Suppose that $c_k$ is an decreasing sequence of non-negative real numbers, such that $c_0=1$ and $c_{k}\leq \frac{1}{2}(c_{k-1}+c_{k+1})$.
Is it true that the generated function of $c_k$ admit an integral representation as below $$ \sum_{k=0}^{\infty}c_kz^k=\int_{\partial\mathbb D} \frac{1}{1-\zeta z}d\mu(\zeta), $$ where $\mu$ is a Borel Probability Measure in $\partial\mathbb D$ ?
Motivation: This question is related a possible slight different solution for the question asked in https://math.stackexchange.com/questions/2188/complex-analysis-question whose the answer, as pointed out by damiano, can be found at the IMC website.