Typically real harmonic functions

Dorff, Michael; Nowak, Maria; Szapiel, Wojciech

doi:10.1216/rmj-2012-42-2-567

Cited by 2 publications

(3 citation statements)

References 18 publications

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“…We denote by S o Hα the subclass of S Hα consisting of u such that c −1 = 0. In [4,10,25], the typically real harmonic mappings are studied. Similar to the typically real functions of analytic functions or harmonic functions, we want to study the complex-valued kernel α−harmonic functions with real coefficients.…”

Section: Introductions and Main Resultsmentioning

confidence: 99%

Some coefficient estimates on real kernel 𝛼-harmonic mappings

Long

Wang

2022

Proc. Amer. Math. Soc.

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We call the solution of a kind of second order homogeneous partial differential equation as real kernel α − \alpha - harmonic mappings. For this class of mappings, we explore its Heinz type inequality. Furthermore, for a subclass of real kernel α − \alpha - harmonic mappings with real coefficients, we estimate their coefficients. At last, we study the extremal function of Schwartz type lemma for the class of real kernel α − \alpha - harmonic mappings.

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Section: Introductions and Main Resultsmentioning

confidence: 99%

Some coefficient estimates on real kernel 𝛼-harmonic mappings

Long

Wang

2022

Proc. Amer. Math. Soc.

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“…The subclass of T H for which g (0) = 0 is denoted by T 0 H . A family of typically real harmonic polynomials that has some interesting geometric properties has been discussed for example in [28] (see also [5]).…”

Section: Lemma C (Methods Of Shearing)mentioning

confidence: 99%

“…and the assumption, we must have |a 2 | ≤5 2 and |a 2 − b 2 | < 2. Since (2a 2 )/3 has to be a half integer by(3.2), a 2 ∈ {0, 3/2, −3/2}.It follows from the inequality |a 2 − b 2 | < 2 with b 2 = 1 2 that either a 2 = 0 or a 2 = 3 2 , since a 2 = −3/2 is not possible.…”

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confidence: 99%

Classification of Univalent Harmonic Mappings on the Unit Disk With Half-Integer Coefficients

Ponnusamy¹,

Qiao²

2014

J. Aust. Math. Soc.

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Let S denote the set of all univalent analytic functions f of the form f (z) = z + ∞ n=2 a n z n on the unit disk |z| < 1. In 1946, Friedman ['Two theorems on Schlicht functions', Duke Math. J. 13 (1946), 171-177] found that the set S Z of those functions in S which have integer coefficients consists of only nine functions. In a recent paper, Hiranuma and Sugawa ['Univalent functions with half-integer coefficients', Comput. Methods Funct. Theory 13(1) (2013), 133-151] proved that the similar set obtained for functions with half-integer coefficients consists of only 21 functions; that is, 12 more functions in addition to these nine functions of Friedman from the set S Z . In this paper, we determine the class of all normalized sensepreserving univalent harmonic mappings f on the unit disk with half-integer coefficients for the analytic and co-analytic parts of f . It is surprising to see that there are only 27 functions out of which only six functions in this class are not conformal. This settles the recent conjecture of the authors. We also prove a general result, which leads to a new conjecture.2010 Mathematics subject classification: primary 30C65; secondary 30C45, 30C20.

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Typically real harmonic functions

Abstract: We consider a class T O H of typically real harmonic functions on the unit disk that contains the class of normalized analytic and typically real functions. We also obtain some partial results about the region of univalence for this class.

Cited by 2 publications

References 18 publications

Some coefficient estimates on real kernel 𝛼-harmonic mappings

Some coefficient estimates on real kernel 𝛼-harmonic mappings

Classification of Univalent Harmonic Mappings on the Unit Disk With Half-Integer Coefficients

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