On the first look this question might sound ridiculous and maybe it is. But as propeller blades act by the same physical laws as wings, and winglets reduce the induced drag by quite a bit, then why are they not put on propeller blades?
I could imagine that it might be a material problem. But are there any more reasons? And why exactly might it be a material problem?
Winglets on wings help because they increase the volume of air on which the wing can act. Extending the wing span would be much more efficient, but when span is restricted or the maximum wing bending moment is limited, winglets bring a small improvement in efficiency at high lift coefficients.
On propellers, however, the winglets would run through air which is already affected by the tips of the propeller. No additional air will be involved, so no efficiency increase will be possible. Please note that propulsive efficiency is increased by accelerating more air by a smaller amount. The formula for the propulsive efficiency $\eta_p$ of an air breathing engine is $$\eta_p = \frac{v_{\infty}}{v_{\infty} + \frac{\Delta v}{2}}$$ where $v_{\infty}$ is the speed of the inflowing air and $\Delta v$ the speed increase of the air affected by the propeller disc. A smaller $\Delta v$ acting on a higher mass flow makes the engine more efficient. This effect is most pronounced when $v_{\infty}$ is low.
The prop tip winglets would operate in a region of high dynamic pressure and generate more friction drag without contributing to the prop's efficiency.
Things become different when the propeller blades have a very low aspect ratio and there is a hard restriction on propeller diameter: This is true for ships, where draught puts a hard limit on propeller size. Here a sort of winglet does indeed help: The Kappel propeller has its tip bent forward and increases efficiency by 3 - 6 percent. Given that marine propellers even for large ships have efficiencies between 50% and 60%, this is a noteworthy increase in efficiency.
By the way: Whoever tells you that winglets reduce induced drag quite a bit has something to sell to you, but I digress.
Hartzell make propellers that appear to have winglets (tips bent aft) called Q-tips. The name seems to refer to them being quieter. I have read that they do the same job as a slightly larger diameter prop. They seem to have no life issues. Since noise costs energy, and a smaller prop does the same job.... it is possible that they are more efficient. The bent tip is small - on the order of a couple of inches.
Pros:
For all of the above reasons, a propeller of the same diameter that is placed into a shroud can potentially create up to 85% more thrust using the same engine, as a propeller that is not shrouded.
Cons:
Boils down to cost, weight, reliability.
Winglets on propellers look like propellers that have gone through "ground strike" and the FAA grounds prop-driven aircraft with propellers damaged due to ground strike.
It's simply more expensive to manufacture, certify, and maintain; even though it does provide improved thrust in take-off position; but potentially more drag during cruise and feather conditions (based off propeller pitch, will be different with different propellers).
BERP tips most likely more effective overall considering take-off, cruise, and feather conditions (feather is the desired pitch during engine failure to prevent wind-mill and drag from the prop free rotating).
The concept of 'winglets' was applied in early days of aviation by the inventions of Henri Coanda, patents: 1937GB191112740, 1910; CA370885, 1937; and to boat propellers; you can have a look, also in ESPACENET, at patents: ES-0444150, Inventor: 'Gonzalo Perez Gomez', and ES-8300608, 1987, same; and ES-0293837_U by 'Ramon Ruiz Fornella', all from: 'Astilleros espanoles'. 'NASA Technical Memorandum 87771', by Milton A. Beheim: 'NASA Research in Aircraft Propulsion', shows a propeller with 'winglets'.
I'd say that one of the reasons for winglets in wing tips is to reduce the marginal vortex, due to the compensation of the difference in pressures between the upper and lower surface of wing, this 'vortex' or 'swirl' increases drag, and thus impairs overall airplane efficiency.
The same reason why there are no winglets on gliders. The majority of propellers tips are small & pointy enough that there is a negligible spillover to cause wingtip vortices that makes the wing inefficient. Prop blades with wide tips like the ones in C130's or Huey helicopter blades may be improved with winglets but such design also adds complexity(cost), weight and forces that may impede the reliability & safety the blades.
