Landau and Bloch theorems for harmonic mappings

Grigoryan, A. L.

doi:10.1080/02781070500369214

Cited by 39 publications

(16 citation statements)

References 6 publications

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“…For harmonic mappings in U, under suitable restriction, results similar to the Bloch and Landau theorems can be obtained, as it is proven in [4,[6][7][8], however, these results are not sharp. Chen and Xiong [9] also investigated the Bloch constant of analytic functions in the unit disk.…”

Section: Introductionmentioning

confidence: 58%

Estimates on Bloch constants for planar harmonic mappings

Liu

2008

Sci. China Ser. A-Math.

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Section: Introductionmentioning

confidence: 58%

Estimates on Bloch constants for planar harmonic mappings

Liu

2008

Sci. China Ser. A-Math.

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“…[15]). Recently, many authors considered Landau's theorem for planar harmonic mappings, see, for example, [4,6,7,10,12,18,19,22] and biharmonic mappings, see [1,5,6,17]. In Theorem 2, we derive an analogous result for planar p-harmonic mappings.…”

Section: Proposition 1 a Mapping F Is P-harmonic In D If And Only Ifmentioning

confidence: 86%

Bloch constant and Landau's theorem for planar p-harmonic mappings

Ponnusamy

Wang

2011

Journal of Mathematical Analysis and Applications

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“…, and 0 ≤ q ≤ 1/n. Then the maxima of the functionals Connections with the strip-mappings (12) or (12 ′ ) lead to the following results.…”

Section: And the Maximum Is Attained By F If And Only If F Or F Is Anmentioning

confidence: 90%

“…In particular, we obtain Coefficient bounds, univalent harmonic mappings and normal families were examined in [4,5,6,9,10,12]. The existence theorems for univalent harmonic mappings with a given dilatation can be found in [9,13].…”

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confidence: 97%

Bounded harmonic mappings

Szapiel

2010

JAMA

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Denote by B(τ) the class of all complex functions of the form f (z) ≡ τ + ∞ n=1 a n (f )z n + b n (f ) z n that are harmonic in the open unit disk D with f (D) ⊂ D.Both B(τ) and some of its closed convex subsets are strongly convex, e.g.,Simple extremal problems in B(τ) may have nontrivial solutions. To find them, we present various methods: the method of subordination, the Poisson integral, and calculus of variations. For instance, by analogy to the known result a n ( (τ)) = {w : |w| ≤ 1 − |τ| 2 }, we find the variability regions a n (B(τ)) = b n (B(τ)) = w :x + cos t √x 2 + 2x cos t + 1 dt .In the case n = 1, |τ| < 2/π (resp., |τ| = 2/π), the extremal functions realizing points of the circle w : |w| = ϕ 1/ϕ −1 (|τ|) are univalent self-mappings of the disk D (resp., univalent mappings of D onto a half unit disk). PreliminariesThe classes of all complex analytic functions and all complex harmonic functions in the disk D = {z ∈ C : |z| < 1} that are bounded there by one have a surprising property. Both of them are strongly convex, and hence their extreme and support points form dense subsets. The geometric nature of the sets considered requires some notions of abstract analysis.Let H be a separated locally convex linear space. Suppose that ∅ = F ⊂ H and write conv F, convF, ext F and spt F to denote the convex hull of F, the closed convex hull of F, the set of all extreme points of F, and the set of all (proper)

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Landau and Bloch theorems for harmonic mappings

Cited by 39 publications

References 6 publications

Estimates on Bloch constants for planar harmonic mappings

Estimates on Bloch constants for planar harmonic mappings

Bloch constant and Landau's theorem for planar p-harmonic mappings

Bounded harmonic mappings

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