Inequalities involving the Khatri-Rao product of maps on positive definite matrices

We shall show several complementary inequalities to Jensen's type inequality involving the Khatri-Rao product of maps on positive definite matrices. They are applied to extend some known inequalities involving powers of the Khatri-Rao product. Besides, we find the best bounds among those we obtain by using the Mond-Pecaric method.