2010
DOI: 10.1007/s11854-010-0012-5
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Bounded harmonic mappings

Abstract: 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 th… Show more

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Cited by 8 publications
(4 citation statements)
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“…For α = 0, there are numerous discussions on coefficient estimates of harmonic mappings in the literature, see for example [8,9,10,12,15,21,29]. We investigate the problem of coefficient estimates on the solutions to (1.1) as follows.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…For α = 0, there are numerous discussions on coefficient estimates of harmonic mappings in the literature, see for example [8,9,10,12,15,21,29]. We investigate the problem of coefficient estimates on the solutions to (1.1) as follows.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…We refer to [5,6,7,8,9,10,13,20,23,28] for further discussion on this topic. In this paper, we generalize Theorem A to higher dimensional case, and give the estimate for the partial derivatives of arbitrary order.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…If D is simply connected, then a harmonic mapping f ( z ) can be written as f(z)=h(z)+trueg(z)¯ with f (0) = h (0), where h ( z ) and g ( z ) are analytic in D (for details, see ). Thus, we have Jf(z)=|h(z)|2|g(z)|2. Suppose that f ( z ) = u ( x , y ) + i v ( x , y ), z = x + i y is a four times continuously differentiable complex‐valued function in a domain Ddouble-struckC.…”
Section: Introductionmentioning
confidence: 99%