# Two (new?) variants of convex functions

I find that the following two types of functions are useful to my research.

(i) We know that a function $f: \mathbb{R}_+^m\rightarrow \mathbb{R}$ is called convex if for all ${\bf x,y}\in \mathbb{R}_+^m$ and all $\lambda_1,\lambda_2\geq 0$ s.t. $\lambda_1+\lambda_2=1$, it holds that $$f(\lambda_1{\bf x}+\lambda_2 {\bf y})\leq \lambda_1f({\bf x})+\lambda_2f({\bf y}).$$ The first type of function is to drop the restriction of $\lambda_1+\lambda_2=1$, i.e. to replace convex combination with conic combination.

(ii) The second type of function: $f({\bf 0})=0$, and for all ${\bf x},{\bf y},{\bf z}\in \mathbb{R}_+^m$ such that ${\bf y}\geq {\bf x}$, it holds that $$f({\bf y})-f({\bf x})\leq f({\bf y}+{\bf z})-f({\bf x}+{\bf z}).$$

Could anyone tell me whether they are new or not? Are there any useful references? Many thanks!