On Riesz type inequalities for harmonic mappings on the unit disk

Kalaj, David

doi:10.1090/tran/7808

Cited by 21 publications

(13 citation statements)

References 18 publications

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“…First, he proved the following, which is one of the two main results in [12]. On the related discussion, we refer to the recent paper [9].…”

Section: Preliminaries and The Statement Of Main Resultsmentioning

confidence: 88%

Norm estimates of the partial derivatives for harmonic and harmonic elliptic mappings

Chen,

Ponnusamy,

Wang

2020

Preprint

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Let f = P [F ] denote the Poisson integral of F in the unit disk D with F being absolutely continuous in the unit circle T and Ḟ ∈ L p (0, 2π), where Ḟ (e it ) = d dt F (e it ) and p ≥ 1. Recently, the author in [12] proved that (1) if f is a harmonic mapping and 1 ≤ p < 2, then f z and f z ∈ B p (D), the classical Bergman spaces of D [12, Theorem 1.2];(2) if f is a harmonic quasiregular mapping and 1 ≤ p ≤ ∞, then f z , f z ∈ H p (D), the classical Hardy spaces of D [12, Theorem 1.3]. These are the main results in [12]. The purpose of this paper is to generalize these two results. First, we prove that, under the same assumptions, [12, Theorem 1.2] is true when 1 ≤ p < ∞. Also, we show that [12, Theorem 1.2] is not true when p = ∞. Second, we demonstrate that [12, Theorem 1.3] still holds true when the assumption f being a harmonic quasiregular mapping is replaced by the weaker one f being a harmonic elliptic mapping.

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“…First, he proved the following, which is one of the two main results in [12]. On the related discussion, we refer to the recent paper [9].…”

Section: Preliminaries and The Statement Of Main Resultsmentioning

confidence: 88%

Norm estimates of the partial derivatives for harmonic and harmonic elliptic mappings

Chen,

Ponnusamy,

Wang

2020

Preprint

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“…Theorem D has been stated and proved in [7] and the proof uses plurisubharmonic function method initiated by Hollenbeck and Verbitsky [5]. The following result due to Frazer is also useful in the proof of Theorem 1.…”

Section: Proofs Of Main Theorems and Related Resultsmentioning

confidence: 99%

“…A positive real-valued function u is called log-subharmonic, if log u is subharmonic. In order to prove our Theorems, we need the following classical inequality of Lozinski [9] which is popularly known as Fejér-Riesz-Lozinski inequality, and also an inequality of Kalaj from [7]. Lemma C. [9] Suppose that Φ is a log-subharmonic function from D to R, such that 2π 0 Φ p (re iθ ) dθ are uniformly bounded with respect to r for some p > 0.…”

Section: Proofs Of Main Theorems and Related Resultsmentioning

confidence: 99%

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Riesz-Fejér Inequalities for Harmonic Functions

2018

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“…They also proved K p ≥ 1 2 cos p π 2p for these p, so the inequality with this K p would be the optimal one. The inequality for 1 < p < 2 depends on an inequality of Kalaj, proved in [8] and Lozinski's inequality from [11].The proof of the first of these inequalities uses the plurisubharmonic method invented in [5]; recent update on this method can be found in [12]. The proof of Riesz-Fejér inequality for p > 2 uses a result of Frazer from [4].…”

Section: Introductionmentioning

confidence: 99%