Reference to this Young Inequality for matrices

Question

In this post: here I saw the usage of this Young Inequality:

$$X^TY+Y^TX\leq \frac{1}{2}(X+SY)^TS^{-1}(X+SY)$$

With S any symmetric positive definite matrix. As far as I understood this could help to avoid bilinear terms in sdp programming if $X$ and $Y$ are the unknowns. Where I can find a reference about this inequality? Or a proof?

Answer 1

I find out the reference for this refinement of the Young Inequality:

https://ieeexplore.ieee.org/abstract/document/7524904/