Some years back the results of a prop efficiency test were published. The efficiency they were testing for was pretty much "fuel economy" for prop airplanes that carry people. So it wasn't exactly all encompassing, something not possible in aeronautics anyway.

The blade area must be sufficient to handle the horsepower of the engine first. And then the tip speed must be kept under mach 1 to exclude that power robber. And then they found that the slower the rpm the more efficient the system across the board. And that when the pitch went below half the diameter, that killed the efficiency. And one of the most important details was that the prop had to operate in the power range of the engine. So...........

The most fan and pitch that the engine could turn in it's highest horsepower range won for that engine at whatever speed the airframe's drag allowed.

Ain't no three word soundbytes gonna work for this question. Everybody always wants a real simple answer for their aerodynamic questions. And they want that infinitely wise three word sentence to apply to everything from paragliders to jumbo jets.

Higher pitch and longer diameter usually give more efficiency. Mostly because those changes slow the prop, and slower is usually more efficient. Until you run out of engine, that is. And any pitch or diameter or combination that won't allow the engine to put out it's power is a loser. Yet that loser prop might be the most efficient prop for some other engine.