Isoperimetric type inequalities for harmonic functions

On Some Riesz and Carleman Type Inequalities for Harmonic Functions in the Unit Disk

“…We are thankful to professor Hedenmalm who drawn attention to his paper [7], where is treated a problem which suggests that the inequalities from Theorem 2.1, which we use in order to prove (1.2), are maybe not sharp. The inequality (1.2) improves similar results for real harmonic functions proved by Kalaj and Meštrović in [10] and by Chen and Ponnusamy and Wang in [4]. We expect that the inequality (1.2) is true for complex harmonic mappings for every p > 1.…”

“…We expect that the inequality (1.2) is true for complex harmonic mappings for every p > 1. Kalaj and Meštrović in [10] proved it for p = 2, namely they obtained that C 2 = 1 2 sin π 8 ≈ 1.3. On the same paper the example f a (z) = ℜ z 1−az , when a ↑ 1 produces the constant C 0 = (5/2) 1/4 ≈ 1.257, so the constant C 2 is not far from the sharp constant.…”

On Riesz type inequalities for harmonic mappings on the unit disk

“…Remark 2.12. The proofs of the same statement for n = 2 and n = 4 can be found in [12] and in [10] respectively (where different approaches used, but applicable only for those two specific cases). The proof here works only for positive integers n ≥ 2, but probably the same estimate is true for every positive number n > 2.…”

“…Remark 2.12. The proofs of the same statement for n = 2 and n = 4 can be found in [12] and in [10] respectively (where different approaches used, but applicable only for those two specific cases).…”

Area integral means, Hardy and weighted Bergman spaces of planar harmonic mappings

“…We refer to [6,8,9,10,12,13,14,15,16] for results related to the theory of analytic Hardy spaces whereas for the harmonic Hardy spaces, the readers may refer to [2,4,11]. In the context of recent investigation and interest on harmonic mappings, it is natural to ask whether Theorem A continues to hold in the setting of planar harmonic mappings over the unit disk.…”