Let $k in {2,3,4,dots}$ and $0<p<1.$ Suppose that $X$ and $Y$ are two independent and identically distributed Binomial random variables with parameters $k$ and $p.$ Let $Z = 2X-Y.$ Consider the function $h_k(p) = P(Z < k).$ Show the following:

a) The function $h_k(p)-p$ is decreasing in $p.$

b) $h_k(p)$ has a unique fixed point $p_k^{ast}$ such that $p_k^{ast} > frac{1}{2}.$

c) $|h_k'(p_k^{ast})| < 1.$

I should probably add that the above three statements are conjectures based on numerical computations for different values of $k.$ I would be very grateful if someone can help me with providing analytic proofs of the above statements (if they are true) or a counterexample in case any of the above statement is incorrect.

Thanks in advance!