On the estimation of moments of sums of independent random variables

Let $\mathscr G$ be the class of real valued functions satisfying conditions (1). It is proved that if $\xi_1,\dots,\xi_n$ are independent random variables such that $\mathbf E\xi_i=0$ and $\mathbf E|\xi_i|^mg(\xi_i)<\infty$ for some integer $m\ge2$ and some $g\in\mathscr G$, $g(\,\cdot\,)\ne|\,\cdot\,|^\delta$, $0\le\delta\le1$, then the inequality (2) holds true; in the case $g(\,\cdot\,)=|\,\cdot\,|^\delta$ a slightly better inequality is proved.