The prime application for the soft starter, and in many cases variable frequency drives on fixed speed applications, is to get the driven load up to full speed to enable it to do some real work. This focus on energy saving by soft starters and VFDs creates a lot of confusion and miss-information.
Soft starters do sometimes include an energy saving algorithm which is designed to reduce the voltage applied to the motor when the motor load is very low. This reduction in voltage will reduce the flux in the iron and reduce the iron loss, which, with today's motors, is already very small. This technology can work, but it can only reduce some of the iron loss of the motor which may only amount to 1 or 2 % of the motor rating, and this will only be realizable when the magnetizing current is higher than the load current.
VFDs do not improve the operating efficiencies of motors, they reduce the losses in machines by reducing the operating speed and frictional losses at the reduced speeds.
VFDs increase the losses in induction motors when operated at rated speed and voltage due to the additional harmonics applied to the stator. Reducing the voltage at rated speed can reduce the iron loss in the same way as the soft starter algorithm can, but at the cost of additional losses in the VFD, (typically 2 - 3%) plus the additional losses in the supply due to the harmonic current drawn from the supply. Harmonic mitigation can reduce the supply losses, but once again at a higher loss (2 - 3%) in the mitigation equipment.
For a fixed speed application, it can be hard to justify the additional expense of a VFD over a correctly designed soft starter system encompassing the correctly designed soft starter and appropriately selected motor.
There is a strong belief in the market that VFDs will benefit of saving energy on all applications, this is incorrect. They can reduce mechanical losses in some applications where there is a period of time operating at reduced demand.