Close-to-Convexity Criteria for Planar Harmonic Mappings

Abstract

We give a criteria for planar harmonic mappings to be univalent close-toconvex which settles a conjecture of P. T. Mocanu.

Improved Bohr’s inequality for locally univalent harmonic mappings

“…There are interesting examples of harmonic mappings f = h + g that are of the form g ′ (z) = ηzh ′ (z) in D for some η with |η| = 1 (cf. [4,13]). Such mappings play a significant role in the theory of harmonic mappings and sometimes with some additional conditions on h. For example, f (z) = z + 1 2 z 2 is extremal for the area minimizing property of harmonic mappings in S 0 H .…”

On a subclass of close-to-convex harmonic mappings

“…In [17], Mocanu conjectured that the functions in the family M(1, −1/2) are univalent in D. In 2011, Bshouty and Lyzzaik [19] proved this conjecture by showing that M(1, −1/2) ⊂ C 0 H . Recently, this result is extended in [20] by proving that M(α, −1/2) ⊂ C 0 H for |α| = 1, furthermore, the authors conjectured that M(1, −1/2) ⊂ S 0, * H , however, Nagpal and Ravichandan [21] proved this conjecture is false by giving a counter-example, and they are established some additional results of the class M(1, −1/2).…”

On a subclass of harmonic close-to-convex mappings

“…In 1984, Clunie and Sheil-Small [6] investigated the class S H and its geometric subclasses. Since then, the class S H and its subclasses have been extensively studied (see [4], [5], [6], [14], [32]). A domain Ω is called starlike with respect to a point z 0 ∈ Ω if the line segment joining z 0 to any point in Ω lies in Ω.…”