What is the smallest $t$ statisfying the inequality: $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$, where $\varepsilon$ is arbitrary small positive number? I believe $t$ must be of the from: $$t = \frac{1}{\lambda}\left(\ln m - q \ln \ln m - \ln \frac{c}{\lambda^q}\right) + o(1) \qquad (\varepsilon \to 0),$$ but I am unable to prove this.
I tried to put $t = \frac{1}{\lambda}\bigl(\ln m + \varphi(m)\bigr)$, where $\varphi(m)$ is the unknown function, and I got: $$e^{\varphi(m)}\left(c \left(\frac{\ln m + \varphi(m)}{\lambda}\right)^q - \varepsilon \right) \geqslant 1. \tag{1}$$
Now if I assume $\varphi(m)>0$ then from (1) I get: $$\varphi(m) \geqslant - q\ln \ln m - \ln \frac{c}{\lambda^q} + \alpha(m),$$ where $\alpha(m) \to 0$ given $\varepsilon \to 0$. Unfortunatelly this is negative for sufficiently large $m$ and assumption $\varphi(m) > 0$ isn't true.
Supposedly we must solve (1) more carefuly (mabe make use of Minkowski inequality), but I don't know how to do this. Any kind of help is greatly appreciated!
Note: Parameters are as follows. $m$ may be as large as we want, $\lambda$ and $c$ are positive, $q$ is nonnegative integer.