I know that
$$\sum c_it^i\leq\frac{\sum\limits_{i=0}^ah_it^i}{1-\sum\limits_{j=1}^bk_jt^{j+1}}$$
(where the inequality means that $c_i$ is less or equal to the $i$th coefficient of what's on the right)
How can I prove that there exists a real number $\alpha$ such that $c_n\leq \alpha^n$ for $n\geq1$
The right-hand side of the inequality is a rational function. Using Partial fraction decomposition, it can be written as a finite sum of terms of the form $$ \frac{A t^k}{(t - B)^l} = \frac{A' t^k}{(1 - t/B)^l} $$ where the $B$'s are the different complex roots of the denominator. Since $1/(1-t/B)^l$ is (apart from some constant factor) the $(l-1)$'th derivative of the geometric series $$ \frac{1}{(1 - t/B)} = \sum_{n=1}^\infty \frac{t^n}{B^n} $$ the desired estimate follows.