Polylogarithms and Subordination of Some Cubic Polynomials
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Let V3(z, f) and σ^(1)3(z, f) be the cubic polynomials representing, respectively, the 3rd de la Vall´ee Poussin mean and the 3rd Ces`aro mean of order 1 of a power series f(z). If K denotes the usual class of convex univalent functions in the open unit disk centered at the origin, we show that, in general, V3(z, f) /< σ^(1)3(z, f), for all f ∈ K. Making use of polylogarithms, we identify a transformation, Λ : K → K , such that V3(z, Λ(f))< σ^(1)3(z, Λ(f)) for all f ∈ K . Here ‘<’ stands for subordination between two analytic functions.
Abstract
Let V3(z, f) and σ^(1)3(z, f) be the cubic polynomials representing, respectively, the 3rd de la Vall´ee Poussin mean and the 3rd Ces`aro mean of order 1 of a power series f(z). If K denotes the usual class of convex univalent functions in the open unit disk centered at the origin, we show that, in general, V3(z, f) /< σ^(1)3(z, f), for all f ∈ K. Making use of polylogarithms, we identify a transformation, Λ : K → K , such that V3(z, Λ(f))< σ^(1)3(z, Λ(f)) for all f ∈ K . Here ‘<’ stands for subordination between two analytic functions.
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