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Abstract
We construct sense-preserving univalent harmonic mappings which map the unit disk onto a domain which is convex in the horizontal direction, but with varying dilatation. Also, we obtain minimal surfaces associated with such harmonic mappings. This solves also a recent problem of Dorff and Muir (Abstr. Appl. Anal. (2014)). In several of the cases, we illustrate mappings together with their minimal surfaces pictorially with the help of \texttt{Mathematica} software.
Results & Lemmas (4)
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Theorem 2.4.
Theorem 2.4. For c ∈[0, 2], and a ∈[−1, 1], let Fc,a = Hc,a + Gc,a ∈S0 H such that Hc,a(z) −Gc,a(z) = kc(z) and ωa(z) = z z + a 1 + az,…
Theorem 2.4. For c ∈[0, 2], and a ∈[−1, 1], let Fc,a = Hc,a + Gc,a ∈S0 H such that Hc,a(z) −Gc,a(z) = kc(z) and ωa(z) = z z + a 1 + az , (2.1) where kc(z) is given by (1.2). Then Fc,a(D) is convex in the horizontal direction, and as c varies from 0 to 2, Fc,a(D) transforms from a strip mapping to a slit mapping.
Theorem 3.1.
Theorem 3.1. Let f1,n = h1,n + g1,n ∈S0 H such that h1,n(z) −g1,n(z) = z 1 −z and ω(z) = g′ 1,n(z) h′ 1,n(z) = zn (n ∈N). (3.1)
Theorem 3.1. Let f1,n = h1,n + g1,n ∈S0 H such that h1,n(z) −g1,n(z) = z 1 −z and ω(z) = g′ 1,n(z) h′ 1,n(z) = zn (n ∈N). (3.1)
Theorem 3.2.
Theorem 3.2. For n ∈N, let f2,n = h2,n + g2,n ∈S0 H such that h2,n(z) −g2,n(z) = z (1 −z)2 and ωn(z) = g′ 0,n(z) h′ 0,n(z) = zn. (3.4)
Theorem 3.2. For n ∈N, let f2,n = h2,n + g2,n ∈S0 H such that h2,n(z) −g2,n(z) = z (1 −z)2 and ωn(z) = g′ 0,n(z) h′ 0,n(z) = zn. (3.4)
Theorem 3.3.
Theorem 3.3. For c ∈[0, 2] and n ∈N, consider the harmonic mappings fc,n = hc,n + gc,n ∈S0 H which satisfy the conditions hc,n(z) −gc,n(z)…
Theorem 3.3. For c ∈[0, 2] and n ∈N, consider the harmonic mappings fc,n = hc,n + gc,n ∈S0 H which satisfy the conditions hc,n(z) −gc,n(z) = kc(z) and g′ c,n(z) = znh′ c,n(z), (3.6)
Function classes studied:
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