Riesz--Fejér type Inequalities for $α$-Harmonic Functions in the Unit Ball

Abstract

In this paper, we establish Riesz--Fejér type inequalities for the $α$-harmonic functions $f=P_α[f^*]$ in $\mathbb B^n$, where $f^*\in L^{p}(\mathbb{S}^{n-1})$ and $1<p<\infty$. More precisely, for $n\geq2$ and $α>-1$, we prove the existence of a constant $\mathcal{C}_{n,p,α}$ such that $\int_{-1}^{1} |f(rη)|^p(1-r^2)^{n-2}\,dr \leq \mathcal{C}_{n,p,α} \int_{\mathbb S^{n-1}}|f^*(ξ)|^p\,dσ(ξ)$. Moreover, in the range $α>\max\left\{ -\frac{n-1}{p},\,n-2-\frac{2(n-1)}{p} \right\}$, 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).