Abstract
The aim of this article is to initiating an exploration of the properties of bi-univalent
functions related to Gegenbauer polynomials. To do so, we introduce a new families
, , , , , , ℷ, ,
and , , , ℷ, ,
of holomorphic and bi-univalent functions.
We derive estimates on the initial coefficients and solve the Fekete-Szego problem of
functions in these families.
Results & Lemmas (10)
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Theorem 2.1.
Theorem 2.1. For 0 ≤ ≤1, 0 ≤ ≤1, 0 ≤ ≤1, ∈ 2, 1j and is a nonzero real constant, let ∈ be in the family , , , ,, ℷ,,. Then | |…
Theorem 2.1. For 0 ≤ ≤1, 0 ≤ ≤1, 0 ≤ ≤1, ∈ 2 , 1j and is a nonzero real constant, let ∈ be in the family , , , , , ℷ, ,
. Then | | ≤ 2| | Γ + + ℷ+ 1
Γ
Γℷ
~2| | •€ 3 + + 1
4 Γ +
Γ ℷ+ 1
Γ + ℷ −2 • + 1
3 + + 1
4 Γ +
Γ ℷ+ 1
Γ + ℷ − Ω, , , , , ℷ
Γ +
Γ + ℷ
Γ + + ℷ+ 1
Γ
Γℷ ƒ €
Corollary 2.1.
Corollary 2.1. For ∈r 2, 1s and is a nonzero real constant, let ∈ be in the family |,, ℷ,,. Then | | ≤ 2| | Γ + + ℷ+ 1 Γ Γℷ ~2| |…
Corollary 2.1. For ∈r 2 , 1s and is a nonzero real constant, let ∈ be in the family |, , ℷ, ,
. Then | | ≤ 2| | Γ + + ℷ+ 1
Γ
Γℷ
~2| | Ž• Γ +
Γ ℷ+ 1
Γ + ℷ −2 • + 1
Γ +
Γ ℷ+ 1
Γ + ℷ − ‘, , ℷ
Γ +
Γ + ℷ
Γ + + ℷ+ 1
Γ
Γℷ
’ •
Theorem 2.2.
Theorem 2.2. For 0 ≤ ≤1, ∈r 2, 1s and is a nonzero real constant, let ∈ be in the family , ,, ℷ,,. Then | | ≤ Γ + + ℷ+ 1 Γ Γℷ |…
Theorem 2.2. For 0 ≤ ≤1, ∈r 2 , 1s and is a nonzero real constant, let ∈ be in the family , , , ℷ, ,
. Then | | ≤ Γ + + ℷ+ 1
Γ
Γℷ
| | ~| | Ž• + 1
Γ +
Γ ℷ+ 1
Γ + ℷ −• 2 + 1
+ 1
Γ +
Γ ℷ+ 1
Γ + ℷ − Υ, , , ℷ
Γ +
Γ + ℷ
Γ + + ℷ+ 1
Γ
Γℷ
’ •
Corollary 2.2.
Corollary 2.2. For ∈r 2, 1s and is a nonzero real constant, let ∈ be in the family ℌ,, ℷ,,. Then | | ≤ Γ + + ℷ+ 1 Γ Γℷ | | ~| | Ž•…
Corollary 2.2. For ∈r 2 , 1s and is a nonzero real constant, let ∈ be in the family ℌ, , ℷ, ,
. Then | | ≤ Γ + + ℷ+ 1
Γ
Γℷ
| | ~| | Ž• Γ +
Γ ℷ+ 1
Γ + ℷ −• 2 + 1
Γ +
Γ ℷ+ 1
Γ + ℷ − ”, , ℷ
Γ +
Γ + ℷ
Γ + + ℷ+ 1
Γ
Γℷ
’ •
Theorem 2.3.
Theorem 2.3. For 0 ≤ ≤1, 0 ≤ ≤1, 0 ≤ ≤1, ∈r 2, 1s, • ∈ℝ and is a nonzero real constant, let ∈ be in the family , , ,, ,, ℷ,,.…
Theorem 2.3. For 0 ≤ ≤1, 0 ≤ ≤1, 0 ≤ ≤1, ∈r 2 , 1s, • ∈ℝ and is a nonzero real constant, let ∈ be in the family , , , , , , ℷ, ,
. Then | −• | ≤ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
Corollary 2.3.
Corollary 2.3. For 0 ≤ ≤1, 0 ≤ ≤1, 0 ≤ ≤1, ∈r 2, 1s and is a nonzero real constant, let ∈ be in the family , , ,, ,, ℷ,,. Then…
Corollary 2.3. For 0 ≤ ≤1, 0 ≤ ≤1, 0 ≤ ≤1, ∈r 2 , 1s and is a nonzero real constant, let ∈ be in the family , , , , , , ℷ, ,
. Then − ≤ 2 | |Γ + + ℷ+ 2
Γ
Γℷ + 1
3 + 2 + 1
4Γ +
Γℷ+ 2
Γ + ℷ
. Putting = 0 and = 1 in Theorem 2.3, we demonstrate the next outcome:
Corollary 2.4.
Corollary 2.4. For ∈r 2, 1s, • ∈ℝ and is a nonzero real constant, let ∈ be in the family |,, ℷ,,. Then −• ≤ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
Corollary 2.4. For ∈r 2 , 1s, • ∈ℝ and is a nonzero real constant, let ∈ be in the family |, , ℷ, ,
. Then −• ≤ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
Theorem 2.4.
Theorem 2.4. For 0 ≤ ≤1, ∈r 2, 1s, • ∈ℝ and is a nonzero real constant, let ∈ be in the family , ,, ℷ,,. Then
Theorem 2.4. For 0 ≤ ≤1, ∈r 2 , 1s, • ∈ℝ and is a nonzero real constant, let ∈ be in the family , , , ℷ, ,
. Then
Corollary 2.5.
Corollary 2.5. For 0 ≤ ≤1, ∈r 2, 1s and is a nonzero real constant, let ∈ be in the family , ,, ℷ,,. Then − ≤ 2 | |Γ + + ℷ+…
Corollary 2.5. For 0 ≤ ≤1, ∈r 2 , 1s and is a nonzero real constant, let ∈ be in the family , , , ℷ, ,
. Then − ≤ 2 | |Γ + + ℷ+ 2
Γ
Γℷ 3 + 1
2 + 1
Γ +
Γℷ+ 2
Γ + ℷ
. Putting = 0 in Theorem 2.4, we demonstrate the next outcome:
Corollary 2.6.
Corollary 2.6. For ∈r 2, 1s, • ∈ℝ and is a nonzero real constant, let ∈ be in the family ℌ,, ℷ,,. Then
Corollary 2.6. For ∈r 2 , 1s, • ∈ℝ and is a nonzero real constant, let ∈ be in the family ℌ, , ℷ, ,
. Then
Definitions (2)
Def 2.1.
Definition 2.1. For 0 ≤ ≤1, 0 ≤ ≤1, 0 ≤ ≤1, ∈r 2, 1s and is a nonzero real constant, a function ∈A is called in the family , , ,…
Definition 2.1. For 0 ≤ ≤1, 0 ≤ ≤1, 0 ≤ ≤1, ∈r 2 , 1s and is a nonzero real constant, a function ∈A is called in the family , , , , , ℷ, ,
if it fulfills the subordinations: t r`O,ℷ Q
u v `O,ℷ
Def 2.2.
Definition 2.2. For 0 ≤ ≤1, ∈r 2, 1s and is a nonzero real constant, a function ∈A is called in the family , ,, ℷ,, if it fulfills…
Definition 2.2. For 0 ≤ ≤1, ∈r 2 , 1s and is a nonzero real constant, a function ∈A is called in the family , , , ℷ, ,
if it fulfills the subordinations: r`O,ℷ Q
u v + 2 + 1
r`O,ℷ Q
u vv
Function classes studied:
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