Filomat 9:8 (015), 1839 1845 DOI 10.98/FIL1508839S Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Coefficient Bounds for a Certain Class of Analytic Bi-Univalent Functions H. M. Srivastava a, Sevtap Sümer Eker b, Rosihan M. Ali c a Department of Mathematics Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada b Department of Mathematics, Faculty of Science, Dicle University, TR-180 Diyarbakır, Turkey c School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia Abstract. In this paper, we introduce investigate a subclass of analytic bi-univalent functions in the open unit disk U. By using the Faber polynomial expansions, we obtain upper bounds for the coefficients of functions belonging to this analytic bi-univalent function class. ome interesting recent developments involving other subclasses of analytic bi-univalent functions are also briefly mentioned. 1. Introduction Let A denote the class of functions f (z) which are analytic in the open unit disk U = {z : z C z < 1} normalized by the following Taylor-Maclaurin series expansion: f (z) = z a n z n. (1.1) Also let S denote the subclass of functions in A which are univalent in U (see, for details, [8]). It is well known that every function f S has an inverse f 1, which is defined by f 1( f (z) ) = z (z U) f ( f 1 (w) ) = w ( w < r 0 ( f ); r 0 ( f ) 1 ), (1.) 4 010 Mathematics Subject Classification. Primary 30C45, 30C50; Secondary 30C80 Keywords. Analytic functions; Univalent functions; Faber polynomials; Bi-Univalent functions; Taylor-Maclaurin series expansion; Coefficient bounds coefficient estimates; Taylor-Maclaurin coefficients. Received: 06 November 013; Accepted: 1 April 014 Communicated by Dragan S. Djordjević Email addresses: harimsri@math.uvic.ca (H. M. Srivastava), sevtaps35@gmail.com (Sevtap Sümer Eker), rosihan@cs.usm.my (Rosihan M. Ali)
H. M. Srivastava et al. / Filomat 9:8 (015), 1839 1845 1840 according to the Koebe One-Quarter Theorem (see, for example, [8]). In fact, the inverse function f 1 is given by f 1 (w) = w a w ( a a ) 3 w 3 ( 5a 3 5a a 3 a 4 ) w 4. (1.3) A function f A is said to be bi-univalent in U if both f (z) f 1 (z) are univalent in U. Let denote the class of analytic bi-univalent functions in U given by the Taylor-Maclaurin series expansion (1.1). Some examples of functions in the class are presented below: ( ) z 1 z, log(1 z), 1 1 z log, 1 z so on. However, the familiar Koebe function is not a member of the class. Other common examples of functions in S such as z z z 1 z are also not members of the class. For a brief history of functions in the class, see [] (see also [4], [14], [18] [5]). In fact, judging by the remarkable flood of papers on the subject (see, for example, [5 7, 9 1, 15 17, 19 1, 3, 6, 7, 9, 30]), the recent pioneering work of Srivastava et al.[] appears to have revived the study of analytic bi-univalent functions in recent years (see also [3], [13] [4]). The object of the present paper is to introduce a new subclass of the function class use the Faber polynomial expansion techniques to derive bounds for the general Taylor-Maclaurin coefficients a n for the functions in this class. We also obtain estimates for the first two coefficients a a 3 of these functions.. Bounds Derivable by the Faber Polynomial Expansion Techniques We begin by introducing the function class N (α,λ) by means of the following definition. Definition. A function f (z) given by (1.1) is said to be in the class N (α,λ) (0 α < 1; λ 0) if the following conditions are satisfied: f R { f (z) λz f (z) } > α (z U; 0 α < 1; λ 0). (.1) By using the Faber polynomial expansions of functions f A of the form (1.1), the coefficients of its inverse map = f 1 may be expressed as follows (see [1] []; see also [1]): where (w) = f 1 (w) = w K n n 1 = ( n 1)!(n 1)! an 1 1 n K n n 1 (a, a 3,, a n ) w n. (.) (( n 1))!(n 3)! an 3 a 3 ( n 3)!(n 4)! an 4 a 4 [ (( n ))!(n 5)! an 5 a5 ( n )a3] ( n 5)!(n 6)! an 6 [a 6 ( n 5)a 3 a 4 ] a n j V j, j 7
H. M. Srivastava et al. / Filomat 9:8 (015), 1839 1845 1841 where such expressions as (for example) are to be interpreted symbolically by Γ(1 n) := ( n)( n 1)( n ) ( n N0 := N {0} (N := {1,, 3, }) ) (.3) V j (7 j n) is a homogeneous polynomial in the variables a, a 3,, a n (see, for details, []). In particular, the first three terms of K n are given below: n 1 K 1 = a, K 3 = 3 ( a a ) 3 K 4 3 = 4 ( 5a 3 5a a 3 a 4 ). In general, an expansion of K p n is given by (see, for details, [1]) where K p n = pa n p(p 1) D n p! (p 3)!3! D3 n Z := {0, ±1, ±, } D p n = D p n (a, a 3, ), alternatively, by (see, for details, [8]) D m n (a 1, a,, a n ) = ( ) m! a µ 1 µ 1! µ n! 1 aµ n n, p! (p n)!n! Dn n (p Z) where a 1 = 1 the sum is taken over all nonnegative integers µ 1,, µ n satisfying the following conditions: µ 1 µ µ n = m It is clear that µ 1 µ nµ n = n. D n n(a 1, a,, a n ) = a n 1. Our first main result is given by Theorem 1 below. Theorem 1. Let f given by (1.1) be in the class N α,λ (0 α < 1 λ 0). If a k = 0 for k n 1, then a n n[1 λ(n 1)] (n N \ {1, }). (.4) Proof. For analytic functions f of the form (1.1), we have f (z) λz f (z) = 1 [1 λ(n 1)]na n z n 1 (.5)
, for its inverse map = f 1, it is seen that (w) λw (w) = 1 = 1 H. M. Srivastava et al. / Filomat 9:8 (015), 1839 1845 184 [1 λ(n 1)]nb n w n 1 [1 λ(n 1)]K n n 1 (a, a 3,, a n ) w n 1. (.6) On the other h, since f N α,λ functions p(z) = 1 c n z n = f 1 N α,λ, by definition, there exist two positive real-part q(w) = 1 d n w n, where so that R ( p(z) ) > 0 R ( q(w) ) > 0 (z, w U), f (z) λz f (z) = α (1 α)p(z) = 1 (1 α) Kn 1 (c 1, c,, c n ) z n (.7) (w) λw (w) = α (1 α)q(w) = 1 (1 α) Kn 1 (d 1, d,, d n ) w n. (.8) Thus, upon comparing the corresponding coefficients in (.5) (.7), we get [1 λ(n 1)]na n = (1 α)k 1 n 1 (c 1, c,, c n 1 ). (.9) Similarly, by using (.6) (.8), we find that so [1 λ(n 1)]K n n 1 (a 1, a,, a n ) = (1 α)k 1 n 1 (d 1, d,, d n 1 ). (.10) We note that, for a k = 0 ( k n 1), we have b n = a n [1 λ(n 1)]na n = (1 α)c n 1 (.11) [1 λ(n 1)]na n = (1 α)d n 1. (.1)
H. M. Srivastava et al. / Filomat 9:8 (015), 1839 1845 1843 Thus, according to the Carathéodory Lemma (see [8]), we also observe that c n d n (n N). Now, taking the moduli in (.11) (.1) applying the Carathéodory Lemma, we obtain a n (1 α) c n 1 n[1 λ(n 1)] = (1 α) d n 1 n[1 λ(n 1)] n[1 λ(n 1)], (.13) which evidently completes the proof of Theorem 1. 3. Estimates for the Initial Coefficients a a 3 In this section, we choose to relax the coefficient restrictions imposed in Theorem 1 derive the resulting estimates for the initial coefficients a a 3 of functions f N α,λ given by the Taylor-Maclaurin series expansion (1.1). We first state the following theorem. Theorem. Let f given by (1.1) be in the class N α,λ (0 α < 1 λ 0). Then a 1 λ λ, 0 α < 3(1 λ) 3(1 λ) 1 α 1 λ, 1 λ λ α < 1 3(1 λ) (3.1) a 3 3(1 λ). (3.) Proof. If we set n = by n = 3 in (.9) (.10), respectively, we obtain (1 λ)a = (1 α)c 1, (3.3) 3(1 λ)a 3 = (1 α)c, (1 λ)a = (1 α)d 1 (3.5) (3.4) 3(1 λ)(a a 3) = (1 α)d. (3.6) Upon dividing both sides of (3.3) or (3.5) by (1λ), if we take their moduli apply the Carathéodory Lemma, we find that a (1 α) c 1 (1 λ) = (1 α) d 1 (1 λ) Now, by adding (3.4) to (3.6), we have 1 α 1 λ. (3.7) 6(1 λ)a = (1 α)(c d ), (3.8)
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