I have constants $\beta$ and $M$ which are positive numbers such that $\eta\leq \frac{\beta}{128M^4}$. We can rewrite the inequality as $\eta = \frac{\theta\beta}{128M^4}$ where $0 < \theta \leq 1$. Let $C = \frac{1 + 128M^4\eta^2}{(1 + \eta \beta)^2} < 1$, $D =\frac{128M^4\eta^2}{(1 + \eta \beta)^2} < 1$ and $ \alpha = C^m + DC\frac{C^m-1}{C-1}$ then $\alpha < 1$ where $m \geq 1$. I am trying to write $m$ in terms of $\theta, \beta, M$ and $\alpha$ but I couldn't find a way to write it.
For $\theta = 1$, we can write $D = C^2 - C^3$, then $\alpha = C^m + C^2 - C^{m + 2}$, but I am not able to solve when $\theta$ is variable.
If you're trying to solve for $m$ in the equation $$ \alpha = C^m + DC\frac{C^m-1}{C-1},$$ then you're only a couple of steps away, since this implies $$ \alpha +\frac{DC}{C-1}= C^m + DC\frac{C^m}{C-1}=C^m\left(1+\frac{1}{C-1}\right).$$