Suppose $c>2$ is a real number, and $x$ solves equation $f(\frac{1}{2}+x,m,2m)=(2m+1)cx$ for some $m>0$ integer number, where $f$ is probability mass function for binomial distribution. Simulations show that it must hold: $Pr(X\geq m+1)\leq \frac{1}{2}+\frac{1}{c}$, where $X$ random variable is distributed binomially with probability $\frac{1}{2}+x$ and population size $2m+1$. Hints how to prove or a counterexample?

Suppose $c>2$ is a real number, and $x$ solves equation $f(\frac{1}{2}+x,m,2m)=(2m+1)cx$ for some $m>0$ integer number, where $f$ is probability mass function for binomial distribution, with probability $\frac{1}{2}+x$ and population size $2m$. 

Simulations show that it must hold: $Pr(X\geq m+1)\leq \frac{1}{2}+\frac{1}{c}$, where $X$ random variable is distributed binomially with probability $\frac{1}{2}+x$ and population size $2m+1$. Hints how to prove the inequality?