I am trying to come up with like three strongly convex function $f\colon\mathbb{R}\to\mathbb{R}$ where the Lipschitz constant $L$ is equal to the strong convexity parameter $u$, i.e. for every $x,y\in\mathbb{R}$ \begin{align} |\nabla f(x)-\nabla f(y)|&\leq L|x-y|, \tag{1}\\ f(\cdot)-\dfrac{u}{2}|\cdot|^2&\;\;\text{is convex, and} \tag{2}\\ L=u \tag{3} \end{align}
I am having a hard time coming up with such functions but I am suspecting that a type of least square might possess the property. Can someone help me come up with such functions where $L=u$? Thanks.