In this paper, we partly solve the generalized Khavinson conjecture in the setting of hyperbolic harmonic mappings in Hardy space. Assume that u=PΩ[ϕ]\documentclass[12pt]{minimal}
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\begin{document}$u=\mathcal {P}_{\Omega }[\phi ]$\end{document} and ϕ∈Lp(∂Ω,ℝ)\documentclass[12pt]{minimal}
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\begin{document}$\phi \in L^{p}(\partial {\Omega }, \mathbb {R})$\end{document}, where p∈[1,∞]\documentclass[12pt]{minimal}
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\begin{document}$p\in [1,\infty ]$\end{document}, PΩ[ϕ]\documentclass[12pt]{minimal}
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\begin{document}$\mathcal {P}_{\Omega }[\phi ]$\end{document} denotes the Poisson integral of ϕ with respect to the hyperbolic Laplacian operator Δh in Ω, and Ω denotes the unit ball Bn\documentclass[12pt]{minimal}
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\begin{document}$\mathbb {B}^{n}$\end{document} or the half-space ℍn\documentclass[12pt]{minimal}
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\begin{document}$\mathbb {H}^{n}$\end{document}. For any x ∈Ω and l∈Sn−1\documentclass[12pt]{minimal}
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\begin{document}$l\in \mathbb {S}^{n-1}$\end{document}, let CΩ,q(x) and CΩ,q(x;l) denote the optimal numbers for the gradient estimate
∇u(x)≤CΩ,q(x)ϕLp(∂Ω,ℝ)\documentclass[12pt]{minimal}
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\begin{document}$ \left |\nabla u(x)\right |\leq \mathbf {C}_{\Omega ,q}(x)\left \|\phi \right \|{\!}_{L^{p}(\partial {\Omega }, \mathbb {R})} $\end{document} and the gradient estimate in the direction l〈∇u(x),l〉≤CΩ,q(x;l)ϕLp(∂Ω,ℝ),\documentclass[12pt]{minimal}
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\begin{document}$ \left |\langle \nabla u(x),l\rangle \right |\leq \mathbf {C}_{\Omega ,q}(x;l)\left \|\phi \right \|{\!}_{L^{p}(\partial {\Omega }, \mathbb {R})}, $\end{document} respectively. Here q is the conjugate of p. If q∈[1,∞]\documentclass[12pt]{minimal}
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\begin{document}$q\in [1,\infty ]$\end{document}, then CBn,q(0)≡CBn,q(0;l)\documentclass[12pt]{minimal}
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\begin{document}$\mathbf {C}_{\mathbb {B}^{n},q}(0)\equiv \mathbf {C}_{\mathbb {B}^{n},q}(0;l)$\end{document} for any l∈Sn−1\documentclass[12pt]{minimal}
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\begin{document}$l\in \mathbb {S}^{n-1}$\end{document}. If q=∞\documentclass[12pt]{minimal}
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\begin{document}$q=\infty $\end{document}, q = 1 or q∈[2K0−1n−1+1,2K0n−1+1]\documentclass[12pt]{minimal}
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\begin{document}$q\in [\frac {2K_{0}-1}{n-1}+1,\frac {2K_{0}}{n-1}+1]$\end{document} with K0∈ℕ\documentclass[12pt]{minimal}
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\begin{document}$K_{0}\in \mathbb {N} $\end{document}, then CBn,q(x)=CBn,q(x;±x|x|)\documentclass[12pt]{minimal}
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\begin{document}$\mathbf {C}_{\mathbb {B}^{n},q}(x)=\mathbf {C}_{\mathbb {B}^{n},q}(x;\pm \frac {x}{|x|})$\end{document} for any x∈Bn∖{0}\documentclass[12pt]{minimal}
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\begin{document}$x\in \mathbb {B}^{n}\backslash \{0\}$\end{document}, and Cℍn,q(x)=Cℍn,q(x;±en)\documentclass[12pt]{minimal}
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\begin{document}$\mathbf {C}_{\mathbb {H}^{n},q}(x)=\mathbf {C}_{\mathbb {H}^{n},q}(x;\pm e_{n})$\end{document} for any x∈ℍn\documentclass[12pt]{minimal}
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\begin{document}$x\in \mathbb {H}^{n}$\end{document}. However, if q∈(1,nn−1)\documentclass[12pt]{minimal}
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\begin{document}$q\in (1,\frac {n}{n-1})$\end{document}, then CBn,q(x)=CBn,q(x;tx)\documentclass[12pt]{minimal}
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\begin{document}$\mathbf {C}_{\mathbb {B}^{n},q}(x)=\mathbf {C}_{\mathbb {B}^{n},q}(x;t_{x})$\end{document} for any x∈Bn∖{0}\documentclass[12pt]{minimal}
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\begin{document}$x\in \mathbb {B}^{n}\backslash \{0\}$\end{document}, and Cℍn,q(x)=Cℍn,q(x;ten)\documentclass[12pt]{minimal}
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\begin{document}$\mathbf {C}_{\mathbb {H}^{n},q}(x)=\mathbf {C}_{\mathbb {H}^{n},q}(x;t_{e_{n}})$\end{document} for any x∈ℍn\documentclass[12pt]{minimal}
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\begin{document}$x\in \mathbb {H}^{n}$\end{document}. Here tw denotes any unit vector in ℝn\documentclass[12pt]{minimal}
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\begin{document}$\mathbb {R}^{n}$\end{document} such that 〈tw,w〉 = 0 for w∈ℝn∖{0}\documentclass[12pt]{minimal}
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\begin{document}$w\in \mathbb {R}^{n}\setminus \{0\}$\end{document}.
