Number of binary gates needed to compute AND and OR of n input bits simultaneously
This paper of Blum &amp; Seysen may be useful:

N.Blum, M. Seysen. Characterization of all Optimal Networks for a Simultaneous Computation of AND and NOR. Acta Inf. 21: 171-181 (1984)  

I have thought that for $x_1\dots x_n\vee \bar{x}_1\dots\bar{x}_n$ $2n-c$ lower bound can be obtained using methods of Blum &amp; Seysen, but it seems this is not the case.