No, the DJI RS 4 Mini is not compatible with the Sony Alpha A9 III.
Sony A9 III body weighs 703g (with battery+card, per Sony's own Help Guide). Paired with a Tamron 150-500mm F5-6.7 Di III VC VXD lens (1,725g for Sony E-mount, per Tamron's own product page) -- a realistic pairing for this sports-flagship body -- combined weight is ~2,428g. DJI RS 4 Mini's stated payload capacity is 2kg (4.4lbs) per DJI's own product page. 2,428g exceeds that capacity by ~21%; the gimbal cannot safely balance or support this combination.
Conditions for compatibility
Verdict assumes a Tamron 150-500mm F5-6.7 Di III VC VXD lens (1,725g for Sony E-mount, per Tamron's own product page) -- a realistic pairing for this sports-flagship body. A9 III body (703g) + this lens = ~2,428g, exceeding DJI RS 4 Mini's 2kg (4.4lbs) payload capacity by ~21%. Body-only or a lighter lens may fit within the limit -- check your specific lens weight before assuming this pairing won't work. See this camera's DJI RS 4 pairing for a gimbal that does fit this exact lens.
What works instead
See all compatible gimbal for the DJI RS 4 Mini →“Weight 1,725g/60.8oz (w/o tripod mount) (Sony), 1,710g/60.3oz (w/o tripod mount) (Fujifilm), 1,720g/60.7oz (w/o tripod mount) (Nikon), tripod mount 155g/5.5oz”
“Mass (Approx.) 703 g (1 lb 8.8 oz) (including battery pack, SD card)”
“2kg/4.4lbs Payload for Camera and Phone It supports a wide range of mainstream mirrorless cameras and lens combinations, lightweight vlog cams,”
Last verified:
Compatibility inferred from spec rule (citations above are manufacturer-verified)