On Koenigs' ratios for iterates of real functions

In a recent note, M. Kuczama [5] has obtained a general result concerning real solutions φ(x) on the interval 0 ≦ x < a ≦∞ of the Schröder functional equation providing the known real function satisfies the following (quite weak) conditions: f(x) is continuous and strictly increasing in ([0 a);(0) = 0 and 0 <f(x) <x for x ∈ (0, a); limx→0+ {f(x)/x} = s; and f(x)/x is monotonic in (0, a).