How would the lens on the RX100 need to change in order to remain fast throughout?

Since the announcements of the Samsung EX2F and Panasonic Lumix LX7 I'v been wondering why the Sony RX100 becomes so much slower with larger focal length while the other two cameras almost preserve their aperture at the lowest focal length.

What differences in lens design enable the Panny and Samsung cameras to achieve this? Can anybody with expertise chime in and possibly also explain why doing something similar in the Sony with its larger sensor would have been difficult?