A basic characteristic of asynchronous motors is that they operate properly only at one rated voltage for a rated frequency and connection. For the 240 volt connection and the rated voltage (usually 50 Hz or 60 Hz) the motor should have only 240 volts applied. The applied voltage is allowed to vary by about 5%, so about 228 to 252 volts would be ok.
If an inverter is designed to control the motor speed, it must keep the voltage proportional to the frequency. If the motor is designed for 240 volts and 50 Hz, it can operate at 120 volts and 25 Hz or 360 volts and 75 Hz. The speed will vary in proportion to the frequency with the voltage having little effect on speed but a more significant effect on current and torque capability. The speed in RPM is equal to (120 X frequency / motor poles) minus slip. Slip at full load is about 3% of the rated 50 Hz speed and is proportional to load torque.
The can probably operate between 10 Hz and 75 Hz if the ratio of voltage to frequency is maintained at 240/50 = 4.8 V/Hz. For the motor to be capable of producing the same torque at all speeds, the ratio will need to be increased somewhat at lower speeds.
This is a basic summary of asynchronous motor speed control by changing frequency. It is not possible to present a the complete theory as an answer to a question here.
It is possible to reduce the speed by reducing just the voltage below the rated voltage, but that method is very limited and much affected by the nature of the load. Increasing the voltage above the rated voltage increases the speed only slightly and is also much affected by the load. Increasing the voltage by much more than 5% will cause excessive current and overheat the motor.
Added Details Regarding Voltage
Increasing voltage increases magnetizing current thus increasing magnetic flux. Increasing flux will allow the motor to develop more torque at a given slip or to have less slip at a given load. Less slip means higher operating speed. At or near the rated voltage, the motor reaches a minimum current for a given torque. Because of magnetic saturation of the iron, the increase in flux for a given increase in current is reduced to the point that the current is increasing faster than the torque is increasing. The current increase increases internal heating with little or no benefit.
Slip is directly proportional to the torque transmitted to the load. Once the load has been accelerated to a stable operating speed, the torque transmitted to the load is the torque that is required to drive the load at that speed. Since the full-load slip of a normal design induction motor is about 3% of rated speed, the motor speed only changes by about 3% between no-load and full-load. The stable operating point is the point at which the speed vs. torque demand curve of the load crosses the speed vs. torque capability curve of the motor. The result of any change in the motor curve is influenced by the magnitude and slope of the load curve.
Increasing Speed Using a VFD
Most variable frequency drives (VFDs) are factory set with the maximum output set to the power frequency in the region where they are sold. However most have configuration settings that allow the maximum frequency to be increased.
In this case, the motor can be connected for either 240 volts or 480 volts and the rated motor frequency is assumed to be 60 Hz. The there are two supply voltages available, 380 and 470. VFDs do not normally have an internal voltage boost feature, so we will assume that the maximum output voltage is limited to the input voltage. Let us first assume that the VFD is set for 50 Hz output and it is desired to operate at the 60 Hz speed with no decrease in torque capability. To do that, connect the 470 volt supply to the VFD and connect the motor for 480 volts. Configure the VFD for normal use with a 480 volt, 60 Hz motor. The input voltage is 2% low, but the motor and VFD should be able to tolerate that with no difficulty.
There are other alternatives. The maximum frequency could be extended above 60 Hz with constant output voltage. That would result in the torque capability dropping as speed increases. The torque capability would follow a constant power curve to about 90 Hz. Above that, torque would be further limited.
If the VFD output current rating is sufficient to supply the motor current required for the 240 volt connection, there are other alternatives using that motor connection.