https://amesweb.info/Torsion/torsion-of-shaft-calculator.aspx
Filled in, for what it matters, just some rough bicycle alike values.
Torque [T] 500
Rotation speed [ω] 60 rpm
Shaft outer radius [c2] 28 mm
Shaft inner radius [c1] 26 mm
Shaft length [L] 100
Modulus of rigidity [G] 300 GPa
= 2 mm wall thickness
Results
Parameter Value
Maximum shear stress [τmax] 56.524
Angle of twist [θ] 0.001
Power requirement [P] 3141.593
Polar moment of inertia [J] 247683.165
Torque [T] 500
Rotation speed [ω] 60 rpm
Shaft outer radius [c2] 28 mm
Shaft inner radius [c1] 25 mm
Shaft length [L] 100
Modulus of rigidity [G] 300 GPa
= 3 mm wall thickness
Maximum shear stress [τmax] 39.783
Angle of twist [θ] 0
Power requirement [P] 3141.593
Polar moment of inertia [J] 351907.072
Maximum shear stress difference is 56.524-39.783 = 16.741
That's 30% less (what you call "minimal") stress for a 50% thicker (what you call "maximal") wall.
Formulas are more accurate than minimal, maximum, and Believe.