Suppose that ${\bf R} \in \mathbb{C}^{n \times n}$ is a Hermitian matrix, and ${\bf D}$ is a diagonal matrix with main diagonal being ${\bf d} \in \mathbb{C}^{n \times 1}$. I am looking for the relationship between: $\lambda_{max}({\bf D}^{H}{\bf R}{\bf D})$ and ${\bf d}^{H}{\bf R}{\bf d}$. Here, $\lambda_{\max}(\cdot)$ denotes the largest eigenvalue.
Now, what I have is as follows: \begin{align} \lambda_{max}({\bf D}^{H}{\bf R}{\bf D}) = & \max\{ {\bf x}^{H}{\bf D}^{H}{\bf R}{\bf D}{\bf x} ~|~ {\bf x}^{H}{\bf x} = 1 \} \\ = & \max\{ {\bf d}^{H}\text{diag}({\bf x})^{H}{\bf R}\text{diag}({\bf x}){\bf d} ~|~ {\bf x}^{H}{\bf x} = 1 \} \\ \geq & \frac{1}{n}{\bf d}^{H}{\bf R}{\bf d}, \end{align} where $\text{diag}({\bf x})$ is a diagonal matrix with main diagonal being ${\bf x}$.
I need an upper bound of $\lambda_{max}({\bf D}^{H}{\bf R}{\bf D})$ in terms of ${\bf d}^{H}{\bf R}{\bf d}$. That is, can we prove: $\lambda_{max}({\bf D}^{H}{\bf R}{\bf D}) \leq {\bf d}^{H}{\bf R}{\bf d}$ ?