Minkowski-type inequalities involving Hardy function and symmetric functions

The Hardy matrix H n ( x , α ) , the Hardy function per H n ( x , α ) and the generalized Vandermonde determinant det H n ( x , α ) are defined in this paper. By means of algebra and analysis theories together with proper hypotheses, we establish the following Minkowski-type inequality involving Hardy function: [ per H n ( x + y , α ) ] 1 | α | ⩾ [ per H n ( x , α ) ] 1 | α | + [ per H n ( y , α ) ] 1 | α | . As applications, our inequality is used to estimate the lower bounds of the increment of a symmetric function. MSC: 26D15, 15A15.