## Sufficient conditions for $\frac{\sum_{k=1}^N {p_k}G(x+e_k)}{G(x)}$ increases/decreases where $\sum_{k=1}^N p_k e_k = 0$ and $\sum_{k=1}^N p_k = 1$

Suppose that $$0 \leq G(x) \leq 1$$, $$G'(x)<0$$, $$G$$ is smooth enough and $$x \in \mathbb{R}$$.

I want to find some neat sufficient conditions for $$H(x) = \frac{\sum_{k=1}^N {p_k}G(x+e_k)}{G(x)}$$ is increasing (decreasing) in $$x$$, where $$\sum_{k=1}^N p_k e_k = 0$$ and $$\sum_{k=1}^N p_k = 1$$, $$\forall N \in \mathbb N$$, for all posible combinations of $$p_k >0$$ and $$e_k$$.

An obvious sufficient condition for $$H(x)$$ to be increasing is $$G”'(x)>0$$. However, the symmetric version ($$G”'(x)<0$$) does not guarantee a decreasing $$H(x)$$ .

(1) Are there any sufficient conditions that are “symmetric” for increasing $$H(x)$$ and decreasing $$H(x)$$?

(2) Are there tighter sufficient conditions, or sufficient and necessary conditions?