Behavior of binomial distribution near its median

We study the behavior of the cumulative distribution function of a binomial random variable with parameters $n$ and $b/(n+c)$ at the point $b-1$ for positive integers $b\le n$ and real $c\in[0,1]$. Our results can be applied directly to the well-known problem about small deviations of sums of independents random variables from their expectations. Moreover, we answer the question about the monotonicity of the Ramanujan function for the binomial distribution posed by Jogdeo and Samuels in 1968.