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Schwarz lemma for harmonic functions in the unit ball
Published online by Cambridge University Press: 03 February 2025
Abstract
Recently, it is proven that positive harmonic functions defined in the unit disc or the upper half-plane in 
$\mathbb{C}$ are contractions in hyperbolic metrics [14]. Furthermore, the same result does not hold in higher dimensions as shown by given counterexamples [16]. In this paper, we shall show that positive (or bounded) harmonic functions defined in the unit ball in 
$\mathbb{R}^{n}$ are Lipschitz in hyperbolic metrics. The involved method in main results allows to establish essential improvements of Schwarz type inequalities for monogenic functions in Clifford analysis [24, 25] and octonionic analysis [21] in a unified approach.
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- © The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.
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