For , I want to prove that

has exactly one zero for .

It is easy to see that (use the thread "Inequality with binomial coefficients"), , and is continuous. This implies a positive number of zeros on the interval . However, I have not been able to prove uniqueness.

The plot in indicates the truth of the statement for . Other values for 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 is negative for . Unfortunately, is a very complex expression. I can post this expression if someone wants to.

Could anyone help me? Many thanks!