On the complexity of monotone circuits for threshold symmetric Boolean functions

The complexity of implementation of a threshold symmetric $n$-place Boolean function with threshold $k = O(1)$ via circuits over the basis $\{\vee,\, \wedge\}$ is shown not to exceed $2 \log_2 k \cdot n + o(n)$. Moreover, the complexity of a threshold-2 function is proved to be $2n+\Theta(\sqrt n)$, and the complexity of a threshold-3 function is shown to be $3n+O(\log n) $, the corresponding lower bounds are put forward.