Some further notes on the matrix equations ATX B + BTXTA = C and ATX B + BTX A = C

Dehghan and Hajarian, [4], investigated the matrix equations ATX B + BTXTA = C and ATX B + BTX A = C providing inequalities for the determinant of the solutions of these equations. In the same paper, the authors presented a lower bound for the product of the eigenvalues of the solutions to these matrix equations. Inspired by their work, we give some generalizations of Dehghan and Hajarian results. Using the theory of the numerical ranges, we present an inequality involving the trace of C when A,B,X are normal matrices satisfying ATB = BAT.