Riesz--Fej\'er type Inequalities for $\alpha$-Harmonic Functions in the Unit Ball

Abstract

In this paper, we establish Riesz--Fej\'er type inequalities for the $\alpha$-harmonic functions $f=P_{\alpha}[f^*]$ in $\mathbb B^n$, where $f^*\in L^{p}(\mathbb{S}^{n-1})$ and $1 -1$, we prove the existence of a constant $\mathcal{C}_{n,p,\alpha}$ such that $\int_{-1}^{1} |f(r\eta)|^p(1-r^2)^{n-2}\,dr \leq \mathcal{C}_{n,p,\alpha} \int_{\mathbb S^{n-1}}|f^*(\xi)|^p\,d\sigma(\xi)$. Moreover, in the range $\alpha>\max\left\{ -\frac{n-1}{p},\,n-2-\frac{2(n-1)}{p} \right\}$, we determine the sharp constant explicitly. The result generalize and extend the corresponding results of Ahmed et al. (J. Math. Anal. Appl., 563:13, 2026), Hu et al. (Anal. Math. 51:15, 2025) and Long (arXiv: 2410.12137).