Constants in the Rosenthal inequality Let $X_1,\ldots,X_n$ be independent with $\mathbf{E}[X_i] = 0$ and $\mathbf{E}[|X_i|^t] < \infty$ for some $t \ge 2$.  Write $X = \sum_{i=1}^n X_i$.  Then we have the family of "Rosenthal-type inequalities":
$$ \mathbf{E}[|X|^t] \le C_1(t)\cdot \left(\sum_{i=1}^n \mathbf{E}[|X_i|^t]\right) + C_2(t)\cdot \left(\sum_{i=1}^n \mathbf{E}[X_i^2]\right)^{t/2} .$$
Henceforth, $c>0$ is some absolute constant.  I have seen in various papers that we can, for example, take $C_1(t) = C_2(t) = (ct/\log(t))^t$.  We can also take $C_1(t) = (ct)^t, C_2(t) = (c\sqrt{t})^t$.  Another option is $C_1(t) = c^t, C_2(t) = c^t\cdot 2^{t^2/4}$.
Is it known whether there is some non-trivial tradeoff curve for the relationship between $C_1(t)$ and $C_2(t)$?  For example, if I'm fine with setting $C_2(t) = (ct^{2/3})^t$, what's the best $C_1(t)$ I can get?
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According to Theorems 6.1 and 6.2, the best constants $C_1(t)$ and $C_2(t)$ are given by
\begin{equation}
    C_1(t)=(c\ga)^t,\quad C_2(t)=(c\sqrt\ga\,e^{t/\ga})^t
\end{equation}
for $\ga\in[1,t]$, up to a universal real constant $c>0$.
E.g., if you want $C_2(t) = (ct^{2/3})^t$, solving the equation $(ct^{2/3})^t=(c\sqrt\ga\,e^{t/\ga})^t$, you get $\ga\sim\dfrac{6t}{\ln t}$ as $t\to\infty$, so that you get $C_1(t)=\Big(\dfrac{ct}{\ln t}\Big)^t$ for some (possibly different) universal real constant $c>0$.
Exact versions of Rosenthal-type bounds for $t\in[2,3]\cup[5,\infty)$ are given in
Theorems 1.3 and 1.5; see also Proposition 1.2.
A: There is a good constant in Rio's Marcinkiewicz-Zygmund type inequality:
$${\bf E} |X|^t \le \left((t-1)\sum_{i=1}^n ({\bf E}|X_i|^t)^{2/t} \right) ^{t/2} $$
for  independent centered $X_i$, $t\ge 2$ . [ 
Rio, E., 2009, Moment Inequalities for Sums of Dependent Random Variables under Projective Conditions, J. Theor. Probab.  22: 146-163. https://doi.org/10.1007/s10959-008-0155-9 ]
