Coefficient problems for a class of analytic functions involving hadamard products

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Abstract

For 0 < α ≤ 1, 0 ≤ λ ≤ 1 and β > 0, let M S(phi; ψ; λ, α, β) be the class of functions defined in the open unit disc D by |arg (λz2f″(z) + zf′(z)/λzg′(z) + (1 - λ)g(z))| < πα/2, z ε D where g(z) = z + b2z2 + b3z 3 + ⋯ is analytic function and satisfies | arg (g(z) * φ (z)/g(z) * ψ(z)) | < πβ/2, z ∈ D for some φ(z) = z + Σn=2 Υnz n and ψ (z) = z + Σn=2 γn=znanalytic in D such that g(z) * ψ(z) # 0, Υn ≥ 0, γn ≥ 0, and Υn > γn (n ≥ 2). For f ∈ M S (φ, ψ; λ, α, β) and given by f(z) = z + a2z2 + a 3z3 + ⋯, a sharp upper bound is obtained for |a3 - μa2 2| when μ ≥ 1.

Original languageEnglish
Pages (from-to)43-47
Number of pages5
JournalActa Mathematica Academiae Paedagogicae Nyiregyhaziensis
Volume21
Issue number1
Publication statusPublished - 2005

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Hadamard Product
Unit Disk
Analytic function
Upper bound
Coefficient
Class

Keywords

  • Analytic functions
  • Functions
  • Normalised
  • Univalent functions

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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abstract = "For 0 < α ≤ 1, 0 ≤ λ ≤ 1 and β > 0, let M S(phi; ψ; λ, α, β) be the class of functions defined in the open unit disc D by |arg (λz2f″(z) + zf′(z)/λzg′(z) + (1 - λ)g(z))| < πα/2, z ε D where g(z) = z + b2z2 + b3z 3 + ⋯ is analytic function and satisfies | arg (g(z) * φ (z)/g(z) * ψ(z)) | < πβ/2, z ∈ D for some φ(z) = z + Σn=2 ∞ Υnz n and ψ (z) = z + Σn=2 ∞ γn=znanalytic in D such that g(z) * ψ(z) # 0, Υn ≥ 0, γn ≥ 0, and Υn > γn (n ≥ 2). For f ∈ M S (φ, ψ; λ, α, β) and given by f(z) = z + a2z2 + a 3z3 + ⋯, a sharp upper bound is obtained for |a3 - μa2 2| when μ ≥ 1.",
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AB - For 0 < α ≤ 1, 0 ≤ λ ≤ 1 and β > 0, let M S(phi; ψ; λ, α, β) be the class of functions defined in the open unit disc D by |arg (λz2f″(z) + zf′(z)/λzg′(z) + (1 - λ)g(z))| < πα/2, z ε D where g(z) = z + b2z2 + b3z 3 + ⋯ is analytic function and satisfies | arg (g(z) * φ (z)/g(z) * ψ(z)) | < πβ/2, z ∈ D for some φ(z) = z + Σn=2 ∞ Υnz n and ψ (z) = z + Σn=2 ∞ γn=znanalytic in D such that g(z) * ψ(z) # 0, Υn ≥ 0, γn ≥ 0, and Υn > γn (n ≥ 2). For f ∈ M S (φ, ψ; λ, α, β) and given by f(z) = z + a2z2 + a 3z3 + ⋯, a sharp upper bound is obtained for |a3 - μa2 2| when μ ≥ 1.

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