Inclusion properties for bi-univalent functions of complex o

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Abstract

In this present investigation, we introduce the new class R of bi-univalent functions defined by using the Tremblay fractional derivative operator. Additionally, we use the Faber polynomial expansions and Fibonacci numbers to derive bounds for the general coefficient an of the bi-univalent function class.

Results & Lemmas (6)

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Theorem 7 Theorem 7 For γ ∈C 0, let f ∈Rµ,ρ Σ,γ ep . If am = 0 (2 ≤m ≤n −1), then |an| ≤|γ| |τ| Γ(µ + 1)Γ(n + ρ) nΓ(ρ + 1)Γ(n + µ) (n ≥3).

Theorem 7 For γ ∈C\{0}, let f ∈Rµ,ρ Σ,γ ep . If am = 0 (2 ≤m ≤n −1), then |an| ≤|γ| |τ| Γ(µ + 1)Γ(n + ρ) nΓ(ρ + 1)Γ(n + µ) (n ≥3).

Corollary 8 Corollary 8 For γ ∈C 0, suppose that f ∈RΣ,γ ep . If am = 0 (2 ≤m ≤n −1), then |an| ≤|γ| |τ| n (n ≥3).

Corollary 8 For γ ∈C\{0}, suppose that f ∈RΣ,γ ep . If am = 0 (2 ≤m ≤n −1), then |an| ≤|γ| |τ| n (n ≥3).

Corollary 9 Corollary 9 Suppose that f ∈RΣ ep . If am = 0 (2 ≤m ≤n −1), then |an| ≤|τ| n (n ≥3).

Corollary 9 Suppose that f ∈RΣ ep . If am = 0 (2 ≤m ≤n −1), then |an| ≤|τ| n (n ≥3).

Theorem 10 Theorem 10 Let f ∈Rµ,ρ Σ,γ ep (γ ∈C 0 ).Then |a2| ≤ min       

Theorem 10 Let f ∈Rµ,ρ Σ,γ ep (γ ∈C\{0}).Then |a2| ≤ min       

Corollary 11 Corollary 11 Let f ∈RΣ,γ ep (γ ∈C 0 ).Then |a2| ≤min ( |γ| |τ| p 3 |γ −4| |τ| + 4, |τ| p |γ| ) and |a3| ≤min

Corollary 11 Let f ∈RΣ,γ ep (γ ∈C\{0}).Then |a2| ≤min ( |γ| |τ| p 3 |γ −4| |τ| + 4 , |τ| p |γ| ) and |a3| ≤min

Corollary 12 Corollary 12 Let f ∈RΣ ep .Then |a2| ≤ |τ| p 9 |τ| + 4 and |a3| ≤ 4 |τ|2 9 |τ| + 4. References [1] Airault H. Symmetric sums associated…

Corollary 12 Let f ∈RΣ ep .Then |a2| ≤ |τ| p 9 |τ| + 4 and |a3| ≤ 4 |τ|2 9 |τ| + 4. References [1] Airault H. Symmetric sums associated to the factorization of Grunsky coef- ﬁcients. In: Conference, Groups and Symmetries, Montreal, Canada, 2007.

Function classes studied: