The $ L_p $-Petty projection inequality for $ s $-concave functions

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.