It is well known that a real-valued function $f: \mathbb{R}^{N\times 1}\rightarrow \mathbb{R}$ is called Lipschitz continuous if there exists a positive real constant $K$ such that, for all real $x_1$ and $x_2$, $$\Vert f(x_1)-f(x_2) \Vert \leq K\Vert x_1-x_2\Vert$$
A differentiable function $f: \mathbb{R}^{N\times 1}\rightarrow \mathbb{R}$ is said to have an L-Lipschitz continuous gradient if there exists a positive real constant $L$ such that, for all real $x_1$ and $x_2$, $$\Vert \nabla f(x_1)-\nabla f(x_2) \Vert \leq L\Vert x_1-x_2\Vert$$.
My question now is:
How to express a differentiable function $f: \mathbb{C}^{N\times 1}\rightarrow \mathbb{R}$ is called Lipschitz continuous or L-Lipschitz continuous gradient? $$\Vert f(x_1)-f(x_2) \Vert \leq K\Vert x_1-x_2\Vert ?$$ $$\Vert \mathcal{D}_{x^*} f(x_1)-\mathcal{D}_{x^*} f(x_2) \Vert \leq L\Vert x_1-x_2\Vert?$$.
Whether the function $$f(x)=\frac{x^H A x}{x^H B x}$$ satisfies Lipschitz continuous or L-Lipschitz continuous gradient?
Any help is highly appreciated. Thanks in advance!
In my opinion, it is still $\|\triangledown f(x_{1})-\triangledown f(x_{1})\|\leq L\|x_{1}-x_{2}\|$, and based on this conclusion, we can have $f(x)$ majorized as $f(x)\leq f(x_{t})+\Re\{\triangledown^{H}f(x_{t})(x-x_{t})\}+L/2\|x-x_{t}\|_{2}^{2}$, which is known as the Descent Lemma, where $x\in\mathbb{C}^{n}$ and $f$ is real-valued function.