Abstract
We consider the convolution of half-plane harmonic mappings with respective dilatations $(z+a)/(1+az)$ and $e^{iθ}z^{n}$, where $-1<a<1$ and $θ\in\mathbb{R},n\in\mathbb{N}$. We prove that such convolutions are locally univalent for $n=1$, which solves an open problem of Dorff et. al (see \cite[Problem~3.26]{Bshouty2010}). Moreover, we provide some numerical computations to illustrate that such convolutions are not univalent for $n\geq 2$.
Results & Lemmas (5)
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Theorem 1.1.
Theorem 1.1. Let f ∈S(H0), i.e., f = h+g ∈KH with h(z)+g(z) = (1+a) z 1−z, and ω(z) = z+a 1+az with a ∈(−1, 1). Then f0 ∗f ∈S0 H and is…
Theorem 1.1. Let f ∈S(H0), i.e., f = h+g ∈KH with h(z)+g(z) = (1+a) z 1−z, and ω(z) = z+a 1+az with a ∈(−1, 1). Then f0 ∗f ∈S0 H and is convex in the direction of the real axis.
Theorem 2.1.
Theorem 2.1. Let f = h + g ∈S(H0) with h + g = (1 + a)z/(1 −z) and the dilatation ω(z) = (z +a)/(1+az), where −1 < a < 1, and f1 = h1 +g1…
Theorem 2.1. Let f = h + g ∈S(H0) with h + g = (1 + a)z/(1 −z) and the dilatation ω(z) = (z +a)/(1+az), where −1 < a < 1, and f1 = h1 +g1 ∈S0(H0) with dilatation ω1(z) = eiθz (θ ∈R). Then f1 ∗f is locally univalent and convex in the horizontal direction. The problem of determining other values of a ∈D and |ϵ| = 1 with ω(z) = ϵ(z + a)/(1 + az) in
Theorem 2.1
Theorem 2.1 remains open. As in the case of [14, Theorem 1.3], establishing the analog of Theorem 2.1 for slanted half-plane mappings is…
Theorem 2.1 remains open. As in the case of [14, Theorem 1.3], establishing the analog of Theorem 2.1 for slanted half-plane mappings is another problem which needs further investigation. Note also that Theorem 2.1 for θ = π is contained in Theorem 1.1. To prove our main result, we need the following lemmas.
Lemma 2.2.
Lemma 2.2. ([4, 5.8 Corollary]) If f = h + g is convex, then g(z1)−g(z2) h(z1)−h(z2) < 1 for all z1, z2 ∈D.
Lemma 2.2. ([4, 5.8 Corollary]) If f = h + g is convex, then g(z1)−g(z2) h(z1)−h(z2) < 1 for all z1, z2 ∈D.
Lemma 2.3.
Lemma 2.3. ([17, Lemma 3]) Let f: D →C be nonconstant and analytic, where f(D) omits some point w ∈ z: Re z < 0. Suppose that bf(eit) =…
Lemma 2.3. ([17, Lemma 3]) Let f : D →C be nonconstant and analytic, where f(D) omits some point w ∈{z : Re z < 0}. Suppose that bf(eit) = limz→eit f(z) exists for all t ∈R (where possibly bf(eit) = ∞). If Re { bf(eit)} ≥0 for all t such that bf(eit) is finite, then Re {f(z)} > 0 for all z ∈D. We remark that Lemma 2.3 is another convenient formulation of maximum modulus theorem for analytic functions.
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