The Liapunov's sufficient condition to the Central Limit Theorem says that if $\exists \delta$ for whom the following expression converges to $0$ then you can apply the CLT to the sequence's standardized partial sums:
$$ \frac{1}{\left(\sqrt{\operatorname{Var}{S_{n}}}\right)^{2+\delta}}\sum_{k=1}^{n}\mathbb{E}(|X_{k}-\mu_{k}|^{2+\delta}) \to 0 \implies \frac{S_{n}-\mathbb{E}(S_{n})}{\sqrt{\operatorname{Var}S_{n}}} \xrightarrow[]{D} \mathcal{N}(0,1) $$
My question is if when we choose $\delta =2$, can i apply Jensen's inequality, $\mathbb{E}(\phi(X)) \geq \phi(\mathbb{E}(X))$ when $\phi$ is a convex function, as in:
\begin{align} \frac{1}{\left(\sqrt{\operatorname{Var}{S_{n}}}\right)^{4}}\sum_{k=1}^{n}\mathbb{E}(|X_{k}-\mu_{k}|^{4}) &=\\ \frac{1}{\left(\sqrt{\operatorname{Var}{S_{n}}}\right)^{4}}\sum_{k=1}^{n}\mathbb{E}((|X_{k}-\mu_{k}|^{2})^{2}) &\geq\\ \frac{1}{\left(\sqrt{\operatorname{Var}{S_{n}}}\right)^{4}}\sum_{k=1}^{n}\mathbb{E}(|X_{k}-\mu_{k}|^{2})^{2} & \\ \end{align}
It's a fairly simple doubt. Thanks in advance for the help.