A clean upper bound for the expectation of a function of a binomial random variable

This is the mean absolute deviation (MAD) for a binomial distribution, divided by $n$. The expectation is hence $$2 \, (1-p)^{n+1-\lceil np \rceil} \, p^{\lceil np \rceil} \, \binom{n-1}{\lceil np \rceil-1} \;.$$ See this paper (Berend & Kontorovich 2013, doi: 10.1016/j.spl.2013.01.023) for bounds and a reference for the above expression.