If $0\leq c_k\leq 1$ and $n$ is any positive integer, then is$$\frac{n\prod_{k=1}^nc_k}{1+n\prod_{k=1}^nc_k}\leq \sum_{k=1}^n\frac{c_k}{1+c_k}?$$It is true for at least one of the $c_k=0$ and is also true if all $c_k'$s are equal. Please suggest if this inequality already exists in the literature.
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