For $\displaystyle n = 1,2,...$, I want to prove that

$\displaystyle f_n(x)=-(2x-1)^n+2nx^{2n-1}(1-x)\sum_{i=0}^{n-1}\binom{n-1}{i}\frac{(-1)^i}{3+2i}\left(\frac{1-x}{x}\right)^{2i}$

has exactly one zero for $\displaystyle x\in\left[\frac{1}{2},1\right]$.

It is easy to see that $\displaystyle f_n\left(\frac{1}{2}\right)>0$ (use the thread "Inequality with binomial coefficients"), $\displaystyle f_n\left(1\right) = -1 <0$, and $\displaystyle f_n(x)$ is continuous. This implies a positive number of zeros on the interval $\displaystyle \left[\frac{1}{2},1\right]$. However, I have not been able to prove uniqueness.

The plot in indicates the truth of the statement for $\displaystyle n = 1, 4, 7, ..., 20$. Other values for $\displaystyle n$ can be easily constructed using the Matlab code in eq.txt. This code also confirms that the number of zeros equals one, which may not be very obvious from the plot.

A potential way to prove the result is to show that $\displaystyle f''_n(x)$ is negative for $\displaystyle x\in\left[\frac{1}{2},1\right]$. Unfortunately, $\displaystyle f''_n(x)$ is a very complex expression. I can post this expression if someone wants to.

Could anyone help me? Many thanks!