Abstract
In this paper, we introduce and investigate a class non-Bazilevic functions that associated by Gegenbauer
Polynomials. The coefficient estimates of functions belonging to this class are derived. Moreover, we obtain the classical
Fekete-Szegö inequality of functions belonging to this class.
Results & Lemmas (8)
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Lemma 2.1. · coeff
Lemma 2.1. Let the Schwarz function w(z) be given by: w(z) = w1z + w2z2 + w3z3 + · · · where z ∈D, then |w1| ≤1 and for t ∈C |w2 −tw2 1| ≤1…
Lemma 2.1. Let the Schwarz function w(z) be given by: w(z) = w1z + w2z2 + w3z3 + · · · where z ∈D, then |w1| ≤1 and for t ∈C |w2 −tw2 1| ≤1 + (|t| −1)|w1|2 ≤max{1, |t|}. The results is sharp for the functions w(z) = z and w(z) = z2. The primary goal of this article is determining the estimates for the initial Taylor-Maclarin coefficients |a2| and |a3| for functions belonging to the class PΣ(α, β, λ, x, y). Furthermore, we examine the corresponding Fekete-Szegö functional problem for functions in t
Theorem 3.1.
Theorem 3.1. Let the function f given by (1.1) be in the class PΣ(α, β, λ, x, y). Then |a2| ≤ 2λt |2 −αβ(x + y)| (3.1) and |a3| ≤ 2λt |3…
Theorem 3.1. Let the function f given by (1.1) be in the class PΣ(α, β, λ, x, y). Then |a2| ≤ 2λt |2 −αβ(x + y)| (3.1) and |a3| ≤ 2λt |3 −αβ(x2 + xy + y2)| max 1, ∆(α, β, λ)t2 −1
Theorem 3.1.
Theorem 3.1. □
Theorem 3.1. □
Corollary 3.1.
Corollary 3.1. Let the function f given by (1.1) be in the class P∗ Σ(β, λ). Then |a2| ≤ 2λt |2 −β| and |a3| ≤ 2λt |3 −β| max 1,…
Corollary 3.1. Let the function f given by (1.1) be in the class P∗ Σ(β, λ). Then |a2| ≤ 2λt |2 −β| and |a3| ≤ 2λt |3 −β| max 1, ∆(β, λ)t2 −1
Corollary 3.2.
Corollary 3.2. Let the function f given by (1.1) be in the class PΣ(α, β, λ, x, y) with α = β = 0. Then |a2| ≤λt, and |a3| ≤2λt 3 max…
Corollary 3.2. Let the function f given by (1.1) be in the class PΣ(α, β, λ, x, y) with α = β = 0. Then |a2| ≤λt, and |a3| ≤2λt 3 max 1, 2(λ + 1)t2 −1
Theorem 4.1.
Theorem 4.1. Let the function f given by (1.1) be in the class PΣ(α, β, λ, x, y). Then for some ζ ∈R, |a3 −ζa2 2| ≤ 2λt |A|, if ζ…
Theorem 4.1. Let the function f given by (1.1) be in the class PΣ(α, β, λ, x, y). Then for some ζ ∈R, |a3 −ζa2 2| ≤ 2λt |A| , if ζ ∈[ζ1, ζ2] λ|2B(λ+1)t2−B−2λRt2−4λζAt2| |AB| , if ζ < [ζ1, ζ2], (4.1) where ζ1 = [2λ(B −R) + 2B]t2 −(2t + 1)B 4λAt2
Corollary 4.1.
Corollary 4.1. Let the function f given by (1.1) be in the class P∗ Σ(β, λ). Then for some ζ ∈R, |a3 −ζa2 2| ≤ 2λt |3−β|, if ζ…
Corollary 4.1. Let the function f given by (1.1) be in the class P∗ Σ(β, λ). Then for some ζ ∈R, |a3 −ζa2 2| ≤ 2λt |3−β|, if ζ ∈[ζ1, ζ2] λ
Corollary 4.2.
Corollary 4.2. Let the function f given by (1.1) be in the class PΣ(α, β, λ, x, y) with α = β = 0. Then for some ζ ∈R, |a3 −ζa2 2| ≤…
Corollary 4.2. Let the function f given by (1.1) be in the class PΣ(α, β, λ, x, y) with α = β = 0. Then for some ζ ∈R, |a3 −ζa2 2| ≤ 2λt 3 , if ζ ∈[ζ1, ζ2] λ 3 ((2λ −3ζ + 2)t2 −1) , if ζ < [ζ1, ζ2], (4.6)
Function classes studied:
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