z ) ( t) z( D f ( z)) tz( D f ( z)) h( z) ( t) D f ( z) td f ( z).,,,,,,,,,,,, ( t) z z ( n )[( n ) ( n )( n ) ] (, n, ) a z C n n n by their equivalent exressions the tz z n n n n n a z ( )[( ) ( )( ) ] C(,, ) n n n n ( t) z ( n ) ( n )( n ) C (, n, ) an z (.4) n n n t z ( n ) ( n )( n ) C (, n, ) an z nz n n Equating the oeffiients of lie ower of z, z z resetively on both sides of (.4), we have ( ) a a, ( ) a a a, ( ) a a a a,, are given in the statement of theorem. fter simlifying, we get a, a, ( ) ( ) ( ) a. ( ) ( )( ) ( )( ) Sine, a a a a, we need to onsider a a a a. Taing into aount (.5) (.) we obtain n n n n a a ( ) ( ) x ( ) ( ) (4 ) x (4 ). ( ) ( ) (.5) (.6) We an assume without loss of generality that 0. For onveniene of notation, we tae ( [0;]) (see Lemma.). lying triangle inequality relaing x by in the right h side of ( ) (.6) using the inequality, it redues to ( ) ( ) 4 a a ( ) ( ) (, ) (0 x ), ( ) 4 (, ). ( ) ( ) DOI: / Page (.7) (.8)

5 On Some Coeffiient Estimates for Certain Sublass of nalyti Multivalent Funtions We assume that the uer bound for (.7) ours at an interior oint of the {(, ) : [0,]} [0,]. Differentiating (.8) artially with reset to, we get 4. ( ) From (.9) we observe that 0 for 0 for fixed with 0. Therefore F (, ) is an inreasing funtion of, whih ontradits our assumtion that the maximum value of ours at an interior oint of the set {(, ) : [0,]} [0,]. So, fixed [0,], we have max (, ) (,) ( ) (say). 0 Therefore relaing by in (.8), we obtain ( ) ( ), ( ) (.9) (.0) () (.) ( ) 0. For otimum value of ( ), onsider ( ) 0. It imlies that. Therefore, the maximum value of () is 8 ( ) 4 ( ) From (.7) (.), we have whih ours at. from the exression (.0), we get whih roves the assertion (.) of Theorem.. 8 ( ) max. 4 ( ) 8 ( ) a, a 4 ( ) Using the same tehnique, we will rove (.). From (.5) an aliation of (.4) we have (.) ( ) a a ( ) ( )( ) ( )( ) ( ) ( ) { (4 ) x (4 ) x (4 )( x ) z} 4( ) ( ) ( )( ) ( )( ) { x(4 )} ( ) ( ) ( ) ( ) ( ) (4 ) x 4( ) ( ) ( )( ) (4 ) x a a (4 )( x ) z 4( ) ( ) (4 ) x ( ) { x(4 )} (.) DOI: / Page

6 On Some Coeffiient Estimates for Certain Sublass of nalyti Multivalent Funtions s earlier, we assume without loss of generality that with 0. lying triangle inequality relaing x by in the right h side of (.) using the fat that, it redues to ( ) ( ) ( ) a a (4 ) 4( ) ( ) ( )( ) (, ), (4 ) 4( ) ( ) ( ) (4 )( ) (4 ) ( ) ( ) ( ) (, ) (4 ) 4( ) ( ) ( )( ) (4 ) 4( ) ( ) ( ) z (4 )( ) (4 ). z (.4) (.5) Suose that (, ) in (.5) attains its maximum at an interior oint (, ) of [0, ] [0,]. Differentiating (.5) artially with reset to, we have Now 0 whih imlies ( ) (4 ) (4 ) (4 ) (4 ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( 4) ( ( ) ) ( ). ( ) ( ) ( ) ( ) 0 (0 ), whih is false sine 0. Thus (, ) attains its maximum on the boundary of [0, ] [0,]. Thus for fixed, we have mx a (, ) (,) ( ) ( say) 0 Therefore, relaing by in (.5) simlifying we get ( ) () ( )( ) ( ) () ( ) ( ) 0. ( ) () For an otimum value of ( ), onsider ( ) 0 whih imlies ( ) () of () ours at ( ). From the exression (.6) we obtain ( ) ( ) 8 ( ) max. ( ) 4 ( ) From (.4) (.8), we have (.6) (.7). Therefore, the maximum value (.8) DOI: / Page

7 On Some Coeffiient Estimates for Certain Sublass of nalyti Multivalent Funtions The roof of Theorem. is thus omleted. Taing t ; 0 in Theorem. we get ( ) 8 ( ) a. a 4 ( ) Corollary.. Let f given by (.) be in the lass C. then Both the inequalities are shar. Putting t 0 in Theorem. we get Corollary.. Let f given by (.) be in the lass ( )( ) a a 8 ( ) ( ) 9( ) 8 ( )( ) a a ( ) ( )( ) S *. Then ( ) a a 8 ( ) 9( ) 8 ( )( ) a a ( ) ( ) Both the inequalities are shar. For, Theorem. redues to the results obtained in t, Corollary.4. [6] Let f given by (.) be in the lass M, ( ). If, then 4 4 a a, 4 a4 a, ( ) ( )[ ( ( ) ) t], ( )( ) ( ) [ ( ( ) ) t], ( )( )( ) 4 ( ) [ (4 ( ) ) t]. 6 Remar.. Here we remar that the results obtained in (orollary, [6]) is omutationally wrong. The estimates a a 5 5 a4 a must be a a 5 5 a4 a Taing t ; 0 in Theorem. we get following Corollary.5. [6] Let f given by (.) be in the lass C. Then Both the inequalities are shar. 5 5 a a a4 a Putting t 0 in Theorem. we get following DOI: / Page

8 On Some Coeffiient Estimates for Certain Sublass of nalyti Multivalent Funtions Corollary.6. [6] Let f given by (.) be in the lass Both the inequalities are shar. S *. Then 5 a a a4 a 4 Referenes []. F. M. l-oboudi, On univalent funtions defined by a generalized Salagean oerator, Int. J. Math. Math. Si., 7(004), []. M. Darus R. W. Ibrahim, Generalization of differential oerator, J. Math. Stat., 4(008), []. M. Darus R. W. Ibrahim, New lasses ontaining generalization of differential oerator, l. Math. Si., (009), [4]. P. L. Duren, Univalent Funtions. Grundlehren der Mathematishen Wissenshaften Fundamental Priniles of Mathematial Sienes]}, 59. Sringer- Verlag, New Yor, 98. [5]. G. M. Golusin, On distortion theorems oeffiients of univalent funtions, Re. Math. [Mat. Sborni] N.S., 9(946) [6]. G. M. Golusin, Some questions in the theory of univalent funtions, Trudy. Mat. Inst. Stelov., 7(949), -. [7]. U. Grener G. Szego, Toelitz forms their aliation, Bereley Los ngeles: Univ. of California Press, 958. [8].. Z. Grisin, Imroved bounds for the differene of the moduli of adaent oeffiients of univalent funtions, in Some Questions in the Modern Theory of Funtions, Sib. Inst. Math, (976), [9]. W. K. Hayman, On suessive oeffiients of univalent funtions, J. London Math. So., 8(96), 8-4. [0]. L. P. Ilina, On the relative growth of adaent oeffiients of univalent funtions, Mat. Zameti, 4(968) 75-7 (in Russian) = Math. Notes 4(968), []. J.. Jenins, On ertain oeffiients of univalent funtions, 960 nalyti funtions Prineton Univ. Press, Prineton, N.J. []. Y. J. Leung, Suessive oeffiients of starlie funtions, Bull. London Math. So.,0(978), []. R. J. Libera E. J. Zlotiewiz, Early oeffiients of the inverse of a regular onvex funtion, Pro. mer. Math. So., 85(98), 5-0. [4]. R. J. Libera E. J. Zlotiewiz, Coeffiients bounds for the inverse of a funtion with derivative in, Pro. mer. Math. So., 87(98), [5]. I. M. Milin, daent oeffiients of univalent funtions, Dol. ad. Nau SSSR., 80(968), (in Russian)=Soviet Math. Dol., 9(969), [6]. T. Panigrahi G. Murugusundaramoorthy, On suessive oeffiient estimate for ertain sublass of analyti funtions, lied Mathematis E-Notes, 6(06), 7-4. [7]. M. S. Robertson, Quasi-subordination oeffiient onetures, Bull. mer. Math. So., 76(970), -9. [8]. S. Rusheweyh, New riteria for univalent funtions, Pro. mer. Math. So., 49(975), [9]. G. S. Salagean, Sublasses of Univalent Funtions. Comlex nalysis-fth Romanian-Finnish seminar, Part (Buharest, 98), 6-7, Leture Notes in Math., 0, Sringer, Berlin, 98. DOI: / Page