On Hermitian positive definite solution of the matrix equation

A conjecture that the nonlinear matrix equation X − ∑ i = 1 m A i ∗ X r A i = Q ( − 1 ≤ r < 0  or  0 < r < 1 ) always has a unique Hermitian positive definite solution is proved. Some bounds of the unique Hermitian positive definite solution are given.