The mixability of elliptical distributions and log-elliptical distributions

The concept of $\phi$-complete mixability and $\phi$-joint mixability was first introduced in Bignozzi and Puccetti (2015), which is an extension of complete and joint mixability. Following Bignozzi and Puccetti (2015), we consider two more general cases of $\phi$ and investigate the $\phi$-joint mixability for elliptical distributions and logarithmic elliptical distributions. Some sufficient conditions for the $\phi$-joint mixability of some distributions are investigated. In addition, a conjecture on the uniqueness of the center of $\phi$-joint mixability and the forms of the densities for some elliptical distributions are given.