FABER POLYNOMIAL COEFFICIENT ESTIMATES FOR A NEW SUBCLASS OF MEROMORPHIC BI-UNIVALENT FUNCTIONS ADNAN GHAZY ALAMOUSH, MASLINA DARUS

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1 Available online at Adv. Inequal. Appl. 216, 216:3 ISSN: FABER POLYNOMIAL COEFFICIENT ESTIMATES FOR A NEW SUBCLASS OF MEROMORPHIC BI-UNIVALENT FUNCTIONS ADNAN GHAZY ALAMOUSH, MASLINA DARUS School of Mathematical Sciences, Faculty of Science Technology, Universiti Kebangsaan Malaysia, Bangi 436 Selangor D. Ehsan, Malaysia Copyright c 216 Alamoush Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited. Abstract. In this article, we introduce a new subclass of meromorphic bi-univalent functions obtain the general coefficient estimates for such functions in this class. For this purpose, we use the Faber polynomial expansions. In certain cases, our estimates improve some of existing coefficient bounds. Keywords: Meromorphic univalent functions; Meromorphic bi-univalent functions; Faber polynomial expansions. 21 AMS Subject Classification: 3C Introduction Let Σ denote the class of meromorphic univalent functions f of the form f (z) = z + b n z n (1.1) defined on the domain = {z : z C 1 < z < }. It is well known that every function f Σ has an inverse f 1, defined by Corresponding author Received August 3, 215 f 1 ( f (z)) = z,(z ), 1

2 2 ADNAN GHAZY ALAMOUSH, MASLINA DARUS f ( f 1 (w)) = w,(m < w <, M > ). For f Σ given by (1.1), the inverse map g = f 1 has the following Faber polynomial expansion: where f 1 (w) = g(w) = w + B n w n = w b b 1 w b 1b + b 2 w 2 b2 1 + b 1b 2 + 2b b 2 + b 3 w = w b n 1 1 n Kn n+1 1 w n, (w ), (1.2) Kn+1 n = nbn 1 b 1 + n(n 1)b n 2 b n(n 1)(n 2)bn 3 2 (b 3 + b 2 1) n(n 1)(n 2)(n 3) + b n 4 (b 4 + 3b 1 b 2 ) + 3! b n k V k, k 5 (1.3) V k with 5 k n is a homogeneous polynomial of degree k in the variables b 1,b 2,...,b n. (See [1], [2], [3]). Analogous to the bi-univalent analytic functions, a function f Σ is said to be meromorphic bi-univalent if f 1 Σ. The class of all meromorphic bi-univalent functions is denoted by Σ M. The coefficient problem was investigated for various subclasses of the meromorphic univalent functions, for example, Schiffer [4] obtained the estimate b for meromorphic univalent functions f Σ with b =. In 1983, Duren [5] obtained the inequality b n 2 n+1 for f Σ with b k =, 1 k < 2 n. For the coefficient of the inverse of meromorphic univalent functions g Σ M, Springer [6] proved that conjectured that B 3 1 B B , B 2n 1 (2n 1)! n!(n 1)! (n = 1,2,...). In 1977, Kubota [7] has proved that the Springer conjecture is true for n = 3,4,5, subsequently Schober [8] obtained sharp bounds for B 2n 1 if 1 n 7. In 27, Kapoor Mishra [9] found the coefficient estimates for the inverse of meromorphic starlike univalent functions of order α in. In 211, Srivastava et al. [1] found sharp bounds for the coefficients of

3 A NEW SUBCLASS OF MEROMORPHIC BI-UNIVALENT FUNCTIONS 3 inverses of starlike univalent functions of order α( α < 1) having m-fold gap series representation. The interests in this direction is increasing. Recently, Hamidi et al. [11] introduced the following class of meromorphic bi-univalent functions f Σ used the Faber polynomial expansions to obtain bounds for the general coefficients b n of meromorphic bi-univalent functions in the class BΣ(α,λ). 2. Preliminaries Definition 2.1. [11] For α < 1 λ 1, let BΣ(α,λ) be the class of meromorphic bi-univalent functions f Σ so that: ( R (1 λ) f (z) ) + λ f (z) > α, z z where the function g is given by (1.2). ( R (1 λ) g(w) ) w + λg (w) > α, w, Theorem 2.1. [11] Let f be given by (1.1). For α < 1 λ 1 if f BΣ(α;λ) b k = ; k n 1,then: b n λ(n + 1) 1 ; n 1, b λ 1, b 1 2λ 1, b 2 3λ 1, b 2 + b b 1 3λ 1. For functions f Σ in the form (1.1), we define the following new linear operator D λ f (z) = f (z) D 1 f (z) = D f (z) = z f (z) = z + [ n] b n z n, D 2 f (z) = D[D f (z)] = z + [ n] 2 b n z n,

4 4 ADNAN GHAZY ALAMOUSH, MASLINA DARUS hence, it can be easily seen that D k f (z) = D[D k 1 f (z)] = z + [ n] k b n z n, (k N ). (3.1) For α < 1, λ k N, let BΣ(k,α,λ) be the class of meromorphic bi-univalent functions f Σ so that ( (1 λ)d k f (z) + λd k+1 ) f (z) R > α z ( (1 λ)d k g(w) + λd k+1 ) g(w) R > α w where z,w the function g = f 1 is given by (1.2). In this paper, motivated by the previous works, we shall use the differential linear operator D k f (z) given above to obtain our results. Similar to the work done by [11], we use the Faber polynomial expansions to obtain bounds for the general coefficients b n of meromorphic biunivalent functions in BΣ(k,α,λ). 3. Main results Our first theorem introduces an upper bound for the coefficients b n of meromorphic biunivalent functions in BΣ(k,α,λ). Theorem 3.1. For α < 1, λ 1, k N, let f be given by (1.1). If f BΣ(α;λ) b k = ;(n 1 k ), then b n [(n + 1)λ 1] k ; n 1. (3.1) Proof. For meromorphic functions f BΣ(k,α,λ) of the form (1.1), we have (1 λ)d k f (z) + λd k+1 f (z) z for its inverse map, g = f 1,we have (1 λ)d k g(w) + λd k+1 g(w) w = 1 + = 1 + [1 λ(n + 1)] k b n, zn+1 (3.2) [1 λ(n + 1)] k B n w n+1

5 A NEW SUBCLASS OF MEROMORPHIC BI-UNIVALENT FUNCTIONS 5 = 1 (1 λ) k b w [1 λ(n + 1)] k 1 n Kn n+1 (b 1,b 2,...,b n ) w n+1. (3.3) On the other h, since f BΣ(k,α,λ) g = f 1 BΣ(k,α,λ), by definition, there exist two positive real part functions p(z) = 1 + q(w) = 1 + where R{p(z)} > R{q(w)} > in so that (1 λ)d k f (z) + λd k+1 f (z) z (1 λ)d k g(w) + λd k+1 g(w) w Note that, by the Caratheodory Lemma (e.g., [5]), c n z n Σ d n w n Σ, = α + (1 α)p(z) = 1 + (1 α) = α + (1 α)q(w) = 1 + (1 α) c n 2 d n 2 (n N). K 1 n(c 1,c 2,...,c n )z n, (3.4) K 1 n(d 1,d 2,...,d n )w n. (3.5) Comparing the corresponding coefficients of (3.2) (3.4), for any n 2, yields (1 λ(n + 1)) k b n = (1 α)k 1 n+1(c 1,c 2,...,c n+1 ), (3.6) similarly, from (3.3) (3.5) we find (1 λ(n + 1)) k B n = (1 α)k 1 n+1(d 1,d 2,...,d n+1 ). (3.7) Note that for b k = ; ( k n 1), we have B n = b n so (1 λ(n + 1)) k b n = (1 α)c n+1, (3.8) (1 λ(n + 1)) k b n = (1 α)d n+1. (3.9)

6 6 ADNAN GHAZY ALAMOUSH, MASLINA DARUS Now taking the absolute values of the above equalities applying the Caratheodory lemma, we obtain b n = (1 α) c n+1 [1 λ(n + 1)] k = (1 α) d n+1 [1 λ(n + 1)] k [λ(n + 1) 1] k. which completes the proof of Theorem 3.1. Remark 3.1. Let k = 1, we have f BΣ(1,α,λ) = BΣ(α,λ), Theorem 3.1 reduced to Theorem 1.1 in [11]. By imposing coefficient restrictions on Theorem 3.1, we capture the initial Taylor-Maclaurin coefficients of functions f in BΣ(k,α,λ) as well as a bound for the coefficient combination of (b 2 + b b 1 ) in the following theorem: Theorem 3.2. For α < 1, λ 1, k N, let f BΣ(k,λ,α) be given by (1.1), then (i) b (ii) b 1 (iii) b 2 (iv) b 2 + b b 1 Proof. Comparing Eqs. (3.2) (3.4) for n =,1,2, we obtain: Also, from (3.3) (3.5), for n = 2 we have: (λ 1) k. (3.1) (2λ 1) k. (3.11) (3λ 1) k. (3.12) (3λ 1) k. (3.13) (1 λ) k b = (1 α)c 1, (3.14) (1 2λ) k b 1 = (1 α)c 2, (3.15) (1 3λ) k b 2 = (1 α)c 3, (3.16) (1 3λ) k (b 2 + b b 1 ) = (1 α)d 3. (3.17) Solving Eqs. (3.14), (3.15), (3.16) (3.17) for b,b 1,b 2 (b 2 + b b 1 ), respectively, taking their absolute values then applying the Caratheodory Lemma, we obtain: b (1 α) c 1 (1 λ) k (λ 1) k,

7 A NEW SUBCLASS OF MEROMORPHIC BI-UNIVALENT FUNCTIONS 7 b 1 (1 α) c 2 (1 2λ) k (2λ 1) k, b 2 (1 α) c 3 (1 3λ) k (3λ 1) k, b 2 + b b 1 (1 α) d 3 (1 3λ) k (3λ 1) k. Remark 3.2 From the above discussion it is understood that the estimates of b,b 1,b 2 (b 2 + b b 1 ) in Theorem 3.2 is the same as the corresponding estimates of Theorem 1.2 in [11], when k = 1. Conflict of Interests The authors declare that there is no conflict of interests. Acknowledgement The authors would like to acknowledge appreciate the financial support received from MO- HE: FRGSTOPDOWN/213/ST6/UKM/1/1. REFERENCES [1] H. Airault, A. Bouali, Differential calculus on the Faber polynomials, Bull. Sci. Math. 13 (26), [2] H. Airault, J. Ren, An algebra of differential operators generating functions on the set of univalent functions, Bull. Sci. Math. 126 (22), [3] P. G. Todorov, On the Faber polynomials of the univalent functions of class Σ, J. Math. Anal. Appl. 162 (1991), [4] M. Schiffer, Sur un problème d, xtrémum de la représentation conforme, Bull. Soc. Math. France. 66 (1938), [5] P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer, New York, (1983). [6] G. Springer, The coefficient problem for schlicht mappings of the exterior of the unit circle, Trans. Amer. Math. Soc. 7 (1951), [7] Y. Kubota, Coefficients of meromorphic univalent functions, Kōdai Math. Sem. Rep. 28 (1976), [8] G. Schober, Coefficients of inverses of meromorphic univalent functions, Proc. Amer. Math. Soc. 67 (1977),

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