Research Article Coefficient Estimates for Two New Subclasses of Biunivalent Functions with respect to Symmetric Points

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1 Function Spaces Volume 2015 Article ID pages Research Article Coefficient Estimates for Two New Subclasses of Biunivalent Functions with respect to Symmetric Points Fahsene AltJnkaya andsibel YalçJn Department of Mathematics Faculty of Arts and Science Uludag University Bursa Turkey Correspondence should be addressed to Şahsene Altınkaya; sahsenealtinkaya@gmail.com Received 1 December 2014; Accepted 9 February 2015 Academic Editor: Alberto Fiorenza Copyright 2015 Ş. Altınkaya and S. Yalçın. 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. We introduce two subclasses of biunivalent functions and find estimates on the coefficients a 2 and a 3 for functions in these new subclasses. Also consequences of the results are pointed out. 1. Introduction and Definitions Let A denote the class of analytic functions in the unit disk that have the form U=z C : z <1 (1) f (z) =z+ a n z n. (2) n=2 Further by S we will denote the class of all functions in A which are univalent in U. The Koebe one-quarter theorem [1]statesthattheimage of U under every function f from S contains a disk of radius (1/4). Thus every such univalent function has an inverse f 1 which satisfies f 1 (f (z)) =z (z U) f(f 1 (w)) =w ( w <r 0 (f) r 0 (f) 1 (3) 4 ) where f 1 (w) =w a 2 w 2 + (2a 2 2 a 3)w 3 (5a 3 2 5a 2a 3 +a 4 )w 4 +. Afunctionf(z) A is said to be biunivalent in U if both f(z) and f 1 (z) are univalent in U.LetΣ denote the class of biunivalent functions defined in the unit disk U. (4) If the functions f and g are analytic in U thenf is said to be subordinate to gwrittenas f (z) g(z) (z U) (5) if there exists a Schwarz function w(z)analytic in Uwith such that w (0) =0 w (z) <1 (z U) (6) f (z) =g(w (z)) (z U). (7) Lewin [2] studied the class of biunivalent functions obtaining the bound 1.51 for modulus of the second coefficient a 2. Subsequently Netanyahu [3] showedthatmax a 2 = 4/3 if f(z) Σ. Brannan and Clunie [4] conjectured that a 2 2 for f Σ. Brannan and Taha [5] introduced certain subclasses of the biunivalent function class Σ similar to the familiar subclasses of univalent functions consisting of strongly starlike starlike and convex functions. They introduced bistarlike functions and obtained estimates on the initial coefficients. Bounds for the initial coefficients of several classes of functions were also investigated in [6 15]. Notmuchisknownabouttheboundsonthegeneral coefficient a n for n 4. In the literature there are only a few works determining the general coefficient bounds a n for the analytic biunivalent functions ([16 20]). The coefficient estimate problem for each of a n (n N \ 1 2; N = )isstillanopenproblem.

2 2 Function Spaces By S (φ) and C(φ) we denote the following classes of functions: S (φ) = f : f A zf (z) f (z) C(φ)=f:f A 1+ zf (z) f (z) φ(z) ;z U φ(z) ;z U. The classes S (φ) and C(φ) are the extensions of classical sets of starlike and convex functions and in such form were defined and studied by Ma and Minda [21]. (8) of starlike In [22] Sakaguchi introduced the class S S functions with respect to symmetric points in Uconsistingof functions f Athatsatisfy the condition Re(zf (z)/(f(z) f( z))) > 0 z U.Similarlyin[23] Wang et al. introduced the class C S of convex functions with respect to symmetric points in U consisting of functions f A that satisfy the condition Re((zf (z)) /(f (z) + f ( z))) > 0 z U.Inthe style of Ma and Minda Ravichandran (see [24]) defined the classes S S (φ) and C S(φ). Afunctionf Ais in the class S S (φ) if andintheclassc S (φ) if 2zf (z) φ(z) z U (9) f (z) f( z) 2(zf (z)) f (z) +f ( z) φ(z) z U. (10) In this paper we introduce two new subclasses of biunivalent functions. Further we find estimates on the coefficients a 2 and a 3 for functions in these subclasses. 2. Coefficient Estimates for the Function Class S SΣ (αh) Definition 1. Let the functions hp : U C be so constrained that min Re (h (z)) Re (p (z)) > 0 h (0) =p(0) =1. (11) Definition 2. Afunctionf Σ is said to be in the class S SΣ (α h) if the following conditions are satisfied: 2[(1 α) zf (z) +αz(zf (z)) ] (1 α) (f (z) f( z))+αz(f (z) +f ( z)) h(u) 2[(1 α) wg (w) +αw(wg (w)) ] (1 α) (g (w) g( w))+αw(g (w) +g ( w)) p(u) where g(w) = f 1 (w). (12) Definition 3. Onenotesthatforα = 0onegetstheclass S S (h) which is defined as follows: 2zf (z) f (z) f( z) h(u) 2wg (w) g (w) g( w) p(u). (13) Theorem 4. Let f given by (2) be in the class S SΣ (α h).then a 2 min + p 2 8 (1+α) 2 min + p 2 8 (1+α) 2 + h + p h + p h 4 (1+2α). (14) Proof. Let f S SΣ (α h) and g be the analytic extension of f 1 to U.Itfollowsfrom(12) that 2(zf (z) +αz 2 f (z)) (1 α) (f (z) f( z))+αz(f (z) +f ( z)) =h(z) (z U) 2(wg (w) +αw 2 g (w)) (1 α)(g (w) g( w))+αw(g (w) +g ( w)) =p(w) (w U) (15) where h(z) and p(w) satisfy the conditions of Definition 1. Furthermore the functions h(z) and p(w) have the following Taylor-Maclaurin series expansions: h (z) =1+h 1 z+h 2 z 2 + (16) p (w) =1+p 1 w+p 2 w 2 + (17) respectively. From (15) wededuce 2 (1+α) a 2 =h 1 (18) 2 (1+2α) a 3 =h 2 (19) 2 (1+α) a 2 =p 1 (20) 2 (1+2α) (2a 2 2 a 3)=p 2. (21) From (18) and (20) we obtain h 1 = p 1 (22) 8 (1+α) 2 a 2 2 =h2 1 +p2 1. (23)

3 Function Spaces 3 By adding (19) to (21)weget 4 (1+2α) a 2 2 =h 2 +p 2. (24) Therefore we find from (23) and (24) that a 2 2 a p 2 8 (1+α) 2 h + p. Subtracting (21) from (19) we have (25) 4 (1+2α) a 3 4(1+2α) a 2 2 =h 2 p 2. (26) then inequalities (14) become a 2 Corollary 7. If we let β min 1+α β 1+2α min (1+α) α β 2 1+2α. φ (z) = 1+(1 2β)z =1+2(1 β)z 1 z +2(1 β)z 2 + (0 β<1) then inequalities (14) become (31) (32) Then upon substituting the value of a 2 2 from (23) and (24) into (26)itfollowsthat We thus find that a 3 = h2 1 +p2 1 8 (1+α) 2 + h 2 p 2 4 (1+2α) a 3 = h 2 +p 2 4 (1+2α) + h 2 p 2 4 (1+2α). 2 h + p 2 8 (1+α) 2 + h 4 (1+2α). This completes the proof of Theorem 4. Taking α=0we get the following. Corollary 5. If f S S (h) then a 2 min + p 2 8 min + p 2 8 Corollary 6. If we let + h + p h + p 8 φ (z) =( 1+z 1 z ) β =1+2βz+2β 2 z 2 + h + p 8 h 4. (27) (28) (29) (0<β 1) (30) a 2 min 1 β 1+α 1 β 1+2α min (1 β)2 (1+α) β 1+2α 1 β 1+2α. (33) Remark 8. Corollaries 6 and 7 provideanimprovementofthe estimate a 3 obtained by Altınkaya and Yalçın [25]. Remark 9. The estimates on the coefficients a 2 and a 3 of Corollaries 6 and 7 are improvement of the estimates in [7]. 3. Coefficient Estimates for the Function Class m SΣ (αh) Definition 10. Afunctionf Σis said to be m SΣ (α h) if the following conditions are satisfied: ( 2zf α (z) f (z) f( z) ) ( 2(zf (z)) 1 α f (z) +f ( z) ) h(z) ( 2wg α (w) g (w) g( w) ) ( 2(wg (w)) 1 α g (w) +g ( w) ) p(w) where g(w) = f 1 (w). (34) We note that for α=1theclassm SΣ (α h) reduces to the class S S (h). Definition 11. One notes that for α = 0onegetstheclass C S (h) which is defined as follows: 2(zf (z)) f (z) +f ( z) h(u) 2(wg (w)) g (w) +g ( w) p(u). (35)

4 4 Function Spaces Theorem 12. Let f given by (2) be in the class m SΣ (α h).then a 2 min + p 2 8 (2 α) 2 h + p 8(3 3α+α 2 ) min + p 2 8 (2 α) 2 + h + p 8 (3 2α) (36) (6 5α + α 2 ) h +(α α2 ) p 8 (3 2α)(3 3α + α 2 ). (37) Proof. Let f m SΣ (α h) and g be the analytic extension of f 1 to U.Wehave ( 2zf α (z) f (z) f( z) ) ( 2(zf (z)) 1 α f (z) +f ( z) ) =1+2(2 α) a 2 z +[2(3 2α) a 3 2α(1 α) a 2 2 ]z2 + ( 2wg α (w) g (w) g( w) ) ( 2(wg (w)) 1 α g (w) +g ( w) ) =1 2(2 α) a 2 w +[2(3 2α) (2a 2 2 a 3) 2α(1 α) a 2 2 ]w2 +. (38) It follows from (34) that ( 2zf α (z) f (z) f( z) ) ( 2(zf (z)) 1 α f (z) +f ( z) ) =h(z) (z U) ( 2wg α (w) g (w) g( w) ) ( 2(wg (w)) 1 α g (w) +g ( w) ) =p(w) (w U) (39) where h(z) and p(w) satisfy the conditions of Definition 1. From (39)we deduce 2 (2 α) a 2 =h 1 (40) 2 (3 2α) a 3 2α(1 α) a 2 2 =h 2 (41) 2 (2 α) a 2 =p 1 (42) 2 (3 2α) (2a 2 2 a 3) 2α(1 α) a 2 2 =p 2. (43) From (40) and (42) we obtain By adding (41) to (43)weget h 1 = p 1 (44) 8 (2 α) 2 a 2 2 =h2 1 +p2 1. (45) 4(3 3α+α 2 )a 2 2 =h 2 +p 2 (46) which gives us the desired estimate on a 2 as asserted in (36). Subtracting (43) from (41) we have 4 (3 2α) a 3 4(3 2α) a 2 2 =h 2 p 2. (47) Then in view of (45) and (46)itfollowsthat a 3 = a 3 = h2 1 +p2 1 8 (2 α) 2 + h 2 p 2 4 (3 2α) h 2 +p 2 4(3 3α+α 2 ) + h 2 p 2 4 (3 2α) as claimed. This completes the proof of Theorem 12. Taking α=0we get the following. Corollary 13. If f C S (h) then a 2 min + p 2 32 min + p Corollary 14. If we let h + p 24 φ (z) =( 1+z 1 z ) β =1+2βz+2β 2 z 2 + then inequalities (36) and (37)become a 2 h + p 24 h 12 β min 2 α β 3 3α+α 2. min (2 α) α β 2 3 3α+α 2. Corollary 15. If we let φ (z) = 1+(1 2β)z =1+2(1 β)z 1 z +2(1 β)z 2 + (0 β<1) (48) (49) (50) (0<β 1) (51) (52) (53)

5 Function Spaces 5 then inequalities (36) and (37)become a 2 min 1 β 2 α 1 β 3 3α+α 2 β)2 min (1 (2 α) β 3 2α 1 β 3 3α+α 2. (54) Remark 16. Corollaries 14 and 15 provide an improvement of the estimate a 3 obtained by Altınkaya and Yalçın [25]. Remark 17. The estimates on the coefficients a 2 and a 3 of Corollaries 14 and 15 areimprovementoftheestimates obtained in [7]. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References [1] P. L. Duren Univalent Functions vol.259ofgrundlehren der Mathematischen Wissenschaften Springer New York NY USA [2] M. Lewin On a coefficient problem for bi-univalent functions Proceedings of the American Mathematical Society vol.18pp [3] E. Netanyahu The minimal distance of the image boundary from the orijin and the second coefficient of a univalent function in z < 1 Archive for Rational Mechanics and Analysisvol.32pp [4] D. A. Brannan and J. G. Clunie Aspects of comtemporary complex analysis in Proceedings of the NATO Advanced Study Institute Held at University of Durham: July Academic Press New York NY USA [5] D.A.BrannanandT.S.Taha Onsomeclassesofbi-univalent functions Studia Universitatis Babeş-Bolyai. Mathematicavol. 31 no. 2 pp [6] S. Altinkaya and S. Yalcin Initial coefficient bounds for a general class of biunivalent functions International Analysis vol Article ID pages [7] O. Crişan Coefficient estimates for certain subclasses of biunivalent functions General Mathematics Notes vol.16no.2 pp [8] B. A. Frasin and M. K. Aouf New subclasses of bi-univalent functions Applied Mathematics Lettersvol.24no.9pp [9] B. S. Keerthi and B. Raja Coefficient inequality for certain new subclasses of analytic bi-univalent functions Theoretical Mathematics and Applicationsvol.3no.1pp [10] S. S. Kumar V. Kumar and V. Ravichandran Estimates for the initial coefficients of Bi-univalent functions [11] N. Magesh and J. Yamini Coefficient bounds for certain subclasses of bi-univalent functions International Mathematical Forumvol.8no.25 28pp [12] R. M. Ali S. K. Lee V. Ravichandran and S. Supramaniam Coefficient estimates for bi-univalent Ma-MINda starlike and convex functions Applied Mathematics Letters vol. 25 no. 3 pp [13] H. M. Srivastava A. K. Mishra and P. Gochhayat Certain subclasses of analytic and bi-univalent functions Applied Mathematics Letters vol. 23 no. 10 pp [14]H.M.SrivastavaS.BulutM.ÇaglarandN.Yagmur Coefficient estimates for a general subclass of analytic and biunivalent functions Filomatvol.27no.5pp [15] Q.-H. Xu Y.-C. Gui and H. M. Srivastava Coefficient estimates for a certain subclass of analytic and bi-univalent functions Applied Mathematics Lettersvol.25no.6pp [16] S. Altinkaya and S. Yalcin Coefficient bounds for a subclass of bi-univalent functions TWMS Pure and Applied Mathematics.Inpress. [17] S. Bulut Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions Comptes Rendus Mathematiquevol.352no.6pp [18] S. G. Hamidi and J. M. Jahangiri Faber polynomial coefficient estimates for analytic bi-close-to-convex functions Comptes Rendus Mathematiquevol.352no.1pp [19] J. M. Jahangiri and S. G. Hamidi Coefficient estimates for certain classes of bi-univalent functions International Journal of Mathematics and Mathematical Sciencesvol.2013ArticleID pages2013. [20] J. M. Jahangiri S. G. Hamidi and S. A. Halim Coefficients of bi-univalent functions with positive real part derivatives Bulletin of the Malaysian Mathematical Sciences Societyvol.37 no.3pp [21] W. C. Ma and D. Minda A unified treatment of some special classes of univalent functions in Proceedings of the Conference on Complex Analysis (Tianjin 1992) Z. Li F. Ren L. Yang and S. Zhang Eds. Conference Proceedings and Lecture Notes in Analysis pp International Press Cambridge Mass USA [22] K. Sakaguchi On a certain univalent mapping the Mathematical Society of Japanvol.11no.1pp [23] Z.-G. Wang C.-Y. Gao and S.-M. Yuan On certain subclasses of close-to-convex and quasi-convex functions with respect to k-symmetric points Mathematical Analysis and Applicationsvol.322no.1pp [24] V. Ravichandran Starlike and convex functions with respect to conjugate points Acta Mathematica: Academiae Paedagogicae Nyıregyháziensisvol.20no.1pp [25] Ş. Altınkaya and S. Yalçın Fekete-Szegö Inequalities for certain classes of bi-univalent functions International Scholarly Research Noticesvol.2014ArticleID pages2014.

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