While gentle driving will in most cases save fuel in a diesel, it may not be true for a petrol engine.
Diesel engines run very very lean. By touching the gas pedal, all you do is making the mixture more rich by injecting more fuel. The air taken by the engine is always the same. So by being as gentle as possible on acceleration, you can save lots of fuel in a diesel by keeping the mixture lean.
With a petrol engine, the more you open the throttle, the more air AND fuel get in, increaing the pressure and most of the time increasing the temperature inside the cylinders. The higher the temperature, the more efficient the combustion (http://en.wikipedia.org/wiki/Thermal_efficiency), the less fuel you use. So by flooring the gas pedal, you get the most efficient acceleration. BUT newer cars will also enrichen the mixture if you floor the gas pedal (to deliver much more power), making you lose any potential gains from the increased temperature. So try to accelerate with 50-75% throttle. Don't floor it and don't baby it.
Don't confuse opening the throttle with fast acceleration. If you keep RPM under control then you can have the throttle open more without accelerating too fast.
It's not just newer cars that will go rich at or near WOT. WOT power enrichment has been around forever. It was programmed into the earliest fuel injection systems, and it's built into carburetors too. Avoiding it is why vacuum gauges were factory-equipped on some carbureted cars.
Running the vehicle at wide open throttle (WOT) will result in the WORST fuel economy -- because
1) at WOT the fuel is enriched (to provide extra engine cooling), and
2) the transmission (especially an automatic) will delay shifts to higher gears.
For most drivers, running in lower gears is THE major reason for poor fuel economy You use lower gears in stop and go traffic, as in the city, or on a clogged freeway. and in maximum acceleration.
I tested this out when I was working in Saudi Arabia back on the '80's. I used to run my vehicle pedal to the medal for a whole tank -- which always resulted in me going half as far on a tank than when I drove more "sedately."
Your best mileage will occur with the driver changing gears at the lowest rpm that doesn't lug the engine, and than running in highest gear as long as possible -- driving around 45 to 55 mph. You can control this easily with a manual transmission, but an automatic transmission won't up-shift as quickly, which is why they usually get worse fuel economy.