We determine the exact optimal bounds $A_p$ and $B_p$ such that $A_p\lbrack E|X|^p + E|Y|^p\rbrack \leq E|X + Y|^p \leq B_p\lbrack E|X|^p + E|Y|^p\rbrack,$ $(p \geq 1)$, whenever $X, Y$ are i.i.d. random variables with mean zero. We give examples of random variables attaining equality or nearly so. Exactly the same bounds $A_p$ and $B_p$ are obtained in the more general case where it is only assumed that $E(X \mid Y) = E(Y \mid X) = 0$.
"Sharp Bounds on the Absolute Moments of a Sum of Two I.I.D. Random Variables." Ann. Probab. 11 (3) 765 - 771, August, 1983. https://doi.org/10.1214/aop/1176993521