How long has he been favoring that arm?
Can you review the tank setup with us, so we can be sure it's all correct for him?
If you have a video of him trying to use his arm, that can help also. He looks very young, too.
I would recommend lowering his basking platform though, for now, so that he doesn't hurt him
self trying to climb up to bask, etc. So you may need to readjust his lights to be closer to him
for awhile also.
How is his appetite?
Looks swollen to me so something serious has happened to it :
a broken bone ?
a dislocation ?
a sprain ? ( unlikely since it's been like this for a while apparently).
You have no option but to get a reptile vet to take xrays and examine the injured arm and then take it from there.
A broken bone can be reset (even a pin inserted in it) and then the arm (and shoulder) immobilized in a mini-lizard sized cast to help it mend), needs to be done surgically under a GA.
A disclocation , can be deduced, see above less the surgery but still under a GA.
A sprain - only takes a few days to come good.