GtkWindowminimize

Asks to minimize the specified window.

Note that you shouldn’t assume the window is definitely minimized afterward, because the windowing system might not support this functionality; other entities (e.g. the user or the window manager) could unminimize it again, or there may not be a window manager in which case minimization isn’t possible, etc.

It’s permitted to call this function before showing a window, in which case the window will be minimized before it ever appears onscreen.

You can track result of this operation via the GdkToplevel:state property.