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I find myself needing to do a trill on fingers 3 and 4, but those fingers are much slower than my usual 2-3 trills. So far I'm practicing that particular trill multiple times per day to increase strength and speed, but I'm wondering if there's any other strengthening exercises that can be recommended for me?

The piece I'm working on is Mozart K331, first movement.

You can have a look at Czerny's School of Velocity and Hanon exercises.

I don't have it in front of me right now , but I believe Hanon #10 is specifically for 3-4. There's probably more on there, too.

My teacher has me doing an exercise that he says will help because my 3-4 and 4-5 trills are weak. You press and hold CDEFG in the right hand. While holding down 4 notes, you lift and play just one. So for example, RH holding CDEF play G, then hold down the G and play F while still holding CDE+ G so you are doing na FG trill. It's not fun but it's helping me with strength and finger independence. If you are a masochist, try it with the left hand too.

Yes to the above.

That exercise, or similar ones, are found in Liszt, Joseffy and other advanced technique books.

They are hard-core, but build technique quickly. One student called them "pumping iron"..

One variation that I find easier is to play a diminished chord and do the same, i.e. hold down all the notes, and play all the notes, either in a row, or back and forth as you describe.

C diminished is C - Eb - Gb - A - C

This spreads the hands out, and has black notes for #2 and #3 fingers, so the natural arch is maintained.

But that feature changes when you do the exercise chromatically, as the next chord is C# diminished.

omg here is something else I have missed, so in the future I am going to be expected to play trills with 3 & 4 or 4 & 5. Will this punishment never end?

Hannon 9 & 11 are also very effective exercises

Trills where you must use 3-4 or 4-5 are so rare that I would not worry about training for them until you are learning a piece that actually demands such an unusual device. There are physical limitations on what the fourth finger can and should do, and trying to fight those anatomical constraints is a recipe for injury.

I used to love doing 2-3 trills, but now I'm finding I like 1-3 trills muh better. Sometimes, though, you can't avoid the awkward 3-4 trill. Slow practice doing these, doing them uneven rhythms, and in short bursts all will help make it feel better. Hanon is good as well. I've never tried the Liszt one, but that looks very interesting, I'll have to give that a try!

Goomer, the piece I'm working on has a lot of trills in it and there are a few places where a 3-4 trill makes mores sense, so now is the time to work on it.

Are there any Czerny exercises in particular that I should practice? I practiced Hannon 9, 10 and 11 today and did the "iron pumping" exercises as stated above.

I'm having a hard time wrapping my head around anywhere that 3-4 or 4-5 trills make more sense other than in polyphonic music where notes held by other fingers make it the only possible option. It's so hard to imagine this in beginner or intermediate repertoire.

I assume this is Var II.
I don't play this piece, but I did work on it a bit years ago.
I wonder if you can substitute different trill fingering. Given that the trills are preceded by a staccato eighth note, you may be able to reposition your hand prior to the start of the trill.
4-3 is yucky, I'll take 3-2, 3-1, 4-2, and probably 5-4 over 4-3 any day.

If you decide to pursue the exercises that involve holding keys down, be careful not to over do it. I was doing some of the Herz exercises for a while and found it resulted in a lot of tension (maybe this is less of an issue with an acoustic than a digital).

I suspected it was Variation II, where there are multiple trills that terminate in a turn (i.e., a lower auxiliary note). In the public domain edition at IMSLP once published by Schirmer, the editor actually suggests 3-2 when that lower auxiliary is a white key and 4-3 when it's a black key.

The fingerings aren't written out for each note, but it's implied that it would be 3-2-3-2-1-2 and 4-3-4-3-2-3, respectively. In that latter case, 3-2-3-1-2-3 would be way more comfortable and controllable for me. (And in all cases, the note following the ornament isn't relevant to the fingering of the ornament itself.)

Wait, on another look I see that editor is inconsistent in the prescribed fingerings even for trills of identical structure. This is a messed-up edition.

Yes, I'm working on Variation II. I'll try it with different hand positions to see if I can get out of the 4-3 trill. I can't play the 4-3 trill up to speed, and it's very uneven when I attempt it.

This is the problem with such exercises. I don't know if you had a teacher knowledgeable in such exercises guiding you, but the bottom line is to do them super-slowly, at first one note every 5 seconds or so, and completely focus upon relaxing and re-relaxing the entire hand structure between each note, and striking each note only after relaxing and focusing upon the movement of that finger.

Most of the work is mental. However, the usual way people do things at the piano, fast and faster, will most definitely lead to much tension.

I ran across a 3-4 trill in a Beethoven sonata. It immediately felt uncomfortable but I figured I needed to work on 3-4 trills so I pushed myself to play it that way. My teacher said it made more sense to play it the easier way, 2-3, so I changed the fingering. In the meantime, he suggested I work on 3-4 and 4-5 just so I have that skill.

I too how switched to use 2-3; but 3-4 is coming along.

And, finger strength? I think you mean relaxation and coordination, right?

That is right out of the Letischetski (never can remember the spelling.) It is very finger centric which is a an older approach. It's an interesting book and available for download but you'll have to search a bit with the right spelling.

Two problems with that exercise, that maybe rocket will address:

Fingers 3 and 4 share a tendon. Independence is not actually possible.

Holding 4 notes keeps your hand still and requires you to move fingers. But trills aren't done with fingers, they are done with forearm rotation. With experience this may be minimal and hence not obvious, but there has to be some or it won't work.

Actually it is all of the above...muscle development and movement.

The fingers and hand move via muscles, people typically have weaker fingers at their non-dominant hand because they do not use those fingers for as many things. And, the fourth and fifth finger on both hands are used less than the second and third, one reason being the fourth is tied with tendons to the third and fifth.

But yes, agility and relaxation and coordination are a large part of it, and keep in mind we are not talking about "bulking up" the finger muscles like a weight lifter, but rather developing under-used muscles.

Almost.

Tendons exist between finger #4 and both its neighbors, #3 and #5, which makes #4 the least agile.

Greater independence is very much possible, not as much as #2 has, but the purpose of many of these exercises to increase the ability and Independence of all fingers.

These advanced technique books...Liszt, Joseffy, Pischna, and others, all have numerous exercises that focus upon this issue.