The number given for the Trent engine is only for a specific engine component and is not representative for the total output of that engine. The 110000 hp for GE90 is plausible.
It is not as natural to compare shaft horsepower for turbofans as it is for piston or turboprop engines, where it is canonical to measure the shaft power delivered to the propeller.
The Trent engine
Looking at the source cited for the Trent engine in the question, the 50000 horsepower figure is "Horsepower generated by 68 high pressure turbine blades". I think that means 50000hp is extracted by the HP turbine. This does not contribute directly to thrust, as most of that power would be used by the compressor. The Trent is actually a three spool engine, I cannot tell if this figure includes the IP turbine or not, but the same goes for the IP as the HP turbine, most of the power extracted is used by the compressor. The fan, which does most of the propulsion, is powered by the low pressure turbine.
Similarly, the high pressure turbine would not contribute directly to the output of a derived turboshaft.
Power of GE90
That leaves the question of what is actually meant by 110000 hp. This could be the shaft power being transferred by both the spools combined, the power delivered to the fan, or the work done in propelling the airplane, i.e. thrust*speed at some chosen conditions.
I would argue the most natural thing is to measure the work done on the air mass propelled by the engine, i.e. the kinetic energy of the air leaving a static engine over one second, for a static engine at full thrust at STP.
Now we can attempt an estimate of this for the GE90-115. For this we need to know the mass flow rate and exit velocity.
According to this the specific thrust for a kg of air is 278 N*s(=m/s) in those conditions for an earlier less powerful GE90 variant. Assuming GE90-115 has the same exit velocities, at 513 kN that would give a mass flow of 513000/278=1845 kg/s. Then the power is 2782 * 1845/2 W, which is approximately 71.3 MW, or 95600 hp.
The exit velocities of the air leaving the core nozzle and bypass nozzle are actually different, which means the above is a slight underestimate. Given that and the fact that our exit velocity is from a speculative source assessing a different GE90 variant, this is not too far off the 110000 figure.