Logarithmic coefficients problems in families related to starlike and convex functions

2018/11/03 by Saminathan Ponnusamy, Ponnusamy, S., Navneet Lal Sharma +3 · 3 citations
Mathematics · Materials Science · #Analytic and geometric function theory #Polymer Synthesis and Characterization #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.1811.01203

Abstract

Let \es be the family of analytic and univalent functions f in the unit disk \D with the normalization f(0)=f'(0)-1=0, and let γ_n(f)=γ_n denote the logarithmic coefficients of f∈ \es. In this paper, we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families \F(c) and \G(δ) of functions f∈ \es defined by \rm Re ( 1+(zf''(z))/(f'(z)) )gt;1-(c)/(2) and \rm Re ( 1+(zf''(z))/(f'(z)) )lt;1+\fracδ2, z∈ \D for some c∈(0,3] and δ∈ (0,1], respectively. We obtain the sharp upper bound for |γ_n| when n=1,2,3 and f belongs to the classes \F(c) and \G(δ), respectively. The paper concludes with the following two conjectures: \beginitemize \item If f∈\F (-1/2), then |γ_n|≤ (1)/(n)(1-\frac12ⁿ⁺¹) for n≥ 1, and ∑_n=1^∞|γ_n|² ≤ (π²)/(6)+(1)/(4) ~\rm Li ₂((1)/(4)) -\rm Li ₂((1)/(2)), where \rm Li₂(x) denotes the dilogarithm function. \item If f∈ \G(δ), then |γ_n| ≤ \fracδ2n(n+1) for n≥ 1. \enditemize

Citations

• On Some Classes of Bounded Univalent Functions

Cited by

• On Coefficient problems for classes S_e^∗ and C_e
• Coefficient Problems for a Ma-Minda Convex Class Associated with the Normalized Arcsine Mapping
• Schwarzian Norm Estimates for Analytic Functions Associated with Convex Functions

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