Assume that $X$ and $Y$ are two arbitrary non-negative random variables. Is the following inequality true for $\alpha\geq1$?

\begin{align} \left(\mathbb{E}\left[X^\alpha\right]-\mathbb{E}\left[X\right]^\alpha\right)^{\frac{1}{\alpha}}+\left(\mathbb{E}\left[Y^\alpha\right]-\mathbb{E}\left[Y\right]^\alpha\right)^{\frac{1}{\alpha}}\geq \left(\mathbb{E}\left[(X+Y)^\alpha\right]-\left(\mathbb{E}\left[X+Y\right]\right)^\alpha\right)^{\frac{1}{\alpha}} \end{align}

Assume that $X$ and $Y$ are two arbitrary non-negative random variables. Is the following inequality true for $1\leq\alpha\leq 2$?

\begin{align} \left(\mathbb{E}\left[X^\alpha\right]-\mathbb{E}\left[X\right]^\alpha\right)^{\frac{1}{\alpha}}+\left(\mathbb{E}\left[Y^\alpha\right]-\mathbb{E}\left[Y\right]^\alpha\right)^{\frac{1}{\alpha}}\geq \left(\mathbb{E}\left[(X+Y)^\alpha\right]-\left(\mathbb{E}\left[X+Y\right]\right)^\alpha\right)^{\frac{1}{\alpha}} \end{align}