you have a few options at hand, depending on how you want it to look... These are all default python / maya functions, and works in previous versions, back to Autodesk Maya 2012.
Dock your Window
You can dock your window to a particular side, which would help you manage your scene in Maya. The easiest way to do this is:
# Import your maya command package
import maya.cmds as cmds
try:
# Try to delete your window then the dock
cmds.deleteUI("yourWnd")
cmds.deleteUI("yourDock")
except:
pass
# Parent your window to this dock and then show the dock instead of the window
cmds.window("yourWnd")
cmds.columnLayout("col")
cmds.button("button1")
cmds.setParent("col")
cmds.dockControl("yourDock",
allowedArea = ["left","right"],
area = "left",
content = "yourWnd",
floating = False)
This will put your UI in an easily accessible location in your Maya Window.
Resizing your window
If you've decided to create a window, you could easily add a button or into your window that resizes your window...
import maya.cmds as cmds
import ctypes
# having imported the ctypes package, you can get viable os system info
user32 = ctypes.windll.user32
# Getting the screen size of your main window
screensize = user32.GetSystemMetrics(0), user32.GetSystemMetrics(1)
cmds.window("yourWnd")
cmds.columnLayout("col")
# A button that allows you to imitate maximizing your window on your mainscreen
cmds.button("maxPlease",
label = "Maximize",
command = """cmds.window("yourWnd",
edit=True,
topLeftCorner = [0,0],
widthHeight = [screensize[0],
screensize[1]])
""")
cmds.setParent("col")
cmds.showWindow("yourWnd")
And these are just basics, you can mix and match the two of these to work your own solution. I highly recommend using ctypes, as it will most definitely help you with resizing to maximize your window.