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\end{split}  \end{equation}  We know that $H_x$ is bounded from above by a constant multiple of $\log $, $\log(n)$,  meaning $H_x \in \mathcal{O}(log(n))$\\ \mathcal{O}(\log(n))$\\  Thus: \[T(n) \in \mathcal{O}(nlog(\sqrt{n}))\]  We use the logarithm identity $log_b(x^d) = dlog_b(x)$ to simplify and yield:  \[T(n) \in \mathcal{O}(nlog(n))\]