On a problem of Bharanedhar and Ponnusamy involving planar harmonic mappings

Abstract

In this paper, we give a negative answer to a problem presented by Bharanedhar and Ponnusamy (Rocky Mountain J. Math. 44: 753-777, 2014) concerning univalency of a class of harmonic mappings. More precisely, we show that for all values of the involved parameter, this class contains a non-univalent function. Moreover, several results on a new subclass of close-to-convex harmonic mappings, which is motivated by work of Ponnusamy and Sairam Kaliraj (Mediterr. J. Math. 12: 647-665, 2015), are obtained.Since the Jacobian of f is given by |h | 2 − |g | 2 , by Lewy's theorem (see [10]), it is locally univalent and sense-preserving if and only if |g | < |h |, or equivalently, the dilatation ω = g /h with h (z) 0 has the property |ω| < 1 in D. The subclass of H that are harmonic, univalent and sense-preserving in D is denoted by S H . Univalent harmonic functions are also called harmonic mappings.The classical family S of analytic univalent and normalized functions in D is a subclass of S H with g(z) ≡ 0. The family of all functions f ∈ S H with the additional property that f z (0) = 0 is denoted by Date: October 16, 2018.

Injectivity of sections of close-to-convex harmonic mappings with functions convex in one direction as analytic part

“…1,Theorem 2]) and a result of T. J. Suffridge [22,Theorem 5.6]. We remark that the original proofs given in the above references are considered for the values of α ∈ [0, 1), which can be extended to the case of α ∈ [−1/2, 0); we skip the details here as the computations are similar (See also [24,Lemma 3 Here…”

Some properties of certain close-to-convex harmonic mappings

“…We recall the natural class of close-to-convex harmonic mappings M(α, ζ, n) which belongs to K 0 H due to Wang et al [50] (see also [44]). Definition 1.1.…”

Some classes of harmonic mappings with analytic part defined by subordination

“…In [1,2,10,11,21,22,25,28,29,31], many authors further investigated various subclasses of S H and obtained some important results. In [15], the authors studied the properties of a subclassS α H of S H , consisting of all univalent antianalytic perturbations of the identity in the unit disk with |b 1 | = α , and in [16], the authors studied the classS α H of all f ∈ S H , such that |b 1 | = α ∈ (0, 1) and h ∈ CV , where CV denotes the well-known family of normalized, univalent functions that are convex.…”