As a horn teacher I have seen quite a few students with various issues with the left hand and arm. I have had some pain over the years too.
The problem most of the time in my opinion is not with the player but with the horn, it needs to be set up more ergonomically. Typically the pinky hook is set in a poor position and the valve paddles are not at an optimal location. A lot of help can be found by added “dimes” to the paddles, moving the pinky ring (an easy repair), or better yet adding “ducks foot,” a “flipper” such as that made by Alexander, or a “strap.”
If you don’t know what a strap is, check out the Clebsch website. I have Alexander flippers on my descant and triple, my double has a Clebsch strap, and my natural horn and early valved horn both have Moosewood straps. Check them out, they can make a huge difference for left and comfort.
UPDATE: One other thing I should have also mentioned is oil the valves! They should move quickly and easily. If they have a heavy, slow action, besides hampering your technique, this of course is another element of what causes left arm problems