I am looking for a reference for the proof of the fact that a twice differentiable function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is strongly convex of constant $\ell$ and has Lipschitz continuous gradient $\nabla f$ with constant $L$ ( we assume $0< \ell< L$) if and only if
$$ I_{d \times d} \ell \le \nabla^2 f(x) \le I_{d \times d} L \quad \forall x \in \mathbb{R}^d $$
where $I_{d \times d}$ is the identity matrix and with the less than or equal sign we mean that $\nabla^2 f(x) - I_{d \times d} \ell $ and $ I_{d \times d}L - \nabla^2 f(x) $ are positive definite matrices.
I see this fact often cited in the convex optimization literature but rarely proved in its entirety.
The condition $$f(y) \ge f(x) + (y-x)^\top \nabla f(x) + \frac{\ell}{2} \|y-x\|^2, \forall x,y$$ is equivalent to $h(x) := f(x) - \frac{\ell}{2} \|x\|^2$ being convex, which in turn is equivalent to $0 \preceq \nabla^2 h = \nabla^2 f - \ell I$.