I am an electrical engineer and currently I am working with an application that involve binary matrix in a previous question. From the previous question, I need to proof that this function is concave with respect to $q$.
This is because then when I fix $m$, $n$ and plot this function with respect to $q$ in the range $0 \le q \le 1$, the function seems to be concave.
$f\left( q \right) = m\sum\limits_{k = 2}^{n} {\left[ {\left( {1 - {{\left[ {1 - kq{{\left( {1 - q} \right)}^{n - 1}}} \right]}^{m - 1}}} \right)\left( {\begin{array}{*{20}{c}} n\\ k \end{array}} \right){q^k}{{\left( {1 - q} \right)}^{n - k}}} \right]} $
Note that $m$ and $n$ are positive integers.
I think that it is very difficult to conduct this proof by using a direct second order derivative test because the first order derivative for the case of $m=n$ has already looks terrible.
$f'\left( q \right) = n\sum\limits_{k = 2}^n {\left( {{q^k}\left( {\begin{array}{*{20}{c}} n\\ k \end{array}} \right)\left( { - k{n^2}q{{(1 - q)}^{n - 2}}{{\left( {1 - kq{{(1 - q)}^{n - 1}}} \right)}^{n - 2}} - kq{{(1 - q)}^{n - 2}}{{\left( {1 - kq{{(1 - q)}^{n - 1}}} \right)}^{n - 2}} - k{{(1 - q)}^{n - 1}}{{\left( {1 - kq{{(1 - q)}^{n - 1}}} \right)}^{n - 2}} + kn{{(1 - q)}^{n - 1}}{{\left( {1 - kq{{(1 - q)}^{n - 1}}} \right)}^{n - 2}} + 2knq{{(1 - q)}^{n - 2}}{{\left( {1 - kq{{(1 - q)}^{n - 1}}} \right)}^{n - 2}}} \right){{(1 - q)}^{n - k}} + k{q^{k - 1}}\left( {\begin{array}{*{20}{c}} n\\ k \end{array}} \right)\left( {1 - {{\left( {1 - kq{{(1 - q)}^{n - 1}}} \right)}^{n - 1}}} \right){{(1 - q)}^{n - k}} + {q^k}\left( {\begin{array}{*{20}{c}} n\\ k \end{array}} \right)\left( {1 - {{\left( {1 - kq{{(1 - q)}^{n - 1}}} \right)}^{n - 1}}} \right)\left( {k{{(1 - q)}^{ - k + n - 1}} - n{{(1 - q)}^{ - k + n - 1}}} \right)} \right)}$
The function definitely doesn't look concave to me (assuming $m\ge2$). Check the behavior of the graph more closely near $q=0$ and $q=1$. Series expansions at those points will show that it is proportional to $q^3$ near $q=0$ and proportional to $(1-q)^4$ near $q=1$, both of which disprove concavity.