For random variable $Z$, let $F_Z$ denote its cdf, i.e., $F_Z(t)=\mathbb{P}(Z\leq t)$. Let $X$ be a binomial distribution with parameters $(n,p)$ and $Y$ a binomial distribution with parameters $(m,p)$.
Let $n,m$ be non-negative integers such that $n>m$. Does the following inequality hold for all $k\in\{1,...,m-1\}$: $$ \frac{1}{m-k}\cdot\frac{F_Y(k)+F_Y(k+1)}{F_Y(k-1)+F_Y(k)} \geq \frac{1}{n-k}\cdot\frac{F_X(k)+F_X(k+1)}{F_X(k-1)+F_X(k)}. \tag{1} $$ I have checked numerically and it seems to hold. References to similarly looking inequalities would also be appreciated.
Note: Based on the relation between incomplete beta functions and cdf's of binomial distributions, I can prove that, for $n,m,k$ as above, the following holds: $$ \frac{1}{m-k}\cdot \frac{F_Y(k)}{F_Y(k-1)} \geq \frac{1}{n-k}\cdot \frac{F_X(k)}{F_{X}(k-1)}. \tag{2} $$ However, I can not adapt my proof of (2) to get (1), so any alternative proof of (2) would also be welcome.
