Solving recurrence relation where the $f(n)$ has some constant factor $k$ where $0 < k < 1$
The error is where you claim that $kn=n^c$ where $0&lt;c&lt;1$.  This is not correct.    Even when $k&lt;1$, it is still the case that $kn=\Theta(n)$; it is not $O(n^c)$ for any $c&lt;1$.  (Check the definition of big-O notation.)