If we have a solenoid electromagnet with an iron core, the magnetic field produced is...

The concept of permeability has next to zero applicability for a nonlinear ferromagnet. It is true that the magnetic induction field $B$ and magnetic force $H$ are related but because of hysteresis the relationship between $B$ (or $M$) and $H$ is not even a simple nonlinear function. But if you insist on examining the quantities the ratio $\mu =B/H$ or differential quotient $\tilde{\mu }=\frac{dB}{dH}$ then your result will depend on the various up or down going branches of the your hysteresis curve. The differential quotient $\frac{dB}{dH}$ is used sometime for analyzing small signals with linearized equations (this is the "next to zero applicability"). The function $B=B(H)$ is monotonically increasing but because of saturation it has an inflection point before saturation and in the limit as $M$ goes to a constant, $\frac{dM}{dH}\to 0$, we have $\mu ,\tilde{\mu }\to {\mu }_0$ which is its minimum for only in the case of a diamagnet it can be less than ${\mu }_0$.