Upper Bound for Exponential Expression.

Question

Let $a>0$.I would like to find an upper bound for the absolute value of the quantity \begin{equation} E(x,a) = \frac{ 2 (1-x)^a -\left(-x^2-x+1\right)^a-\left(x^2-x+1\right)^a}{\sqrt{(1-x)^{a} (1-(1-x)^a)}} \end{equation} when $x>0$ is close to $0$.

The series expansion in Mathematica gives \begin{equation} \left(\sqrt{a}-a^{3/2}\right) x^{7/2}+O\left(x^{9/2}\right) \end{equation}

Is there a way to show that $|E(x,a)| \leq C x^2$, where $C>0$ is a constant for all $a \in (0,+\infty)$ ?

Answer 1

The maximum of $a^{1/2}-a^{3/2}$ is $\frac 2{3\sqrt3} \approx 0.385$. As long as $x \lt 1$ we can take $C$ to be this value because $x^{7/2} \lt x^2$ if $x \lt 1$. You didn't specify how close $x$ is to zero, so we can take it small enough that the later terms in the expansion don't matter.

Answer 2

Take the square of your function. You can get rid of that pesky square root and take L'hopital rule, and show easily that the limit exist for any fixed $a$...