Faber polynomial coefficient bounds for a subclass of bi-univalent functions

Stud. Univ. Babeş-Bolyai Math. 6(06), No., 37 Faber polynomial coefficient bounds for a subclass of bi-univalent functions Şahsene Altınkaya Sibel Yalçın Abstract. In this ork, considering a general subclass of bi-univalent functions using the Faber polynomials, e obtain coefficient expansions for functions in this class. In certain cases, our estimates improve some of those existing coefficient bounds. Mathematics Subject Classification (00): 30C5, 30C50. Keyords: Analytic univalent functions, bi-univalent functions, Faber polynomials.. Introduction Let A denote the class of functions f U = { : < } ith in the form f() = + hich are analytic in the open unit disk a n n. (.) Let S be the subclass of A consisting of the form (.) hich are also univalent in U let P be the class of functions ϕ() = + ϕ n n that are analytic in U satisfy the condition Re (ϕ()) > 0 in U. By the Caratheodory s lemma (e.g., see []) e have ϕ n. The Koebe one-quarter theorem [] states that the image of U under every function f from S contains a disk of radius. Thus every such univalent function has an inverse f hich satisfies f ( f () ) =, n= f (f ()) =, ( U) ( < r 0 (f), r 0 (f) ),

38 Şahsene Altınkaya Sibel Yalçın here f () = a + ( a a 3 ) 3 ( 5a 3 5a a 3 + a ) +. A function f A is said to be bi-univalent in U if both f f are univalent in U. For a brief history interesting examples in the class Σ, see [7]. Lein [0] studied the class of bi-univalent functions, obtaining the bound.5 for modulus of the second coefficient a. Netanyahu [] shoed that max a = 3 if f Σ. Subsequently, Brannan Clunie [7] conjectured that a for f Σ. Brannan Taha [8] introduced certain subclasses of the bi-univalent function class Σ similar to the familiar subclasses. Recently, many authors investigated bounds for various subclasses of bi-univalent functions ([5], [0], [3], [8], [9], [], [], [7], [8], [9]). The Faber polynomials introduced by Faber [] play an important role in various areas of mathematical sciences, especially in geometric function theory. Grunsky [] succeeded in establishing a set of conditions for a given function hich are necessary in their totality sufficient for the univalency of this function, in these conditions the coefficients of the Faber polynomials play an important role. Schiffer [5] gave a differential equations for univalent functions solving certain extremum problems ith respect to coefficients of such functions; in this differential equation appears again a polynomial hich is just the derivative of a Faber polynomial (Schaeffer-Spencer [6]). Not much is knon about the bounds on the general coefficient a n for n. In the literature, there are only a fe orks determining the general coefficient bounds a n for the analytic bi-univalent functions ([6], [9], [5], [6], [7]). The coefficient estimate problem for each of a n ( n N\ {, } ; N = {,, 3,...}) is still an open problem. For f () F () analytic in U, e say that f subordinate to F, ritten f F, if there exists a Schar function u() = c n n ith u() < in U, such that f () = F (u ()). For the Schar function u () e note that c n <. (e.g. see Duren []). A function f Σ is said to be B Σ (µ, λ, ϕ), λ µ 0, if the folloing subordination hold ( ) µ ( ) µ f() f() ( λ) + λf () ϕ () (.) ( ) µ ( ) µ g() g() ( λ) + λg () ϕ () (.3) here g () = f (). In this paper, e use the Faber polynomial expansions to obtain bounds for the general coefficients a n of bi-univalent functions in B Σ (µ, λ, ϕ) as ell as providing estimates for the initial coefficients of these functions.

. Main results Faber polynomial coefficient bounds 39 Using the Faber polynomial expansion of functions f A of the form (.), the coefficients of its inverse map g = f may be expressed as, [3], here g () = f () = + n K n n= n (a, a 3,...) n, K n n = ( n + )! (n )! an + [ ( n + )]! (n 3)! an 3 a 3 + ( n + 3)! (n )! an a [ + [ ( n + )]! (n 5)! an 5 a5 + ( n + ) a 3] + ( n + 5)! (n 6)! an 6 [a 6 + ( n + 5) a 3 a ] + a n j V j, j 7 (.) such that V j ith 7 j n is a homogeneous polynomial in the variables a, a 3,..., a n []. In particular, the first three terms of K n n are K 3 K 3 K = a, = a a 3, (.) 3 = ( 5a 3 5a a 3 + a ). In general, for any p N, an expansion of K p n is as, [3], Kn p p (p ) = pa n + En + here En p = En p (a, a 3,...) by [], E m n (a, a,..., a n ) = p! (p 3)!3! E3 n +... + p! (p n)!n! En n, (.3) m! (a ) µ... (a n ) µn, (.) µ!...µ n! m= hile a =, the sum is taken over all nonnegative integers µ,..., µ n satisfying Evidently, E n n (a, a,..., a n ) = a n, []. µ + µ +... + µ n = m, (.5) µ + µ +... + nµ n = n.

0 Şahsene Altınkaya Sibel Yalçın Theorem.. For then λ µ 0, let f B Σ (µ, λ, ϕ). If a m = 0 ; m n, a n Proof. Let functions f given by (.). We have µ + (n )λ ; n (.6) ( ) µ ( ) µ f() f() ( λ) + λf () = + F n (a, a 3,..., a n ) a n n, (.7) ( ) µ ( ) µ g() g() ( λ) + λg () = + F n (A, A 3,..., A n ) a n n here n= F = (µ + λ)a, (.8) [ ] µ F = (µ + λ) a + a 3, [ ] (µ ) (µ ) F 3 = (µ + 3λ) a 3 + (µ ) a a 3 + a, 3! In general, (see [9]). On the other h, the inequalities (.) (.3) imply the existence of to positive real part functions u () = + c n n v () = + d n n here Reu () > 0 Rev () > 0 in P so that ( ) µ ( ) µ f() f() ( λ) + λf () = ϕ(u()) (.9) here ( ) µ ( ) µ g() g() ( λ) + λg () = ϕ(v()) (.0) ϕ(u()) = + k= n ϕ k En k (c, c,..., c n ) n, (.) n ϕ(v()) = + ϕ k En k (d, d,..., d n ) n. (.) k= Comparing the corresponding coefficients of (.9) (.) yields n [µ + (n )λ] a n = ϕ k En k (c, c,..., c n ), n (.3) k=

Faber polynomial coefficient bounds similarly, from (.0) (.) e obtain n [µ + (n )λ] b n = ϕ k En k (d, d,..., d n ), n. (.) k= Note that for a m = 0 ; m n e have b n = a n so [µ + (n )λ] a n = ϕ c n [µ + (n )λ] a n = ϕ d n No taking the absolute values of either of the above to equations using the facts that ϕ, c n d n, e obtain a n ϕ c n µ + (n )λ = ϕ d n µ + (n )λ µ + (n )λ. (.5) Theorem.. Let f B Σ (µ, λ, ϕ), λ µ 0. Then { } (i) a min µ + λ, 8 (µ + λ) (µ + ) { } (ii) a 3 min (µ + λ) + µ + λ, 8 (µ + λ) (µ + ) + (.6) µ + λ Proof. Replacing n by 3 in (.3) (.), respectively, e find that (µ + λ)a = ϕ c, (.7) [ ] µ (µ + λ) a + a 3 = ϕ c + ϕ c, (.8) (µ + +λ)a = ϕ d, (.9) [ ] µ + 3 (µ + λ) a a 3 = ϕ d + ϕ d (.0) From (.7) or (.9) e obtain a ϕ c µ + λ = ϕ d µ + λ µ + λ. (.) Adding (.8) to (.0) implies (µ + λ) (µ + ) a ( = ϕ (c + d ) + ϕ c + d ) or, equivalently, 8 a (µ + λ) (µ + ). (.) Next, in order to find the bound on the coefficient a 3, e subtract (.0) from (.8). We thus get (µ + λ) ( a 3 a ( ) = ϕ (c d ) + ϕ c d ) (.3) or a 3 = a + ϕ (c d ) a + (.) (µ + λ) µ + λ

Şahsene Altınkaya Sibel Yalçın Upon substituting the value of a from (.) (.) into (.), it follos that a 3 (µ + λ) + µ + λ a 3 8 (µ + λ) (µ + ) + µ + λ. If e put λ = in Theorem., e obtain the folloing consequence. Corollary.3. Let f B Σ (µ, ϕ), µ 0. Then a µ + a 3 (µ + ) + µ + Remark.. The above estimates for a a 3 sho that Corollary.3 is an improvement of the estimates given in Prema Keerthi ([3], Theorem 3.) Bulut ([9], Corollary 3). If e put µ = in Theorem., e obtain the folloing consequence. Corollary.5. Let f B Σ (λ, ϕ), λ. Then a λ + a 3 (λ + ) + + λ Remark.6. The above estimates for a a 3 sho that Corollary.5 is an improvement of the estimates given in Bulut ([9], Corollary ). References [] Airault, H., Symmetric sums associated to the factoriation of Grunsky coefficients, in Conference, Groups Symmetries, Montreal, Canada, 007. [] Airault, H., Remarks on Faber polynomials, Int. Math. Forum, 3(008), no. 9, 9-56. [3] Airault, H., Bouali, H., Differential calculus on the Faber polynomials, Bulletin des Sciences Mathematiques, 30(006), no. 3, 79-. [] Airault, H., Ren, J., An algebra of differential operators generating functions on the set of univalent functions, Bulletin des Sciences Mathematiques, 6(00), no. 5, 33-367. [5] Altinkaya, Ş., Yalcin, S., Initial coefficient bounds for a general class of bi-univalent functions, International Journal of Analysis, Article ID 86787, (0), pp. [6] Altinkaya, Ş., Yalcin, S., Coefficient bounds for a subclass of bi-univalent functions, TWMS Journal of Pure Applied Mathematics, 6(05), no., 80-85.

Faber polynomial coefficient bounds 3 [7] Brannan, D.A., Clunie, J., Aspects of contemporary complex analysis, Proceedings of the NATO Advanced Study Instute Held at University of Durham, Ne York: Academic Press, 979. [8] Brannan, D.A., Taha, T.S., On some classes of bi-univalent functions, Stud. Univ. Babeş-Bolyai Math., 3(986), no., 70-77. [9] Bulut, S., Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions, C.R. Acad. Sci. Paris, Ser. I, 35(0), no., 79-8. [0] Crişan, O., Coefficient estimates certain subclasses of bi-univalent functions, Gen. Math. Notes, 6(03), no., 93-00. [] Duren, P.L., Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Springer, Ne York, USA, 59(983). [] Faber, G., Über polynomische entickelungen, Math. Ann., 57(903), no. 3, 569-573. [3] Frasin, B.A., Aouf, M.K., Ne subclasses of bi-univalent functions, Applied Mathematics Letters, (0), 569-573. [] Grunsky, H., Koeffiientenbedingungen für schlict abbildende meromorphe funktionen, Math. Zeit., 5(939), 9-6. [5] Hamidi, S.G., Halim, S.A., Jahangiri, J.M., Faber polynomial coefficient estimates for meromorphic bi-starlike functions, International Journal of Mathematics Mathematical Sciences, 03, Article ID 9859, p. [6] Hamidi, S.G., Jahangiri, J.M., Faber polynomial coefficient estimates for analytic biclose-to-convex functions, C.R. Acad. Sci. Paris, Ser.I, 35(0), no., 7-0. [7] Jahangiri, J.M., Hamidi, S.G., Coefficient estimates for certain classes of bi-univalent functions, International Journal of Mathematics Mathematical Sciences, 03, Article ID 90560, p. [8] Keerthi, B.S., Raja, B., Coefficient inequality for certain ne subclasses of analytic biunivalent functions, Theoretical Mathematics Applications, 3(03), no., -0. [9] Li, X.F., Wang, A.P., To ne subclasses of bi-univalent functions, Internat. Math. Forum, 7(0), 95-50. [0] Lein, M., On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 8(967), 63-68. [] Magesh, N., Yamini, J., Coefficient bounds for a certain subclass of bi-univalent functions, International Mathematical Forum, 8(03), no. 7, 337-3. [] Netanyahu, E., The minimal distance of the image boundary from the origin the second coefficient of a univalent function in <, Archive for Rational Mechanics Analysis, 3(969), 00-. [3] Prema, S., Keerthi, B.S., Coefficient bounds for certain subclasses of analytic functions, J. Math. Anal., (03), no., -7. [] Poral, S., Darus, M., On a ne subclass of bi-univalent functions, J. Egypt. Math. Soc., (03), no. 3, 90-93. [5] Schiffer, M., A method of variation ithin the family of simple functions, Proc. London Math. Soc., (938), no., 3-9. [6] Schaeffer, A.C., Spencer, D.C., The coefficients of schlict functions, Duke Math. J., 0(93), 6-635. [7] Srivastava, H.M., Mishra, A.K., Gochhayat, P., Certain subclasses of analytic biunivalent functions, Applied Mathematics Letters, 3(00), no. 0, 88-9.

Şahsene Altınkaya Sibel Yalçın [8] Srivastava, H.M., Bulut, S., Çağlar, M., Yağmur, N., Coefficient estimates for a general subclass of analytic bi-univalent functions, Filomat, 7(03), no. 5, 83-8. [9] Xu, Q.H., Gui, Y.C., Srivastava, H.M., Coefficient estimates for a certain subclass of analytic bi-univalent functions, Applied Mathematics Letters, 5(0), 990-99. Şahsene Altınkaya Department of Mathematics Faculty of Arts Science Uludag University, Bursa, Turkey e-mail: sahsene@uludag.edu.tr Sibel Yalçın Department of Mathematics Faculty of Arts Science Uludag University, Bursa, Turkey e-mail: syalcin@uludag.edu.tr