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:
Registry evidence (4)
Family memberships and relations in the registry that this paper supports.
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