Logarithmic Coefficients and a Coefficient Conjecture for Univalent Functions

الأصل · EN

Let U(λ) denote the family of analytic functions f(z), f(0)=0=f'(0)-1, in the unit disk, which satisfy the condition | (z/f(z))²f'(z)-1 |<λ for some 0<λ≤ 1. The logarithmic coefficients γₙ of f are defined by the formula (f(z)/z)=2∑ₙ₌₁∞ γₙzⁿ. In a recent paper, the present authors proposed a conjecture that if f∈ U(λ) for some 0<λ≤ 1, then |aₙ|≤ ∑ₖ₌₀ⁿ⁻¹λᵏ for n≥ 2 and provided a new proof for the case n=2. One of the aims of this article is to present a proof of this conjecture for n=3, 4 and an elegant proof of the inequality for n=2, with equality for f(z)=z/[(1+z)(1+λz)]. In addition, the authors prove the following sharp inequality for f∈U(λ): ∑ₙ₌₁∞|γₙ|² ≤ 1/4(π²6+2 Li₂(λ)+ Li₂(λ²)), where Li₂ denotes the dilogarithm function. Furthermore, the authors prove two such new inequalities satisfied by the corresponding logarithmic coefficients of some other subfamilies of S.