Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2013, Article ID 190560, pages http://dx.doi.org/10.1155/2013/190560 Research Article Coefficient Estimates for Certain Classes of Bi-Univalent Functions Jay M Jahangiri1 and Samaneh G Hamidi2 Department of Mathematical Sciences, Kent State University, Burton, OH 44021-9500, USA Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia Correspondence should be addressed to Jay M Jahangiri; jjahangi@kent.edu Received 29 April 2013; Revised 27 July 2013; Accepted 31 July 2013 Academic Editor: Heinrich Begehr Copyright © 2013 J M Jahangiri and S G Hamidi 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 A function analytic in the open unit disk D is said to be bi-univalent in D if both the function and its inverse map are univalent there The bi-univalency condition imposed on the functions analytic in D makes the behavior of their coefficients unpredictable Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions We use Faber polynomial expansions of bi-univalent functions to obtain estimates for their general coefficients subject to certain gap series as well as providing bounds for early coefficients of such functions Introduction Let A denote the class of functions 𝑓 which are analytic in the open unit disk D := {𝑧 ∈ C : |𝑧| < 1} and normalized by ∞ 𝑓 (𝑧) = 𝑧 + ∑ 𝑎𝑛 𝑧𝑛 (1) 𝑛=2 Let S denote the class of functions 𝑓 ∈ A that are univalent in D and let P be the class of functions 𝑝(𝑧) = 𝑛 + ∑∞ 𝑛=1 𝑝𝑛 𝑧 that are analytic in D and satisfy the condition Re(𝑝(𝑧)) > in D By the Caratheodory lemma (e.g., see [1]) we have |𝑝𝑛 | ≤ For ≤ 𝛼 < and 𝜆 ≥ we let D(𝛼; 𝜆) denote the family of analytic functions 𝑓 ∈ A so that Re ((1 − 𝜆) 𝑓 (𝑧) + 𝜆𝑓 (𝑧)) > 𝛼, 𝑧 𝑧 ∈ D (2) We note that D(0; 1) is the class of bounded boundary turning functions and also that D(𝛼; 𝜆) ⊂ D(𝛽; 𝜆) if ≤ 𝛽 < 𝛼 For 𝑓 ∈ A, the class D(𝛼; 𝜆) ⊂ S and was first defined and investigated by Ding et al [2] It is well known that every function 𝑓 ∈ S has an inverse 𝑓−1 satisfying 𝑓−1 (𝑓(𝑧)) = 𝑧 for 𝑧 ∈ D and 𝑓(𝑓−1 (𝑤)) = 𝑤 for |𝑤| < 1/4, according to Kobe One Quarter Theorem (e.g., see [1]) A function 𝑓 ∈ A is said to be bi-univalent in D if both 𝑓 ∈ S and 𝑔 = 𝑓−1 ∈ S Finding bounds for the coefficients of classes of bi-univalent functions dates back to 1967 (see Lewin [3]) But the interest on the bounds for the coefficients of classes of bi-univalent functions picked up by the publications of Brannan and Taha [4], Srivastava et al [5], Frasin and Aouf [6], Ali et al [7], and Hamidi et al [8] The bi-univalency condition imposed on the functions 𝑓 ∈ A makes the behavior of their coefficients unpredictable Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions, as Ali et al [7] also remarked that finding the bounds for |𝑎𝑛 | when 𝑛 ≥ is an open problem Here in this paper we let 𝑓 ∈ D(𝛼; 𝜆) and 𝑔 = 𝑓−1 ∈ D(𝛼; 𝜆) and use the Faber polynomial coefficient expansions to provide bounds for the general coefficients |𝑎𝑛 | of such functions with a given gap series We also obtain estimates for the first two coefficients |𝑎2 | and |𝑎3 | of these functions as well as providing an estimate for their coefficient body (𝑎2 , 𝑎3 ) The bounds provided in this paper prove to be better than those estimates provided by Srivastava et al [5] and Frasin and Aouf [6] 2 International Journal of Mathematics and Mathematical Sciences Main Results Using the Faber polynomial expansion of functions 𝑓 ∈ A of the form (1), the coefficients of its inverse map 𝑔 = 𝑓−1 may be expressed as, [9], ∞ −𝑛 𝑔 (𝑤) = 𝑓−1 (𝑤) = 𝑤 + ∑ 𝐾𝑛−1 (𝑎2 , 𝑎3 , ) 𝑤𝑛 , 𝑛 𝑛=2 (3) (1 − 𝛼) ; 𝑎𝑛 ≤ + (𝑛 − 1) 𝜆 𝑓 (𝑧) + 𝜆𝑓 (𝑧) 𝑧 (10) = + ∑ (1 + (𝑛 − 1) 𝜆) 𝑎𝑛 𝑧𝑛−1 , 𝑛=2 (−𝑛)! + 𝑎𝑛−3 𝑎 (2 (−𝑛 + 1))! (𝑛 − 3)! and for its inverse map, 𝑔 = 𝑓−1 , we have (−𝑛)! 𝑎𝑛−4 𝑎 (−2𝑛 + 3)! (𝑛 − 4)! (1 − 𝜆) (−𝑛)! + 𝑎𝑛−5 [𝑎5 + (−𝑛 + 2) 𝑎32 ] (2 (−𝑛 + 2))! (𝑛 − 5)! (4) 𝑔 (𝑤) + 𝜆𝑔 (𝑤) 𝑤 ∞ = + ∑ (1 + (𝑛 − 1) 𝜆) 𝑏𝑛 𝑤𝑛−1 𝑛=2 (11) ∞ (−𝑛)! + 𝑎𝑛−6 [𝑎6 + (−2𝑛 + 5) 𝑎3 𝑎4 ] (−2𝑛 + 5)! (𝑛 − 6)! + (9) ∞ (−𝑛)! 𝑎𝑛−1 (−2𝑛 + 1)! (𝑛 − 1)! + 𝑛 ≥ Proof For analytic functions 𝑓 of the form (1) we have (1 − 𝜆) where −𝑛 = 𝐾𝑛−1 Theorem For ≤ 𝛼 < and 𝜆 ≥ let 𝑓 ∈ D(𝛼; 𝜆) and 𝑔 ∈ D(𝛼; 𝜆) If 𝑎𝑘 = 0; ≤ 𝑘 ≤ 𝑛 − 1, then = + ∑ (1 + (𝑛 − 1) 𝜆) 𝑛=2 𝑛−𝑗 ∑ 𝑎2 𝑉𝑗 , 𝑗≥7 −𝑛 × 𝐾𝑛−1 (𝑎2 , 𝑎3 , , 𝑎𝑛 ) 𝑤𝑛−1 𝑛 such that 𝑉𝑗 with ≤ 𝑗 ≤ 𝑛 is a homogeneous polynomial in the variables 𝑎2 , 𝑎3 , , 𝑎𝑛 [10] In particular, the first three −𝑛 are terms of 𝐾𝑛−1 −2 𝐾 = −𝑎2 , (1 − 𝜆) −3 𝐾 = 2𝑎22 − 𝑎3 , (5) In general, for any 𝑝 ∈ N, an expansion of 𝐾𝑛𝑝 is as, [9, page 183], 𝑝 (𝑝 − 1) 𝑝! 𝐷3 𝐷𝑛 + 𝐾𝑛𝑝 = 𝑝𝑎𝑛 + (𝑝 − 3)!3! 𝑛 𝑝! 𝐷𝑛 , (𝑝 − 𝑛)!𝑛! 𝑛 ∞ 𝜇 Evidently, 𝐷𝑛𝑛 (𝑎1 , 𝑎2 , , 𝑎𝑠+𝑚 ) = 𝑎1𝑛 , [13] 𝑛 (𝑐1 , 𝑐2 , , 𝑐𝑛 ) 𝑧 , 𝑔 (𝑤) + 𝜆𝑔 (𝑤) 𝑤 ∞ (13) 𝑛=1 (6) Comparing the corresponding coefficients of (10) and (12) yields (14) and similarly, from (11) and (13) we obtain 𝜇 (7) while 𝑎1 = 1, and the sum is taken over all nonnegative integers 𝜇1 , , 𝜇𝑛 satisfying 𝜇1 + 2𝜇2 + ⋅ ⋅ ⋅ + 𝑛𝜇𝑛 = 𝑛 𝛼) ∑ 𝐾𝑛1 𝑛=1 (𝑐1 , 𝑐2 , , 𝑐𝑛−1 ) , (1 + 𝜆 (𝑛 − 1)) 𝑎𝑛 = (1 − 𝛼) 𝐾𝑛−1 𝑚!(𝑎1 ) ⋅ ⋅ ⋅ (𝑎𝑛 ) 𝑛 , 𝜇1 ! ⋅ ⋅ ⋅ 𝜇𝑛 ! 𝑚=1 𝜇1 + 𝜇2 + ⋅ ⋅ ⋅ + 𝜇𝑛 = 𝑚, (1 − 𝜆) (12) ∞ = + (1 − 𝛼) ∑ 𝐾𝑛1 (𝑑1 , 𝑑2 , , 𝑑𝑛 ) 𝑤𝑛 where 𝐷𝑛𝑝 = 𝐷𝑛𝑝 (𝑎2 , 𝑎3 , ) and by [11] or [12], 𝐷𝑛𝑚 (𝑎1 , 𝑎2 , , 𝑎𝑛 ) = ∑ 𝑓 (𝑧) + 𝜆𝑓 (𝑧) 𝑧 = + (1 − −4 𝐾 = − (5𝑎23 − 5𝑎2 𝑎3 + 𝑎4 ) + ⋅⋅⋅ + On the other hand, since 𝑓 ∈ D(𝛼; 𝜆) and 𝑔 = 𝑓−1 ∈ D(𝛼; 𝜆), by definition, there exist two positive real part −𝑛 −𝑛 and 𝑞(𝑤) = + ∑∞ functions 𝑝(𝑧) = + ∑∞ 𝑛=1 𝑐𝑛 𝑧 𝑛=1 𝑑𝑛 𝑤 where Re 𝑝(𝑧) > and Re 𝑞(𝑤) > in D so that (8) −𝑛 (𝑏0 , 𝑏1 , , 𝑏𝑛 ) (1 + (𝑛 − 1) 𝜆) 𝐾𝑛−1 𝑛 = (1 − 𝛼) 𝐾𝑛−1 (15) (𝑑1 , 𝑑2 , , 𝑑𝑛−1 ) Note that for 𝑎𝑘 = 0; ≤ 𝑘 ≤ 𝑛 − we have 𝑏𝑛 = −𝑎𝑛 and so (1 + (𝑛 − 1) 𝜆) 𝑎𝑛 = (1 − 𝛼) 𝑐𝑛−1 , − (1 + (𝑛 − 1) 𝜆) 𝑎𝑛 = (1 − 𝛼) 𝑑𝑛−1 (16) International Journal of Mathematics and Mathematical Sciences Now taking the absolute values of either of the above two equations and applying the Caratheodory lemma, we obtain (1 − 𝛼) 𝑐𝑛−1 (1 − 𝛼) 𝑑𝑛−1 (1 − 𝛼) = ≤ 𝑎𝑛 ≤ |1 + (𝑛 − 1) 𝜆| |1 + (𝑛 − 1) 𝜆| + (𝑛 − 1) 𝜆 (17) Dividing (20) by (1 + 2𝜆), taking the absolute values of both sides, and applying the Caratheodory lemma yield (1 − 𝛼) 𝑐2 (1 − 𝛼) ≤ 𝑎3 = + 2𝜆 + 2𝜆 Dividing (22) by (1 + 2𝜆), taking the absolute values of both sides, and applying the Caratheodory lemma, we obtain (1 − 𝛼) 𝑎3 − 2𝑎22 ≤ + 2𝜆 Theorem For ≤ 𝛼 < and 𝜆 ≥ let 𝑓 ∈ D(𝛼; 𝜆) and 𝑔 ∈ D(𝛼; 𝜆) Then one has the following (i) (1 − 𝛼) + 2𝜆 − 𝜆2 { √ ; , ≤ 𝛼 < { { + 2𝜆 (1 + 2𝜆) 𝑎2 ≤ { + 2𝜆 − 𝜆 { { (1 − 𝛼) , ≤ 𝛼 < (1 + 2𝜆) { 1+𝜆 (ii) (1 − 𝛼) 𝑎3 ≤ + 2𝜆 (iii) (1 − 𝛼) 𝑎3 − 2𝑎22 ≤ + 2𝜆 (28) (29) Corollary For ≤ 𝛼 < let 𝑓 ∈ D(𝛼; 1) and 𝑔 ∈ D(𝛼; 1) Then one has the following (18) (i) Proof Replacing 𝑛 by and in (14) and (15), respectively, we deduce (ii) { (1 − 𝛼) , {√ 𝑎2 ≤ { { − 𝛼, { 0≤𝛼< ; ≤ 𝛼 < (30) (1 − 𝛼) 𝑎3 ≤ (1 + 𝜆) 𝑎2 = (1 − 𝛼) 𝑐1 , (19) (1 + 2𝜆) 𝑎3 = (1 − 𝛼) 𝑐2 , (20) Remark The above two estimates for |𝑎2 | and |𝑎3 | show that the bounds given in Theorem are better than those given by Srivastava et al ([5, page 1191, Theorem 2] and Frasin and Aouf [6, page 1572, Theorem 3.2]) − (1 + 𝜆) 𝑎2 = (1 − 𝛼) 𝑑1 , (21) References (22) [1] P L Duren, Univalent Functions, vol 259 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, NY, USA, 1983 [2] S S Ding, Y Ling, and G J Bao, “Some properties of a class of analytic functions,” Journal of Mathematical Analysis and Applications, vol 195, no 1, pp 71–81, 1995 [3] M Lewin, “On a coefficient problem for bi-univalent functions,” Proceedings of the American Mathematical Society, vol 18, pp 63–68, 1967 [4] D A Brannan and T S Taha, “On some classes of bi-univalent functions,” Studia Universitatis Babes¸-Bolyai Mathematica, vol 31, no 2, pp 70–77, 1986 [5] 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 1188–1192, 2010 [6] B A Frasin and M K Aouf, “New subclasses of bi-univalent functions,” Applied Mathematics Letters, vol 24, no 9, pp 1569– 1573, 2011 [7] 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 344–351, 2012 [8] S G Hamidi, S A Halim, and J M Jahangiri, “Faber polynomial coefficient estimates for meromorphic bi-starlike functions,” International Journal of Mathematics and Mathematical Sciences, vol 2013, Article ID 498159, pages, 2013 [9] H Airault and A Bouali, “Differential calculus on the Faber polynomials,” Bulletin des Sciences Math´ematiques, vol 130, no 3, pp 179–222, 2006 (1 + 2𝜆) (2𝑎22 − 𝑎3 ) = (1 − 𝛼) 𝑑2 Dividing (19) or (21) by (1 + 𝜆), taking their absolute values, and applying the Caratheodory lemma, we obtain (1 − 𝛼) 𝑐1 (1 − 𝛼) 𝑑1 (1 − 𝛼) = ≤ 𝑎2 ≤ 1+𝜆 1+𝜆 1+𝜆 (23) Adding (20) to (22) implies (1 + 2𝜆) 𝑎22 = (1 − 𝛼) (𝑐2 + 𝑑2 ) (24) or 𝑎22 = (1 − 𝛼) (𝑐2 + 𝑑2 ) (1 + 2𝜆) (25) An application of Caratheodory lemma followed by taking the square roots yields √2 (1 − 𝛼) (𝑐2 + 𝑑2 ) √2 (1 − 𝛼) ≤ 𝑎2 ≤ (1 + 2𝜆) + 2𝜆 (26) Now the bounds given in Theorem (i) for |𝑎2 | follow upon noting that if (1 + 2𝜆 − 𝜆2 )/2(1 + 2𝜆) ≤ 𝛼 < 1, then (1 − 𝛼) √2 (1 − 𝛼) ≤ 1+𝜆 + 2𝜆 (27) [10] H Airault and J Ren, “An algebra of differential operators and generating functions on the set of univalent functions,” Bulletin des Sciences Math´ematiques, vol 126, no 5, pp 343–367, 2002 [11] P G Todorov, “On the Faber polynomials of the univalent functions of class Σ,” Journal of Mathematical Analysis and Applications, vol 162, no 1, pp 268–276, 1991 [12] H Airault, “Symmetric sums associated to the factorization of Grunsky coefficients,” in Conference, Groups and Symmetries, Montreal, Canada, April 2007 [13] H Airault, “Remarks on Faber polynomials,” International Mathematical Forum Journal for Theory and Applications, vol 3, no 9-12, pp 449–456, 2008 International Journal of Mathematics and Mathematical Sciences Copyright of International Journal of Mathematics & Mathematical Sciences is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use