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Abstract
In this paper, we obtain upper bounds for the first two Taylor-Maclaurin || and
|| for two new families Υ
, ; and Υ
∗ , ; of holomorphic and -fold
symmetric bi-univalent functions defined in the open unit disk . Further, we point out
several certain special cases for our results.
Results & Lemmas (5)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1.1 · coeff
Lemma 1.1 [3]. If ℎ∈Q, then |R(| ≤2 for each T ∈ℕ, where Q is the family of all UO-ℎ . > 0, ∈. Here ℎ = 1 + R + R + ⋯, ∈.…
Lemma 1.1 [3]. If ℎ∈Q, then |R(| ≤2 for each T ∈ℕ, where Q is the family of all UO-ℎ . > 0, ∈. Here ℎ = 1 + R + R + ⋯, ∈. 2. Coefficient Bounds for the Function Family Υ , ;
Theorem 2.1.
Theorem 2.1. Let ∈Υ, ; 0 < ≤1, ≥0, 0 ≤ ≤1, ∈ℕ be given by (1.3). Then || ≤ 2 =@2 2 + 1 + 2@ −1 + + 2 +…
Theorem 2.1. Let ∈Υ , ; 0 < ≤1, ≥0, 0 ≤ ≤1, ∈ℕ be given by (1.3). Then || ≤ 2 =@2 2 + 1 + 2@ −1 + + 2 + 1A + − 1 + 2.3 and || ≤ 4 1 + + 2 + 1 . 2.4
Corollary 2.1. · coeff
Corollary 2.1. Let ∈Υ, ; 0 < ≤1, ≥0, 0 ≤ ≤1 be given by (1.1). Then || ≤ 2 =@2 2 + 1 + 2@ −1 + 3 + 1A + −1 1 + …
Corollary 2.1. Let ∈Υ , ; 0 < ≤1, ≥0, 0 ≤ ≤1 be given by (1.1). Then || ≤ 2 =@2 2 + 1 + 2@ −1 + 3 + 1A + −1 1 + and |8| ≤ 4 1 + + 2 + 1 . 3. Coefficient Bounds for the Function Family Υ ∗ , ;
Theorem 3.1.
Theorem 3.1. Let ∈Υ ∗, ; 0 ≤ < 1, ≥0, 0 ≤ ≤1, ∈ℕ, be given by (1.3). Then || ≤2 m 1 − 1 + n-n + 2 1 + o. + −1 1…
Theorem 3.1. Let ∈Υ ∗ , ; 0 ≤ < 1, ≥0, 0 ≤ ≤1, ∈ℕ, be given by (1.3). Then || ≤2 m 1 − 1 + n-n + 2 1 + o. + −1 1 + o 3.3 and || ≤ 2 + 1 1 − -n + 1 + o. + 1 − -n + 1 + 2o. . 3.4
Corollary 3.1. · coeff
Corollary 3.1. Let ∈Υ ∗, ; 0 ≤ < 1, ≥0, 0 ≤ ≤1, be given by (1.1). Then || ≤2m 1 − 2 + 1 + 2 −1 + 3 + 1 and |8| ≤4 1…
Corollary 3.1. Let ∈Υ ∗ , ; 0 ≤ < 1, ≥0, 0 ≤ ≤1, be given by (1.1). Then || ≤2m 1 − 2 + 1 + 2 −1 + 3 + 1 and |8| ≤4 1 − + 1 + 1 − 2 + 1 . 4. Conclusion This paper has introduced a new subfamilies Υ , ; and Υ ∗ , ; of < and find estimates on the coefficients || and || for functions in each of these new
Definitions (2)
Def 2.1.
Definition 2.1. A function ∈< given by (1.3) is said to be in the family Υ, ; 0 ≤ ≤1, ≥0,0 ≤ ≤1 if it satisfies the following…
Definition 2.1. A function ∈< given by (1.3) is said to be in the family Υ , ; 0 ≤ ≤1, ≥0,0 ≤ ≤1 if it satisfies the following conditions: Xarg Z 1 − G 1 − $ + 01 + " ′ 5K + " + ′ $ + 1 − ]X < ^ 2 , ∈ 2.1
Def 3.1.
Definition 3.1. A function ∈< given by (1.3) is said to be in the family Υ ∗, ; 0 ≤ < 1, ≥0, 0 ≤ ≤1 if it satisfies the following…
Definition 3.1. A function ∈< given by (1.3) is said to be in the family Υ ∗ , ; 0 ≤ < 1, ≥0, 0 ≤ ≤1 if it satisfies the following conditions: UO i 1 − G 1 − ′ + 01 + " ′ 5K + " + ′ ′ + 1 − j > , 3.1 and UO i 1 − G 1 − /6′ /
Function classes studied:
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