How do I show this : $0<c_k < \frac{1}{p^{k-1}}$ for $h(t)= a+\sum_{k\geq1}c_{k}t^k$ ,$k\geq1$?

Question

let $h$ be a function defined in the neighborhood of $t=0$ with this formula: $$h(t)= a+\sum_{k\geq1}c_{k}t^k$$ with $a=1+p >1$ , $h$ satisfy the following functional equation: $$h\bigl(\frac{t}a \bigr)h'(t)=ah'(at)~,$$ My question here is :

How do I show this : $0<c_k < \dfrac{1}{p^{k-1}}$, $k\geq1$ ?