• Panuganti Srinivasa Sai

we investigate several inequalities related to the Heinz mean for positive semidefinite matrices. Motivated by a question posed by Bourin concerning the validity of certain unitarily invariant norm inequalities, we present results that provide partial affirmative answers in specific cases. In particular, we discuss inequalities involving the trace norm and norms of the real and imaginary parts of matrix expressions associated with the Heinz mean. We also review refinements of the Cauchy–Schwarz norm inequality for operators and present related trace inequalities inspired by the work of Ando, Hiai, and Plevnik. These results contribute to the ongoing study of matrix inequalities and operator means. he Heinz mean is a fundamental matrix mean that interpolates between the arithmetic and geometric means for positive definite matrices. In this paper, we study several inequalities involving the Heinz mean and the trace norm. By employing properties of unitarily invariant norms and convexity arguments, we establish new trace norm inequalities and refine existing bounds related to the Heinz mean. These results provide sharper estimates and extend known inequalities in matrix analysis and operator theory..

Heinz mean; Matrix inequalities; Trace norm; Unitarily invariant norms; Positive semidefinite matrices

"Some Inequalities Related to the Heinz Mean and Trace Norms", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.13, Issue 1, page no.a551-a557, January-2026, Available :http://www.jetir.org/papers/JETIR2601067.pdf

"Some Inequalities Related to the Heinz Mean and Trace Norms", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.13, Issue 1, page no. ppa551-a557, January-2026, Available at : http://www.jetir.org/papers/JETIR2601067.pdf