$f_{n,p}(k)$ is the probability mass function of a binomial distribution with parameters $n$ and $p$ i.e, for $k \in \{0,1,2, \cdots,n\}$, $f_{n,p}(k) = \binom{n}{k}p^k(1-p)^{n-k}$. Let $F_{n,p}$ be the corresponding cumulative distribution function, i.e. $F_{n, p}(k) = \sum_{i=0}^k f_{n,p}(k)$ for $k \in \{0,1,2,\cdots,n\}.$ Consider the function $h_n(p)$ defined for $0\leq p \leq 1$ as follows: \begin{equation} h_n(p) = \sum_{k=1}^n f_{n,p}(k)F_{n,p}(k-1). \end{equation} After plotting the function for different values of $n,$ I think that the following two properties hold for $h_n(p):$
$h_n(p) \leq h_{n+1}(p)$ for all $0\leq p \leq 1$ and $n= 1,2,3, ...\, .$
$h_n(p)$ is a concave function for $n= 1,2,3, ...\,.$
Am trying to prove the above two statements analytically. I would really appreciate any help with the proofs.