Bounding the umber of factorizations into irreducibles in quadratic number rings

I was wondering about imaginary quadratic rings.

Let the class number of the ring $R(p)$ be $c>1$.

and let the norm be $N(x) = a^2 + b^2 p$ where $p$ is a prime.

Let $f(x)$ be the number of factorizations into irreducibles of $x$.

Let $N(x) > N(y)$ and let $g(N(x))$ be the largest value in $f(s)$ for any $N(s)<N(x)$.

Then I wonder if this is true :

$$f(x) < g(N(x)) + c$$