Let $f(x)$ be a continuous Lipschitz gradient, that is $\|f'(x)-f'(y)\| \leq \lambda\|x-y\|$ where $\|\cdot\|$ is Euclidean norm and $x, y \in \mathbb{R^n}$. If $x$ can be partitioned into $x=\begin{bmatrix}u\\v\end{bmatrix}$ where $u \in \mathbb{R}^m$ and $v \in \mathbb{R}^p$, we have the following as continuous Lipschitz gradient statement $$ \| f'(u,v)-f'(r,s) \| \leq \lambda \|\begin{bmatrix}u-r\\v-s\end{bmatrix}\| $$ Can we put any assumption on the function or variables to get a bound based on the norms of $\|u-r\|$ and $\|v-s\|$?
I have tried by using $\|a-b\|^2 \leq 2\|a\|^2 +2\|b\|^2$ $$ \| f'(u,v)-f'(r,s) \|^2=\| f'(u,v)-f'(u,s)+f'(u,s)-f'(r,s) \|^2 \leq 2\| f'(u,v)-f'(u,s) \|^2 + 2\|f'(u,s)-f'(r,s) \|^2 \leq 2\lambda\|u-r\|^2+2\lambda\|v-s\|^2 $$ However, my last expression is in terms of square of the difference of the gradient and summation of expression of variables norms. Can I have a bound on the difference of the gradient based on the multiplication of $\|u-r\|$ and $\|v-s\|$ by assuming some new conditions on $f$? Something like $$ \| f'(u,v)-f'(r,s) \| \leq g(\|u-r\|\|v-s\|) $$