A Schwarz lemma for hyperbolic harmonic mappings in the unit ball
Assume that p[1,∞ ] and u=P_h[φ ], where φ L^p(S^n-1,R^n) and u(0) = 0. Then we obtain the sharp inequality u(x) vert G_p(x vert )φ Vert_L^p for some smooth function G_p vanishing at 0. Moreover, we obtain an explicit form of the sharp constant C_p in the inequality Du(0) Vert C_pφ Vert C_pφ Vert_L^p . These two results generalize and extend some known results from the harmonic mapping theory (D. Kalaj, Complex Anal. Oper. Theory 12 (2018), 545–554, Theorem 2.1) and the hyperbolic harmonic theory (B. Burgeth, Manuscripta Math. 77 (1992), 283–291, Theorem 1). , number=3, journal=MATHEMATICA SCANDINAVICA, author=Chen, Jiaolong and Kalaj, David, year=2021, month=Nov.