We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation.
Show successively that $Y ^ { 2 }$ and $| Y | ^ { 3 }$ have finite expectation, and that
$$\mathrm { E } \left( Y ^ { 2 } \right) \leq \left( \mathrm { E } \left( Y ^ { 4 } \right) \right) ^ { 1 / 2 } \quad \text { then } \quad \mathrm { E } \left( | Y | ^ { 3 } \right) \leq \left( \mathrm { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 }$$