Let $(x_{i,j})$ be a double sequence of nonnegative real numbers, and $ 0< p<1$. 

I would like to know whether one can bound from above the sum
\begin{equation}
\sum_{i,j} x_{i,j}^p
\end{equation}
in terms of 
\begin{equation}
\sum_{i}\Big(\sum_{j} x_{i,j}\Big)^p \quad \text{and} \quad \sum_{j}\Big(\sum_{i} x_{i,j}\Big)^p \quad ?
\end{equation}
The bound does not have to be tight, any upper bound will do. 

My first guess was
\begin{equation}
\sum_{i,j} x_{i,j}^p \leq \sum_{i}\Big(\sum_{j} x_{i,j}\Big)^p + \sum_{j}\Big(\sum_{i} x_{i,j}\Big)^p  + \sum_{i}\Big(\sum_{j} x_{i,j}\Big)^p \cdot \sum_{j}\Big(\sum_{i} x_{i,j}\Big)^p
\end{equation}
but I was unsuccessful in proving it so far.