The Fekete-Szego Coefficient Inequality For a New Class of m-Fold Symmetric Bi-Univalent Functions Satisfying Subordination Condition

Theorem 5 [2] Let f given by (4) be in the class SΣm(α, λ), 0 < α ≤1. Then |am+1| ≤ 2α m(1 −λ)√α + 1 and |a2m+1| ≤ α m(1 −λ) + 2(m + 1)α2 m2(1 −λ)2 .

Theorem 6 [2] Let f given by (4) be in the class SΣm(β, λ), 0 ≤β < 1. Then |am+1| ≤ p 2(1 −β) m(1 −λ) and |a2m+1| ≤ (1 −β) m(1 −λ) + 2(m + 1)α2 m2(1 −λ)2 . 7

Theorem 7 Let f given by (5) be in the class Sλ Σm(ϕ). Then |am+1| ≤ B1 √B1 m(1 −λ) p |B2 1 −2B2| + B1 (16) and |a2m+1| ≤   

Corollary 8 Let f given by (5) be in the class SΣm(ϕ). Then |am+1| ≤ B1 √B1 m p |B2 1 −2B2| + B1 (28) and |a2m+1| ≤ (  m + 1 −m B1

Corollary 9 If the function f ∈Σ is in the class of Sλ Σ1(ϕ) = Gϕ,ϕ Σ (γ), then |a2| ≤ B1 √B1 (1 −λ) p |B2 1 −2B2| + B1 (30) and |a3| ≤ 

Corollary 10 If the function f ∈Σ is in the class of Sλ Σ1(  1+z 1−z α ), then |a2| ≤ 2α (1 −λ) (32) and |a3| ≤ ( 4α2

Corollary 11 If the function f ∈Σ is in the class of Sλ Σ1( 1+(1−2β)z 1−z ), then |a2| ≤ 2(1 −β) (1 −λ)√2β + 1 (34) and |a3| ≤ ( 4α2 (1−λ)2 , for α ≥1−λ

Corollary 12 If the function f ∈Σ is in the class of S0 Σ1(ϕ) then |a2| ≤ B1 √B1 p |B2 1 −2B2| + B1 (36) |a3| ≤ (  2 − 1 B1 

Corollary 13 [18] Let the function f(z) given by the equality (1) be in the class SS∗ Σ(β, λ), 0 ≤β < 1 and 0 ≤λ < 1. Then |a2| ≤2 p (1 −β) (1 −λ) and |a3| ≤4(1 −β)2 (1 −λ)2 + (1 −β) (1 −λ).

Corollary 14 [18] Let the function f(z) given by the equality (1) be in the class SS∗ Σ(α, λ), 0 < α ≤1 and 0 ≤λ < 1. Then |a2| ≤ 2α (1 −λ)√1 + α and |a3| ≤ 4α2 (1 −λ)2 + α (1 −λ). For one-fold symmetric bi-univalent functions and λ = 0, Theorem 5 reduces to Corollary which were proven earlier by Murugunsundaramoorthy et al. [18]

Corollary 15 Let f given by (4) be in the class S∗ Σ(α) (0 < α ≤1). Then |a2| ≤ 2α √α + 1 and |a3| ≤4α2 + α. Here, in this study, we will spesify the theorem concerning the Fekete- Szeg¨o inequality for the class Sλ Σm(ϕ).To improve the result, especially Theorem 2.1, we consider Fekete-Szeg¨o inequality for the class Sλ Σm(ϕ) . This kind of studies has been made by many authors. The results regarding this problem are given in the works of [6], [11], [14], [22]. The conclutions given in the stud

Theorem 16 Let f given by (4) be in the class Sλ Σm(ϕ). Then 11

Corollary 17 Let f given by (4) be in the class SΣm(ϕ). Then a2m+1 −γa2 m+1 ≤  B1 2m for 0 ≤|h(γ)| < 1 4m 2B1 |h(γ)| for |h(γ)| ≥ 1

Corollary 18 If the function f ∈Σ is in the class of Sλ Σ1(ϕ) = Gϕ,ϕ Σm (γ) , then we get a3 −γa2 2 ≤ ( B1 4(1−λ) for 0 ≤|h(γ)| < 1 4(1−λ) 4B1 |h(γ)|

Corollary 19 If the function f ∈Σ is in the class of S0 Σ1(ϕ) then a3 −γa2 2 ≤  B1 4 for 0 ≤|h(γ)| < 1 4 4B1 |h(γ)| for |h(γ)| ≥1 4

Corollary 20 Let f given by (5) be in the class Sλ Σm(ϕ). Then a2m+1 −γa2 m+1 ≤ ( B1 4m(1−λ) for 0 ≤|h(γ)| < 1 4m(1−λ) 4B1 |h(γ)| for |h(γ)| ≥

Corollary 21 Let f given by (5) be in the class Sϕ,λ Σ,1 . Then a3 −a2 2 ≤ B1 4(1 −λ). Also, if we choose λ = 0, then we have a3 −a2 2 ≤B1 4 . Conclusion 13