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

Abstract

In this paper, we establish Riesz--Fejér type inequalities for the $\alpha$-harmonic functions $f=P_{\alpha }[f^{\ast }]$ in ${\mathbb{B}}^n$, where $f^{\ast }\in L^p({\mathbb{S}}^{n-1})$ and $1<p<\infty$. More precisely, for $n\ge 2$ and $\alpha >-1$, we prove the existence of a constant ${\mathcal{C}}_{n,p,\alpha }$ such that ${\int }_{-1}^1|f(r\eta ){|}^p(1-r^2{)}^{n-2}dr\le {\mathcal{C}}_{n,p,\alpha }{\int }_{{\mathbb{S}}^{n-1}}|f^{\ast }(\xi ){|}^{p}d\sigma (\xi )$. 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).