I'm studying the lecture notes by Philippe Clément about Gradient Flows in Metric Spaces.
Now the following problem arises ($X$ is a Hilbert space):
Definition: Let $\phi:X \to (-\infty, \infty]$ be proper (that is it does not only attain $\infty$), lower semicontinuous and $\alpha$-convex for some $\alpha \in \mathbf R$, that is $\phi - \alpha e$ is convex where $e(x) = \frac12 |x|^2$. Define \begin{equation} \psi(y) := \begin{cases} \frac1{2h} |y - x|^2 + \phi(y), &\text{if $y \in D(\phi)$},\\ \infty &\text{otherwise}. \end{cases} \end{equation} this function has a global minimizer (is a lemma) which will be denoted by $J_h x$. Further let $h \in I_\alpha$, that is $I_\alpha = (0, \infty)$ if $\alpha \geq 0$ and $(0, |\alpha|^{-1})$ if $\alpha < 0$. Now we set $$\phi_h(x) = \psi(J_h x).$$
Now the claim is that $\phi_{h_1}(x) \leq \phi_{h_2}(x)$ for $0 < h_2 < h_1 \leq h_\alpha$ and $x \in X$ where $h_\alpha$ is $1$ for nonnegative $\alpha$ and $\frac1{2|\alpha|}$ for negative $\alpha$. Why is this true? I can't seem to figure out, but probably I'm missing something simple. They claim that it should follow from the information which I have given.
It's actually not too difficult to see that. Maybe just because it's buried into the notations.
Let $y_1=J_{h_1}x$ and $y_2=J_{h_2}x$. Strictly speaking, $\psi(y)$ also depends on $h$ and you are varying $h$ later. So a better notation would be $\psi_h(y)$, which will be used below to make things clear. Because $y_1$ is a minimizer of $\psi_{h_1}(y)$ \begin{eqnarray} \phi_{h_1}(x)&=\psi_{h_1}(y_1)\le \psi_{h_1}(y_2)=\frac{1}{2h_1}|y_2-x|^2+\phi(y_2)\\ &\le \frac{1}{2h_2}|y_2-x|^2+\phi(y_2)=\psi_{h_2}(y_2)=\phi_{h_2}(x). \end{eqnarray} The last inequality is because, of course, $h_2<h_1$.
I haven't figured out why $h_1$ has to be $\le h_\alpha$, but I guess it has something to do with where the global maximum of $\psi_h$ is obtained. So please check the lemma and you might find the answer.