Degree powers and number of stars in graphs with a forbidden broom

Given a graph $G$ with degree sequence $d_1,\dots, d_n$ and a positive integer $r$, let $e_r(G)=\sum_{i=1}^n d_i^r$. We denote by $\mathrm{ex}_r(n,F)$ the largest value of $e_r(G)$ among $n$-vertex $F$-free graphs $G$, and by $\mathrm{ex}(n,S_r,G)$ the largest number of stars $S_r$ in $n$-vertex $F$-free graphs. The \textit{broom} $B(\ell,s)$ is the graph obtained from an $\ell$-vertex path by adding $s$ new leaves connected to a penultimate vertex $v$ of the path. We determine $\mathrm{ex}_r(n,B(\ell,s))$ for $r\ge 2$, any $\ell,s$ and sufficiently large $n$, proving a conjecture of Lan, Liu, Qin and Shi. We also determine $\mathrm{ex}(n,S_r,B(\ell,s))$ for $r\ge 2$, any $\ell,s$ and sufficiently large $n$.