Geodesic
On the estimation of moments of sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 2, pp. 383-386

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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. 
@article{TVP_1974_19_2_a11,
     author = {V. V. Sazonov},
     title = {On the estimation of moments of sums of independent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {383--386},
     publisher = {mathdoc},
     volume = {19},
     number = {2},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1974_19_2_a11/}
}
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V. V. Sazonov. On the estimation of moments of sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 2, pp. 383-386. http://geodesic.mathdoc.fr/item/TVP_1974_19_2_a11/