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?