Let $A \in \mathbb{R}^{n \times n}$ be a matrix whose entries are nonnegative. Also, let $d=\text{diag}(A)$. Is $\frac{1}{n}\mathbf{1}^{\top}\text{diag}(A)\mathbf{1} - \frac{1}{n^2}\mathbf{1}^{\top}A\mathbf{1} \geq 0$ where $\mathbf{1}$ is a vector of all one? If not under what condition on $A$ it would be true?
My try
$$ \text{trace}(\frac{1}{n}\mathbf{1}^{\top}\text{diag}(A)\mathbf{1} - \frac{1}{n^2}\mathbf{1}^{\top}A\mathbf{1})=\frac{1}{n}\text{trace}(\mathbf{1}\mathbf{1}^{\top}\text{diag}(A))-\frac{1}{n^2}\text{trace}(\mathbf{1}\mathbf{1}^{\top}A) $$