matrices - Understanding Kantorovich's inequality

I'm looking for a proof of the Kantorovich inequality, namely:

$$\langle Ax,x\rangle \langle A^{-1}x,x\rangle \leq \frac{1}{4}\left(K(A)+\frac{1}{K(A)}\right)$$
Where $K(A)= \lVert A\rVert_2 \lVert A^{-1}\rVert_2$ and $A$ is an Hermitian positive definite matrix and $x$ a vector with the accurate size. Or alternatively

$$\langle Ax,x\rangle \langle A^{-1}x,x\rangle \leq \frac{1}{4}\left(\left(\frac{\beta}{\alpha}\right)^{2}+\left(\frac{\alpha}{\beta}\right)^{2}\right)$$
where $0 < \alpha = \lambda_1 \leq \cdots \leq \lambda_n = \beta$ are the eigenvalues of $A$.

There are a lot of proofs on internet but this is the one that I found easier to understand. Nevertheless, I'm stuck figuring out something:

They use $f,g: [\alpha,\beta]\to \mathbb{R}$ two convex function with $f$ positive and $f(t)\leq g(t)$ for every $t \in [\alpha, \beta]$. Then they claim that $F=f(A)$ and $G=g(A)$ are well defined and hermitian positive definite.

I don't even understand why they mean by $f(A)$.

If any of you guys can suggest me alternative documentation (other proof) or help me with this one, I would be very grateful