Some Bohr-type inequalities for several subclasses of harmonic functions
Jianying Zhou, Wanqing Hou, Boyong Long
TL;DR
The paper addresses Bohr-type inequalities for complex-valued harmonic functions on the unit disk, extending the classical Bohr phenomenon from analytic to several normalized harmonic subclasses. It develops sharp Bohr-type radii for inequalities of the form $|f(z)|+ obreak olinebreak olinebreak olinebreak olinebreak olinebreak extstyleigl( olinebreak olinebreak olinebreak igl|a_nigr|+|b_n|igr)|z|^n ight)$ (and variants including the area term $S_r/ pi$) across classes such as $ olinebreak ilde{G^0_H}(eta)$, $ olinebreak W^0_H( abla)$, and $ olinebreak olinebreak olinebreak olinebreak abla$-based families $ olinebreak \\mathcal{G}^k_H( abla)$. The main method combines sharp coefficient bounds, subordination arguments, and extremal constructions to produce explicit radii $r_1,\,r_2,\,r_3,\,r_4,\,r_5,\,r_6$ at which the inequalities hold, with proofs of sharpness via extremal functions like $f_k(z)=z+2 extstyleigl( rac{1}{1+(N-1) abla}igr)z^N$. These results generalize the classical Bohr radius from analytic to several harmonic settings, providing concrete, verifiable Bohr radii for a broad spectrum of harmonic subclasses and establishing the significance of the harmonic Bohr phenomenon in complex analysis.
Abstract
In this article, Bohr type inequalities for some complex valued harmonic functions defined on the unit disk are given. All the results are sharp.