Understanding the definition of the order of an entire function in Ahlfors's Complex Analysis
Note that
$$F(\rho) = \sup_{r \geqslant \rho} \frac{\log\log M(r)}{\log r}$$
is a non-increasing function of $\rho$, hence $\lim\limits_{\rho\to\infty} F(\rho) = \inf\limits_{\rho > R} F(\rho).$
By the definition of the limes superior, for every $\mu < \lambda$, with $\varepsilon = \frac{\lambda-\mu}{3}$ there are arbitrarily large radii $r$ with
$$\frac{\log \log M(r)}{\log r} > \lambda - \varepsilon = \mu + 2\varepsilon,$$
and that inequality is equivalent to
$$M(r) > e^{r^{\mu+2\varepsilon}},$$
so for every $\mu < \lambda$, there is an $\varepsilon > 0$, such that there is no $\rho_0$ with
$$M(r) \leqslant e^{r^{\mu+\varepsilon}}$$
for all $r \geqslant \rho_0$, indeed
$$M(r)e^{-r^{\mu+\varepsilon}}$$ is unbounded for all small enough $\varepsilon > 0$ then.