We study the "functionalization" of the $L_p$-projection body and two related important inequalities in geometry. On the class of $s$-concave functions, a general function counterpart of the $L_p$-projection body is introduced. In addition, the $L_p$-Petty projection inequality and the $L_p$ isoperimetric inequality are established on this class. Finally, we show that the $L_p$-Petty projection inequality strengthens the $L_p$ isoperimetric inequality on the same class.