If $\,f: \mathbb{R}^2 \to \mathbb{R}, (x_1, x_2) \mapsto f(x_1, x_2)$ is convex in the $x_2$ axis for all $x_1$, and $A, B \subseteq \mathbb{R}^2$ are both non-empty, is the following true?
$$\frac{1}{|A|}\sum_{\mathbf{x} \in A}\frac{\partial f}{\partial x_2}(x_1, x_2) < \frac{1}{|B|}\sum_{\mathbf{x} \in B}\frac{\partial f}{\partial x_2}(x_1, x_2)\iff$$ $$\underset{\alpha \in \mathbb{R}}{\text{argmin}}\sum_{\mathbf{x} \in A}f(x_1, x_2 + \alpha) > \underset{\beta \in \mathbb{R}}{\text{argmin}}\sum_{\mathbf{x} \in B}f(x_1, x_2 + \beta)$$