A subclass of pseudo-type meromorphic bi-univalent functions

In this paper, In the present article, a new subclass of pseudo-type meromorphic bi-univalent functions is defined on △={z |:z∈C and 1<|z|<∞}, we derive estimates on the initial coefficient |b₀|, |b₁| and |b₂|. Relevant connections of the new results with various well-known results are indicated. Motivated by the earlier work of ( Srivastava , Janani ), in the present paper, we introduce a new subclasses of the class Σ′ and the estimates for the coefficients |b₀|,|b₁| and |b₂| are investigated. Some new consequences of the new results are also pointed out.


Introduction
Let A denote the class of functions f (z) of the form: Furthermore, bi-univalency concept is extended to the class of meromorphic functions de…ned on 4 = fz : z 2 C; 1 < jzj < 1g. For this aim, let denote the class of meromorphic univalent functions g of the form de…ned on the domain 4. It is well known that every function g 2 has an inverse g 1 = h, de…ned by g 1 (g(z)) = z (z 2 4); and g 1 (g(w)) = w (M < jwj < 1; M > 0); where (3) A simple computation shows that Comparing the initial coe¢ cients in (4), we …nd that The family of all meromorphic bi-univalent functions is denoted by 0 . Estimates on the co-e¢ cient of meromorphic univalent functions were investigated by some researchers recently; for example, Schi¤er [11] obtained the estimate jb 2 j < 3 2 for meromorphic univalent functions f 2 S with b 0 = 0. Also, Duren [12] obtained the inequality jb 2 j < 2 n+1 for f 2 S with b k = 0; 1 k n 2 . Springer [8] used variational methods to prove that proved that jB 3 j < 1 and jB 3 + and conjectured that jB 2n 1 j (2n 2)! n!(n 1)! (n = 1; 2; :::): Later on, Kubota [16] has proved that the Springer conjecture is true for n = 3;4;5. Furthermore Schober [7] obtained sharp bounds for jB 2n 1 j if 1 n 7. Recently. Kapoor and Mishra [5] found the coe¢ cient estimates for a class consisting of inverses of meromorphic starlike univalent functions of order in U .

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In 2013, Babalola [10] de…ned a new subclass -pseudo starlike function of order 0 < 1 satisfying the analytic condition In particular, Babalola [10] proved that all -pseudo-starlike functions are Bazilevic of type 1 1 and order 1 and are univalent in open unit disk U . Motivated by the earlier work of ( [9], [15]), in the present paper, we introduce a new subclasses of the class 0 and the estimates for the coe¢ cients jb 0 j; jb 1 j and jb 2 j are investigated. Some new consequences of the new results are also pointed out.

Coefficient Bounds for the Function Class
such that minf<(h(z)); <(p(z))g > 0; z 2 4: A function g(z) 2 0 given by (2) is said to be in the class 0 h;p ( ; ) if the following conditions are satis…ed: and where g 2 0 and 2 C n f0g and the function h is given by (3).
So it is easy to verify that the functions h(z) and p(z) satisfy the hypotheses of De…nition 2.1. If f 2 0 ( ; ). Then 1; z 2 4); and ; where g(z) 2 0 and 2 C n f0g and the function h is given by (3).
(2) If we take So it is easy to verify that the functions h(z) and p(z) satisfy the hypotheses of De…nition 2.1. If f 2 0 ( ; ). Then 1; w 2 4); where g 2 0 and 2 C n f0g and the function h is given by (3).
Theorem 2.1. Let g(z) be given by (2) be in the class 0 ( ; ). Then and and Proof. Let g 2 0 ( ; ). Then, by De…nition 2.1 of meromorphically bi-univalent function class 0 ( ; ), the conditions (6) and (7) can be rewritten as follows: and respectively. Here, and in what follows,the functions h(z) 2 P and p(w) 2 P have the following forms: and upon substituting from (13) and (14) into (11) and (12), respectively, and equating the coe¢ cients, we get and (15) and (18), we …nd that Adding (16) and (19), we get that is, From (23) and (25) we get the desired estimate on the coe¢ cient jb 0 j as asserted in (8).
Next, in order to …nd the bound on jb 0 j, by subtracting the equation (16) from the equation (19), we get that is, By squaring and adding (16) and (19), using (22) in the computation leads to that is, (29) From (26) and (28) we get the desired estimate on the coe¢ cient jb 1 j as asserted in (9).
In order to …nd the estimate jb 2 j, consider the sum of (17) and (20), we have Subtracting (20) from (17) with h 1 = p 1 , we obtain This evidently completes the proof of Theorem 2.1.
If we take = 1 in Theorem 2.1, we get the following Corollary.