What is the effect of increasing supply frequency on hysterisis and eddy current losses in a transformer?

For a GIVEN flux level, ie. if flux density is assumed to be constant, the losses increase with an increase in frequency. Hysteresis losses are proportional to frequency. Eddy-current losses are proportional to the square of frequency. 

However, when flux is not assumed to be constant whole scenario is changed. If voltage is constant then if freq. increases flux density B will decrease since B=v/f. Hysterisis losses are also proportional to (B)^n (exponent n is approximately taken as 1.6), so they should decrease. However eddy current losses should remain constant as they are prop. to B^2. 

Generally the flux is assumed to be constant and the first case has more validity.