Logarithmic coefficients and a coefficient conjecture for univalent functions

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Research Article

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Monatshefte fur Mathematik


Let U(λ) denote the family of analytic functions f(z), f(0) = 0 = f′(0) - 1 , in the unit disk D, which satisfy the condition | (z/ f(z)) 2f′(z) - 1 | OpenSPiltSPi λ for some 0 OpenSPiltSPi λ≤ 1. The logarithmic coefficients γn of f are defined by the formula log(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 OpenSPiltSPi 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) : (Formula Presented), where Li 2 denotes the dilogarithm function. Furthermore, the authors prove two such new inequalities satisfied by the corresponding logarithmic coefficients of some other subfamilies of S.

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