Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 280183, 5 pages doi:10.1155/2008/280183 ResearchArticleTheRadiusofStarlikenessoftheCertain Classesofp-ValentFunctionsDefinedby Multiplier Transformations Mugur Acu, 1 Yas¸ar Polato ˜ glu, 2 and Emel Yavuz 2 1 Department of Mathematics, ”Lucian Blaga” University of Sibiu, 5-7 Ion Ratiu Street, Sibiu 550012, Romania 2 Department of Mathematics and Computer Science, TC ˙ Istanbul K ¨ ult ¨ ur University, ˙ Istanbul 34156, Turkey Correspondence should be addressed to Emel Yavuz, e.yavuz@iku.edu.tr Received 12 November 2007; Accepted 02 January 2008 Recommended by Narendra Kumar K. Govil The aim of this paper is to give theradiusofstarlikenessofthecertainclassesofp-valentfunctionsdefinedbymultiplier transformations. The results are obtained by using techniques of Robertson 1953,1963 which was used by Bernardi 1970, Libera 1971, Livingstone 1966,andGoel1972. Copyright q 2008 Mugur Acu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let H be the class of analytic functions in the open unit disc D {z ∈ C ||z| < 1} and Ha, n be the subclasses of H consisting ofthefunctionsofthe form fza a n z n a n1 z n1 ···. Let Ap, n denote the class offunctions fz normalized by fzz p ∞ knp a k z k p, n ∈ N : 1, 2, 3, 1.1 which are analytic in the open unit disc D. In particular, we set Ap, 1 : A p , A1, 1 : A A 1 . 1.2 If fz and gz are analytic in D, we say that fz is subordinate to gz, written symbolically as f ≺ g or fz ≺ gzz ∈ D. 1.3 2 Journal of Inequalities and Applications If there exists a Schwarz function wz which is analytic in D with w00, |wz| < 1 such that fzgwz, z ∈ D. For two analytic functions fz and Fz, we say that Fz is superordinate to fz if fz is subordinate to Fz. For integer n ≥ 1, let Ωn denote the class offunctions wz which are regular in D and satisfy the conditions w00, |wz| < 1, and wzz n φz for all z ∈ D,whereφz is regular and analytic in D and satisfies |φz| < 1 for every z ∈ D. Also, let P{p, n denote the class offunctions pzp ∞ kn p k z k which are regular in D and satisfy the conditions p0p,Repz > 0 for all z ∈ D.Wenotethatifpz ∈Pp, n,then pzp 1 − wz 1 wz 1 − z n φz 1 z n φz 1.4 for some functions wz ∈ Ωn and every z ∈ D. Definition 1.1. Let fz ∈Ap, n for m ∈ N 0 N ∪{0}, λ ≥ 0, l>0, one defines themultiplier transformations I p m, λ, l on Ap, n bythe following infinite series: I p m, λ, lfz : z p ∞ kpn p λk − pl p l m a k z k . 1.5 It follows that I p 0,λ,lfzfz, p lI p 2,λ,lfz p1 − λl I p 1,λ,lfzλzI p 1,λ,lfz , I p m 1 ,λ,l I p m 2 ,λ,l fz I p m 2 ,λ,l I p m 1 ,λ,l fz 1.6 for all integers m 1 , m 2 . Remark 1.2. This multiplier transformation was introduced by C ˘ atas¸ 1.Forp 1, l 0, λ ≥ 0, the operator D m λ : I 1 m, λ, 0 was introduced by Al-Oboudi 2 which reduces to the S ˘ al ˘ agean differantial operator 3.Forλ 1, the operator I m l : I 1 m, 1,l was studied recently by Cho and Srivastava 4 and Cho and Kim 5. The operator I m : I 1 m, 1, 1 was studied by Urale- gaddi and Somanatha 6 and the operator I p m, l : I p m, 1,l was investigated recently by Sivaprasad Kumar et al. 7. Definition 1.3 see 1.Letϕz be analytic in D and ϕ01. A function fz ∈Ap, n is said to be in the class A p m, λ, l, n; ϕ if it satisfies the following subordination: I p m 1,λ,lfz I p m, λ, lfz ≺ ϕzz ∈ D. 1.7 Definition 1.4. Theradiusofstarlikenessofthe class A p m, λ, l, n, ϕ is definedbythe following. For each fz ∈A p m, λ, l, n; ϕ,letrf be the supremum of all numbers r such that f D r is starlike with respect to the origin. Then theradiusofstarlikeness for A p m, λ, l, n; ϕ is r st A p m, λ, l, n; ϕ inf f∈A p m,λ,l,n,ϕ rf. 1.8 Mugur Acu et al. 3 Theorem 1.5. Let fz ∈Ap, n and λ>0,thenfz belongs to the class A p m, λ, l, n; χ if and only if Fz, definedby Fz p l λz p1−λl/λ z 0 ζ p1−λl/λ−1 fζdζ z p ∞ kpn p l p l k − pλ a k z k , 1.9 belongs to the class A p m 1,λ,l,n; χ. This theorem was proved by C ˘ atas¸ 1. 2. Main result Theorem 2.1. Theradiusofstarlikenessofthe class A p m, λ, l, n, φ is r st ⎛ ⎜ ⎝ p l λp n λ 2 p n 2 p l p l − 2λp ⎞ ⎟ ⎠ 1/n . 2.1 This radius is sharp because the extremal function is f ∗ z λ p l z p c p c − pz n 1 z n 2p/n1 ,c p1 − λl λ . 2.2 Proof. If we take c p1 − λl/λ, then the function Fz in Theorem 1.5 can be written in the form Fz p l λz c z 0 ζ c−1 fζdζ. 2.3 If we take the logarithmic derivative from 2.3 and after simple calculations, we g et z F z Fz z c fz − c z 0 ζ c−1 fζdζ z 0 ζ c−1 fζdζ . 2.4 Since Fz is starlike, hence there exists a function wz ∈ Ωn such that z F z Fz z c fz − c z 0 ζ c−1 fζdζ z 0 ζ c−1 fζdζ p 1 − wz 1 wz . 2.5 Solving for fz, fz c pc − pwz 1 wz z c z 0 ζ c−1 fζdζ. 2.6 4 Journal of Inequalities and Applications Taking the logarithmic derivative from 2.6,weget z f z fz p 1 − wz 1 wz b − 1 zw z 1 wz 1 bwz , 2.7 where b c − p/c p. To show that fz is starlike in |z| <r 0 , we must show that Re z f z fz > 0 2.8 for |z| <r 0 . This condition is equivalent to 1 − bRe zw z 1 wz 1 bwz ≤ Re p 1 − wz 1 wz . 2.9 On the other hand, we have the following relations: Re p 1 − wz 1 wz p 1 − wz 2 1 wz 2 , 1 − bRe zw z 1 wz 1 bwz ≤ 1 − b zw z 1 wz 1 bwz , zw z ≤ n|z| n 1 −|z| 2n 1 − wz 2 2.10 Golusin inequality, 8. Therefore, the inequality 2.9 will be satisfied if n1 − b|z| n 1 wz 1 bwz 1 − wz 2 1 −|z| 2n ≤ p 1 − wz 2 1 wz 2 . 2.11 Simplifying and writing |z| r,weobtain n1 − br n 1 − r 2n ≤ p 1 bwz 1 wz . 2.12 Since |wz|≤|z| n r n , p|1 bwz/1 wz|≥p1 br n /1 r n so that 2.12 will be satisfied if n1 − br n 1 − r 2n <p 1 br n 1 r n . 2.13 The inequality 2.13 can be written in the following form: p − 1 − bp nr n − bpr 2n > 0, 2.14 which gives the required root r 0 ofthe theorem. To see that the result is sharp, consider the function Fzz p /1 z n 2p/n .Forthis function, we have f ∗ z λ p l z p c pc − pz n 1 z n 2p/n1 , z f ∗ z f ∗ z p − 1 − bp nz n − pbz 2n 1 z n 2p/n1 . 2.15 So that zf ∗ z/f ∗ z 0for|z| r 0 . Thus, fz is not starlike in any circle |z| <rif r>r 0 . Mugur Acu et al. 5 Remark 2.2. If we give special values to m, λ, l, n, we obtain theradiusofstarlikeness for the corresponding integral operators. Acknowledgment This paper was supported by GAR 20/2007. References 1 A. C ˘ atas¸, On certainclassesofp-valentfunctionsdefinedbymultiplier transformations, in Proceedings ofthe International Symposium on Geometric Function Theory and Applications, ˙ Istanbul, Turkey, August 2007. 2 F. M. Al-Oboudi, On univalent functionsdefinedby a generalized S ˘ al ˘ agean operator, International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 27, pp. 14291436, 2004. 3 G. S. S ˘ al ˘ agean, Subclasses of univalent functions, in Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Math, pp. 362372, Springer, Berlin, Germany, 1983. 4 N. E. Cho and H. M. Srivastava, Argument estimates ofcertain analytic functionsdefinedby a class ofmultiplier transformations, Mathematical and Computer Modelling, vol. 37, no. 1-2, pp. 3949, 2003. 5 N. E. Cho and T. H. Kim, Multiplier transformations and strongly close-to-convex functions, Bulletin ofthe Korean Mathematical Society, vol. 40, no. 3, pp. 399410, 2003. 6 B. A. Uralegaddi and C. Somanatha, Certainclassesof univalent functions, in Current Topics in Analytic Function Theory, pp. 371374, World Scientific, Singapore, 1992. 7 S. Sivaprasad Kumar, H. C. Taneja, and V. Ravichandran, Classesof multivalent functionsdefinedby Dziok-Srivastava linear operator and multiplier transformation, Kyungpook Mathematical Journal,vol. 46, no. 1, pp. 97109, 2006. 8 G. M. Golusin, Geometrische Funktionentheorie,vol.31ofHochschulb ¨ ucher f ¨ ur Mathematik,VEBDeutscher Verlag der Wissenschaften, Berlin, Germany, 1957. . 2008 Recommended by Narendra Kumar K. Govil The aim of this paper is to give the radius of starlikeness of the certain classes of p-valent functions defined by multiplier transformations. The results. Corporation Journal of Inequalities and Applications Volume 2008, Article ID 280183, 5 pages doi:10.1155/2008/280183 Research Article The Radius of Starlikeness of the Certain Classesofp-ValentFunctionsDefinedby Multiplier. cited. 1. Introduction Let H be the class of analytic functions in the open unit disc D {z ∈ C ||z| < 1} and Ha, n be the subclasses of H consisting of the functions of the form fza a n z n