Conclusion: Overall, the Xperia Z1 Compact is the best smartphone among the all three. What's good about the Z1 Compact is that you get the same hardware as the Xperia Z1 (barring the display) in a much handy form-factor. The Z1 Compact has the trademark Sony Omnibalance design, which means straight lines, glass on the front and rear, an aluminum frame and a big round power button on the edge. The device offers an IPS LCD display with decent viewing angles, color reproduction, and contrast. Qualcomm's Snapdragon 800 quad-core processor clocked at 2.2GHz, 2GB of RAM and 16GB of internal storage, also makes it a truly desirable Android smartphone.
However, this is not to say that the One Mini and the S4 Mini do not have things that make them appealing. These two are pretty decent devices but Sony's miniature version leads the rest.