Let $n,m$ be positive integers with $1 \le m < n$. For each pair of integers $z,r$ with $1 \le z \le n$ and $0 \le r \le z$, define the function $$ u_{z,r}(x) := \Pr[\operatorname{Bin}(n-z,\,x)\ge m-r] \;-\; \frac{r}{z}, \qquad x\in[0,1]. $$ Here $\operatorname{Bin}(N,x)$ denotes a binomial random variable with $N$ trials and success probability $x$.
Two special cases are: $$ u_{1,0}(x) = \Pr[\operatorname{Bin}(n-1,x)\ge m], $$ and $$ u_{n,m}(x) = 1 - \frac{m}{n}, $$ since $\operatorname{Bin}(0,x)=0$.
Question
Is it true that for all $x\in[0,1]$ and all admissible $z$, $r$, $$ u_{z,r}(x) \le \max\{u_{1,0}(x),\,u_{n,m}(x)\}. $$
Numerically, it has held for every example that I have examined. However, an analytical proof eludes me (I have only been able to show partial results, applying to part but not all of the parameter space).
Any proof or counterexample would be very welcome.
