Minimum Diameter of the shaft

Shaft size is affected by torque, not horsepower. But changes in horsepower and speed affect torque.so we can say an increase in horsepower would require more torque ,decrease in rpm.
methods:
1) make the shaft large enough and strong enough to drive the specified load without breaking.thus we can say the minimum shaft diameter can withstand at least two times the rated torque of the motor.
2) calculate the minimum diameter needed to control torsional deflection (twisting).