# An elementary proof of Davies' inequality

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 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,

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.