Ma-Minda φ-classes studied in this paper:
Abstract
In this work, coefficient bounds for the functions in the class S(λ, 𝝓) were obtained. This work was concluded by
determining Fekete-Szego functional and the Hankel determinant.
Results & Lemmas (5)
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Lemma 1
Lemma 1: If 𝜔 𝜖 Ω, 𝜔(𝑧) = ∑ 𝑐𝑛𝑧𝑛 ∞ 𝑛=1, (z 𝜖 𝑼), then |𝑐𝑛| ≤1 n = 1, 2, …, |𝑐2| ≤1 − |𝑐1|2
Lemma 1: If 𝜔 𝜖 Ω, 𝜔(𝑧) = ∑ 𝑐𝑛𝑧𝑛 ∞ 𝑛=1 , (z 𝜖 𝑼), then |𝑐𝑛| ≤1 n = 1, 2, … , |𝑐2| ≤1 − |𝑐1|2
Theorem 1
Theorem 1: If f(z) belongs to the class S(λ, 𝝓), then |𝑎2| ≤ 1 2(1+ 𝜆), |𝑎3| ≤ 1 4(1+2𝜆) + 1+3𝜆 8(1+ 𝜆)2(1+2𝜆), |𝑎4| ≤ 13 72(1+3𝜆) + 1+7𝜆…
Theorem 1: If f(z) belongs to the class S(λ, 𝝓), then |𝑎2| ≤ 1 2(1+ 𝜆), |𝑎3| ≤ 1 4(1+2𝜆) + 1+3𝜆 8(1+ 𝜆)2(1+2𝜆), |𝑎4| ≤ 13 72(1+3𝜆) + 1+7𝜆 24(1+ 𝜆)3(1+2𝜆) + 1+5 𝜆
Corollary 1
Corollary 1: If f(z) belongs to the class S(0, 𝝓), then |𝑎2| ≤ 1 2, |𝑎3| ≤ 3 8, |𝑎4| ≤ 59 144, |𝑎5| ≤ 635 1152.
Corollary 1: If f(z) belongs to the class S(0, 𝝓), then |𝑎2| ≤ 1 2, |𝑎3| ≤ 3 8, |𝑎4| ≤ 59 144, |𝑎5| ≤ 635 1152.
Corollary 2
Corollary 2: If f(z) belongs to the class S(1, 𝝓), then |𝑎2| ≤ 1 4, |𝑎3| ≤ 1 8, |𝑎4| ≤ 59 576, |𝑎5| ≤ 179 1440.
Corollary 2: If f(z) belongs to the class S(1, 𝝓), then |𝑎2| ≤ 1 4, |𝑎3| ≤ 1 8, |𝑎4| ≤ 59 576, |𝑎5| ≤ 179 1440.
Theorem 2
Theorem 2 (Fekete – Szegӧ Inequality): If f(z) belongs to the class S(λ, 𝝓), then
Theorem 2 (Fekete – Szegӧ Inequality): If f(z) belongs to the class S(λ, 𝝓), then
Definitions (2)
Def 1
Definition 1: The set P is the set of all functions of the form: 𝑃(𝑧) = 1 + 𝑝1𝑧+ 𝑝2𝑧2 + ⋯+ 𝑝𝑛𝑧𝑛+ ⋯= 1 + ∑ 𝑝𝑛𝑧𝑛 ∞ 𝑛=1, that are analytic in…
Definition 1: The set P is the set of all functions of the form: 𝑃(𝑧) = 1 + 𝑝1𝑧+ 𝑝2𝑧2 + ⋯+ 𝑝𝑛𝑧𝑛+ ⋯= 1 + ∑ 𝑝𝑛𝑧𝑛 ∞ 𝑛=1 , that are analytic in U, and such that for z 𝜖U, Re(P(z)) > 0 (Duren, 1983; Goodman, 1983). Definition 2: Let h(z) be a sigmoid function and 𝜙(𝑧) = 2ℎ(𝑧) = 2
Def 3
Definition 3: A function f ϵ A is said to be in the class S(λ, 𝝓), 0 ≤ λ ≤ 1, if the following subordination holds: (1 − 𝜆) 𝑧𝑓′(𝑧) 𝑓(𝑧) +…
Definition 3: A function f ϵ A is said to be in the class S(λ, 𝝓), 0 ≤ λ ≤ 1, if the following subordination holds: (1 − 𝜆) 𝑧𝑓′(𝑧) 𝑓(𝑧) + 𝜆(1 + 𝑧𝑓′′(𝑧) 𝑓′(𝑧) ) ≺ 𝜙(𝑧)
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