Geodesic
An inequality for moments of a~random variable
Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 2, pp. 402-403

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Let $X$ be a random variable. For any $r>0$, we set $\beta_r=\mathbf E|X|^r$. The following inequality is proved: $$ \beta_r^{1/r}\le\gamma^{1/r-1/s}\beta_s^{1/s}\quad(r) $$ where $\gamma=\mathbf P(X\ne0)$. This inequality is optimal in a certain sense. 
@article{TVP_1975_20_2_a15,
     author = {V. V. Petrov},
     title = {An inequality for moments of a~random variable},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {402--403},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1975_20_2_a15/}
}
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V. V. Petrov. An inequality for moments of a~random variable. Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 2, pp. 402-403. http://geodesic.mathdoc.fr/item/TVP_1975_20_2_a15/