Upper bound on quadratic forms

Question

Say I have a quadratic form $x^\dagger H x$ with a hermitian matrix $H$ with $x \in \mathbb{C}^n$ and $H \in \mathbb{C}^{n \times n}$. In other words $H = H^\dagger$ where $H^\dagger$ is the conjugate transpose of $H$ and so is $x^\dagger$ for the vector $x$. I know Rayleigh-Ritz principle guarantees an upper bound for the quadratic form $x^\dagger H x \le \lambda_{max} (H)$ where $\lambda_{max}$ is the largest eigenvalue in spectrum of $H$. Is there a different upper bound lower than $\lambda_{max}$ that exists for such quadratic forms ? If such a bound exists, a short simple proof for the same (or a literature discussing the proof) would be of great help . Note that in the problem spectrum of $H$ is finite.