Coefficient Estimates for Meromorphic Bi-Univalent Functions

Lemma 1. [7, Theorem 3, p. 80] The coefficient cn of a function p ∈P satisfy the sharp inequality |cn| ≤2 (n ≥1). 2. COEFFICIENT ESTIMATES In this section, certain subclasses of the class ΣB of meromorphic bi-univalent functions are introduced and estimates on the coefficient b0 and b1 for functions in these subclasses are obtained. Definition 1. A function g given by series expansion (2) is a meromorphic starlike bi-univalent functions of order α, 0 ≤α < 1, if Re zg′(z) g(z)  > α (z ∈∆), and

Theorem 1. If the function g given by (2) is a meromorphic starlike bi-univalent function of order α, 0 ≤α < 1, then the coefficients b0 and b1 satisfy the inequalities |b0| ≤2(1 −α), and |b1| ≤(1 −α) p 4α2 −8α + 5.

Theorem 2. If the function g given by (2) is in the class eΣ∗ B(α), 0 < α ≤1, then the coefficients b0 and b1 satisfy the inequalities |b0| ≤2α, and |b1| ≤ √ 5 α2.

Lemma 1 again gives the estimates |ci| = |di| ≤2 for i = 1,2, and using these in the above equation immediately yields |b2 1| ≤α2(α −1)2 + α2 + 2α2(α −1)+ 2α4+ 2α4 = 5α4. This shows that |b1| ≤ √ 5 α2. □

Theorem 3. Let β > 0 and 0 < α ≤1. If g ∈ΣB B(β,α), then the coefficients b0 and b1 satisfy the inequalities |b0| ≤ 2α 1 −β , and |b1| ≤ 2α2 (1 −β)(2 −β) p 2(1 −β)(2 −β)+ 1.

Theorem 4. If g given by (2) is in the class Σ∗ B(α), 0 < α ≤1, and b0 = 0, then |b1| ≤α.

Theorem 5. Let g ∈eΣB B(α,β), where α > 0 and 0 < β ≤1. Then |b1| ≤2β 2 2 −α .