Runtime analysis of a recursive algorithm with a tricky amount of work per recursive call
I can see three issues with what you have.


There are some inaccurracies in your sums. The outer one needs rounding of the upper boundary, the inner needs a $-1$.
$\displaystyle \sum_{i=1}^{\log(\sqrt{n})} \frac{1}{i} \neq \log(\log(\sqrt{n}))$

The true value of the sum is $H_{\log(\sqrt{n})}$ (will change slightly if you fix the sums). It's true that the difference vanishes in $\Theta$ if you go that route, but better not write equality where it does not hold.
You dropped the recursion at the end! You should have

$\qquad \displaystyle T(n) = T(2/3 \cdot  n) + \dots$


From there, unfold the recurrence:

$\qquad\begin{align*}
  T(n) &amp;= T(2/3 \cdot n) + f(n) \\
   &amp;= T(4/9 \cdot n) + f(2/3 \cdot n) + f(n) \\
   &amp;\dots
\end{align*}$

Spot a pattern, guess the solution and prove it correct by induction! This part is well covered by our reference question in case you have trouble.