Hindawi Publishing 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 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 the radius of starlikeness of the certain classes of p-valent functions defined by multiplier transformations. The results are obtained by using techniques of Robertson 1953,1963 which was used by Bernardi 1970, Libera 1971, Livingstone 1966,andGoel1972. 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 Ha, n be the subclasses of H consisting of the functions of the form fza  a n z n  a n1 z n1 ···. Let Ap, n denote the class of functions fz normalized by fzz p  ∞  knp 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 Ap, 1 : A p , A1, 1 : A  A 1 . 1.2 If fz and gz are analytic in D, we say that fz is subordinate to gz, written symbolically as f ≺ g or fz ≺ gzz ∈ D. 1.3 2 Journal of Inequalities and Applications If there exists a Schwarz function wz which is analytic in D with w00, |wz| < 1 such that fzgwz, z ∈ D. For two analytic functions fz and Fz, we say that Fz is superordinate to fz if fz is subordinate to Fz. For integer n ≥ 1, let Ωn denote the class of functions wz which are regular in D and satisfy the conditions w00, |wz| < 1, and wzz n φz for all z ∈ D,whereφz is regular and analytic in D and satisfies |φz| < 1 for every z ∈ D. Also, let P{p, n denote the class of functions pzp   ∞ kn p k z k which are regular in D and satisfy the conditions p0p,Repz > 0 for all z ∈ D.Wenotethatifpz ∈Pp, n,then pzp 1 − wz 1  wz  1 − z n φz 1  z n φz 1.4 for some functions wz ∈ Ωn and every z ∈ D. Definition 1.1. Let fz ∈Ap, n for m ∈ N 0  N ∪{0}, λ ≥ 0, l>0, one defines the multiplier transformations I p m, λ, l on Ap, n by the following infinite series: I p m, λ, lfz : z p  ∞  kpn  p  λk − pl p  l  m a k z k . 1.5 It follows that I p 0,λ,lfzfz, p  lI p 2,λ,lfz  p1 − λl  I p 1,λ,lfzλzI p 1,λ,lfz   , I p  m 1 ,λ,l  I p  m 2 ,λ,l  fz   I p  m 2 ,λ,l  I p  m 1 ,λ,l  fz  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.Letϕz be analytic in D and ϕ01. A function fz ∈Ap, 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,lfz I p m, λ, lfz ≺ ϕzz ∈ D. 1.7 Definition 1.4. The radius of starlikeness of the class A p m, λ, l, n, ϕ is defined by the following. For each fz ∈A p m, λ, l, n; ϕ,letrf be the supremum of all numbers r such that f D r  is starlike with respect to the origin. Then the radius of starlikeness 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 rf. 1.8 Mugur Acu et al. 3 Theorem 1.5. Let fz ∈Ap, n and λ>0,thenfz belongs to the class A p m, λ, l, n; χ if and only if Fz, defined by Fz p  l λz p1−λl/λ  z 0 ζ p1−λl/λ−1 fζdζ  z p  ∞  kpn  p  l p  l k − pλ  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. The radius of starlikeness of the class A p m, λ, l, n, φ is r st  ⎛ ⎜ ⎝ p  l λp  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 − pz n   1  z n  2p/n1 ,c p1 − λl λ . 2.2 Proof. If we take c p1 − λl/λ, then the function Fz in Theorem 1.5 can be written in the form Fz p  l λz c  z 0 ζ c−1 fζdζ. 2.3 If we take the logarithmic derivative from 2.3 and after simple calculations, we g et z F  z Fz  z c fz − c  z 0 ζ c−1 fζdζ  z 0 ζ c−1 fζdζ . 2.4 Since Fz is starlike, hence there exists a function wz ∈ Ωn such that z F  z Fz  z c fz − c  z 0 ζ c−1 fζdζ  z 0 ζ c−1 fζdζ  p 1 − wz 1  wz . 2.5 Solving for fz, fz c  pc − pwz  1  wz  z c  z 0 ζ c−1 fζdζ. 2.6 4 Journal of Inequalities and Applications Taking the logarithmic derivative from 2.6,weget z f  z fz  p 1 − wz 1  wz b − 1 zw  z  1  wz  1  bwz  , 2.7 where b c − p/c  p. To show that fz is starlike in |z| <r 0 , we must show that Re  z f  z fz  > 0 2.8 for |z| <r 0 . This condition is equivalent to 1 − bRe  zw  z  1  wz  1  bwz   ≤ Re  p 1 − wz 1  wz  . 2.9 On the other hand, we have the following relations: Re  p 1 − wz 1  wz   p 1 −   wz   2   1  wz   2 , 1 − bRe  zw  z  1  wz  1  bwz   ≤ 1 − b   zw  z     1  wz     1  bwz   ,   zw  z   ≤ n|z| n 1 −|z| 2n  1 −   wz   2  2.10 Golusin inequality, 8. Therefore, the inequality 2.9 will be satisfied if n1 − b|z| n   1  wz     1  bwz   1 −   wz   2 1 −|z| 2n ≤ p 1 −   wz   2   1  wz   2 . 2.11 Simplifying and writing |z|  r,weobtain n1 − br n 1 − r 2n ≤ p     1  bwz 1  wz     . 2.12 Since |wz|≤|z| n  r n , p|1  bwz/1  wz|≥p1  br n /1  r n  so that 2.12 will be satisfied if n1 − br 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 − bp  nr n − bpr 2n > 0, 2.14 which gives the required root r 0 of the theorem. To see that the result is sharp, consider the function Fzz p /1  z n  2p/n .Forthis function, we have f ∗ z λ p  l z p  c  pc − pz n   1  z n  2p/n1 , z f  ∗ z f ∗ z  p − 1 − bp  nz n − pbz 2n  1  z n  2p/n1 . 2.15 So that zf  ∗ z/f ∗ z  0for|z|  r 0 . Thus, fz 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 the radius of starlikeness for the corresponding integral operators. Acknowledgment This paper was supported by GAR 20/2007. References 1 A. C ˘ atas¸, On certain classes of p-valent functions defined by multiplier transformations, in Proceedings of the International Symposium on Geometric Function Theory and Applications, ˙ Istanbul, Turkey, August 2007. 2 F. M. Al-Oboudi, On univalent functions defined by 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. 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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 Ha, n be the subclasses of H consisting of the functions of the form fza  a n z n 