How to show that this expression is $O(n\frac{\ln\delta}{\delta})$ 
Find $p \in (0,1)$ such that $n \Big( \big( p + (1-p)^{1+\delta} \big) + (1-p)\big( p + (1-p)^{1+\delta}\big)^{\delta} \Big)$ is $O(n\frac{\ln\delta}{\delta})$

I have no idea to find the value $p$. Can anyone give me a hint? Thank you in advance!
 A: In what follows I assume that $\delta \ge 1$, because for $\delta \le 0$, $\ln \delta$ is not well-defined, for $0< \delta <1$, $\ln \delta$ is negative and thus not an appropriate bound for big Oh notation.
Also note that since $n$ only appears once in the expression multiplying everything from the left, and only appears once in the big Oh upper bound $O(n \frac{\ln \delta}{\delta})$, the problem reduces immediately to finding a $p \in (0,1)$ such that $$\Big( \big( p + (1-p)^{1+\delta} \big) + (1-p)\big( p + (1-p)^{1+\delta}\big)^{\delta} \Big) \in O(\frac{\ln\delta}{\delta}).$$
With these considerations in mind, the problem becomes somewhat easier.
Define $g_{\delta}(p)$ to be $g_{\delta}(p):= (p + (1-p)^{1+\delta})$. Thus we can rewrite the above as $$\left[ g_{\delta}(p) + (1-p)(g_{\delta}(p))^{\delta} \right] \in O(\frac{\ln \delta}{\delta}) $$
Since $\delta >1$, we have that the $(1-p)(g_{\delta}(p))^{\delta}$ term dominates the $g_{\delta}(p)$ term as $t \to \infty$, i.e. if we had that $$(1-p)(g_{\delta}(p))^{\delta} \in O(\frac{\ln \delta}{\delta}) $$ it would follow that $$g_{\delta}(p)\in o(\frac{\ln \delta}{\delta}) $$ and therefore the first term can be discarded/ignored in this analysis.
Since $$a \cdot f(\delta) \in O(\theta(\delta)) \iff f(\delta) \in O(\theta(\delta)),$$ for any $a \in \mathbb{R}$ and any functions $f, \theta$ of $\delta$, the problem reduces to finding $p \in (0,1)$ such that $$(g_{\delta}(p))^{\delta} \in O(\frac{\ln \delta}{\delta}).$$
If one multiplied out $(p +(1-p)^{1+\delta})^{\delta}$, for $\delta > 1$ and as $\delta \to \infty$, we would have that the term of order $x^{(1 + \delta)\cdot \delta}$ would dominate everything else.
In other words, the problem really reduces to finding $p \in (0,1)$ such that $$(1-p)^{(1+\delta)\cdot \delta} \in O(\frac{\ln \delta}{\delta}) $$
I hope this helps.
