Radius Constants For the Cesàro Operator of a Class of Harmonic Mappings Defined by a Third-Order Differential Inequality

Source abstract

For λ ≥ δ ≥ 0, let R⁰_H(λ, δ) denote the class of normalized harmonic mappings f = h + ḡ in the open unit disk D satisfying the third-order differential inequality Re{h′(z) + λ z h″(z) + δ z² h‴(z)} > |g′(z) + λ z g″(z) + δ z² g‴(z)|, whose members are known to be close-to-convex in D. In this paper, the image of harmonic mappings under the Cesàro operator Cf = Ch + conj(Cg) is studied. Under four natural families of coefficient conditions, among them the sharp coefficient bounds of the class K⁰_H of convex harmonic mappings and the sharp bounds of the class R⁰_H(λ, δ) itself, explicit radii r ∈ (0, 1) are determined such that the normalized Cesàro operator 2Cf belongs to R⁰_H(λ, δ) in the disk D_r. The radii obtained are optimal with respect to the coefficient-based method employed, namely the underlying sufficient coefficient condition combined with the triangle inequality. In particular, it is shown that the normalized Cesàro operator f ↦ 2Cf maps R⁰_H(λ, δ) into itself in an explicitly determined disk. Each radius is characterized as the unique root in (0, 1) of an explicit polynomial or logarithmic-polynomial equation, and the radii corresponding to the four coefficient conditions are completely ordered. The results extend recent radius constants for the Cesàro operator to a class generated by a third-order differential inequality.