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Abstract
Making use of Gegenbauer polynomials, we initiate and explore
two sets of normalized regular and bi-univalent (or bi-Schlicht) functions in
D = {z ∈C : |z| < 1} linked with Gegenbauer polynomials.
We investi-
gate certain coefficients bounds and the Fekete-Szeg¨o functional for functions
in these families.
We also present few interesting observations and provide
relevant connections of the results investigated.
Results & Lemmas (8)
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.
Theorem 2.1.
Theorem 2.1. Let 0 ≤γ ≤1, µ ≥γ, 1/2 < x ≤1 and α a nonzero real constant. If a function g of the form (1.1) ∈SSα Σ(x, γ, µ), then |d2| ≤…
Theorem 2.1. Let 0 ≤γ ≤1, µ ≥γ, 1/2 < x ≤1 and α a nonzero real constant. If a function g of the form (1.1) ∈SSα Σ(x, γ, µ), then |d2| ≤ 2|α|x √ 2x p |(1 −γ + 2µ)2(1 −2x2) + 2((γ −1)2 −4µ(µ −1))αx2| , (2.1) |d3| ≤ 4α2x2 (1 −γ + 2µ)2 + |α|x
Theorem 2.1.
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Theorem 2.1. □ Setting γ = 0 in the above theorem, we have
Theorem 2.1. □ Setting γ = 0 in the above theorem, we have
Corollary 2.2.
Corollary 2.2. Let µ ≥0, 1/2 < x ≤1 and α a nonzero real constant. If a function g of the form (1.1) ∈SLα Σ(x, µ), then |d2| ≤ 2|α|x √ 2x p…
Corollary 2.2. Let µ ≥0, 1/2 < x ≤1 and α a nonzero real constant. If a function g of the form (1.1) ∈SLα Σ(x, µ), then |d2| ≤ 2|α|x √ 2x p |(2µ + 1)2(1 −2x2) −2(4µ(µ −1) −1)αx2| , |d3| ≤ 4α2x2 (2µ + 1)2 + |α|x 3µ + 1
Corollary 2.4.
Corollary 2.4. Let 0 ≤γ ≤1, 1/2 < x ≤1 and α a nonzero real constant. If a function g of the form (1.1) ∈SM α Σ(x, γ), then |d2| ≤ 2|α|x √…
Corollary 2.4. Let 0 ≤γ ≤1, 1/2 < x ≤1 and α a nonzero real constant. If a function g of the form (1.1) ∈SM α Σ(x, γ), then |d2| ≤ 2|α|x √ 2x p |(3 −γ)2(1 −2x2) + 2(γ −1)2αx2| , |d3| ≤ 4α2x2 (3 −γ)2 + |α|x 4 −γ and for δ ∈R
Theorem 3.1.
Theorem 3.1. Let 0 ≤γ ≤1, µ ≥γ, 1/2 < x ≤1 and α a nonzero real constant. If a function g of the form (1.1) ∈RBα Σ(x, γ, µ), then |d2| ≤…
Theorem 3.1. Let 0 ≤γ ≤1, µ ≥γ, 1/2 < x ≤1 and α a nonzero real constant. If a function g of the form (1.1) ∈RBα Σ(x, γ, µ), then |d2| ≤ |α|x √ 2x p |(1 −γ + µ)2(1 −2x2) + (2γ2 −3γ + 1 + 2µ −2µ2)αx2| , (3.1) |d3| ≤ (αx)2 (1 −γ + µ)2 + 2|α|x
Corollary 3.2.
Corollary 3.2. Let µ ≥0, 1/2 < x ≤1 and α a nonzero real constant. If a function g of the form (1.1) ∈RP α Σ(x, µ), then |d2| ≤ |α|x √ 2x p…
Corollary 3.2. Let µ ≥0, 1/2 < x ≤1 and α a nonzero real constant. If a function g of the form (1.1) ∈RP α Σ(x, µ), then |d2| ≤ |α|x √ 2x p |(1 + µ)2(1 −2x2) + (1 + 2µ −2µ2)αx2| , |d3| ≤ α2x2 (1 + µ)2 + 2|α|x 3(1 + 2µ)
Corollary 3.4.
Corollary 3.4. If a function g of the form (1.1) ∈RQα Σ(x, γ), then |d2| ≤ |α|x √ 2x p |(2 −γ)2(1 −2x2) + (2γ2 −3γ + 1)αx2|, |d3| ≤ (αx)2…
Corollary 3.4. If a function g of the form (1.1) ∈RQα Σ(x, γ), then |d2| ≤ |α|x √ 2x p |(2 −γ)2(1 −2x2) + (2γ2 −3γ + 1)αx2| , |d3| ≤ (αx)2 (2 −γ)2 + 2|α|x 3(3 −γ) and for δ ∈R,
Corollary 3.5.
Corollary 3.5. Let µ ≥1, 1/2 < x ≤1 and α a nonzero real constant. If a function g of the form (1.1) ∈RNα Σ(x, µ), then |d2| ≤ |α|x √ 2x p…
Corollary 3.5. Let µ ≥1, 1/2 < x ≤1 and α a nonzero real constant. If a function g of the form (1.1) ∈RNα Σ(x, µ), then |d2| ≤ |α|x √ 2x p |µ2(1 −2x2) + 2µ(1 −µ)αx2| , |d3| ≤α2x2 µ2 + |α|x 3µ and for δ ∈R,
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