A map $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ is $m$-strongly monotone if $$ (x-y)^{\sf T}((f(x)-f(y)) \geq m \|x-y\|_2^2 $$ for $m > 0$ and is $\delta$-cocoercive if $$ (x-y)^{\sf T}((f(x)-f(y)) \geq \delta \|f(x)-f(y)\|_2^2 $$ for $\delta > 0$. I am working on a proof where I need instead the following property: $$ (x-y)^{\sf T}((f(x)-f(y)) \geq m \|x-y\|_2^2 + \delta \|f(x)-f(y)\|_2^2\,. $$ for $m, \delta > 0$.
Is there a standard name for this property, or some results? I can see that it is implied by $f$ being both $2m$-strongly monotone and $2\delta$-cocoercive, but this ends up not being useful for my problem. Moreover, I can see that it is stronger than $f$ being both $m$-strongly monotone and $\delta$-cocoercive.
The property you stated is equivalent to $f$ being strongly monotone and Lipschitz continuous; searching for this combination of terms will bring up a number of papers. It doesn't have a single-word name, since "Lipschitz strongly monotone" is short enough, and self-descriptive.
Here's a justification. If $f$ is $L$-Lipschitz and $m$-strongly monotone, then $$(x-y)^{\sf T}((f(x)-f(y)) \geq m \|x-y\|_2^2 \ge mL^{-2}\|f(x)-f(y)\|_2^2$$ hence $$(x-y)^{\sf T}((f(x)-f(y)) \geq \frac{m}{2} \|x-y\|_2^2 + \frac{m}{2L^{2}}\|f(x)-f(y)\|_2^2$$
Conversely, if $$(x-y)^{\sf T}((f(x)-f(y)) \geq m \|x-y\|_2^2 + \delta \|f(x)-f(y)\|_2^2\,$$ then $$\|x-y\|_2 \|f(x)-f(y)\|_2 \geq \delta \|f(x)-f(y)\|_2^2\,$$ hence $$\|f(x)-f(y)\|_2 \le \delta^{-1} \|x-y\|_2$$