In the paper Lipschitz continuity of functions of operators in the Schatten classes, Davies proved the following matrix inequality.

Let $a_i,b_i>0$ for $1\leq i\leq n$ and $A$ be an $n\times n$ matrix, and

$B_{ij}=\frac{a_i-b_j}{a_i+b_j}A_{ij}$.

Then it holds that $\|B\|_p\leq \gamma_p\|A\|_p$ for $1<p<\infty$ and

$\gamma_p=\begin{cases}cp \quad\quad\quad\quad\quad\mbox{if $2\leq p<\infty$}\\ cp/(p-1) \quad\quad\mbox{if $1<p\leq 2$}\end{cases}$,

where $c\geq 1$ is an absolute constant and $\|\cdot \|_p$ is the Schattern $p$-norm.

The proof crucially relies on a fact about Volterra operators in Hilbert space in an old book "Theory and Applications of Volterra Operators in Hilbert space".

Is there a more elementary proof about this inequality? "An elementary proof" means a matrix-analytic proof, which only involves operators in finite dimensional space, but not the operators in infinite dimensional Hilbert space, like Volterra operators.