Sector preserving functions on sector matrices

Subject Areas : Analyze

Amir ghasem Ghazanfari ¹ , Somayeh Malekinejad ^{2 *} , Hamideh Mohammadzadehkan ³

1 - Department of MathematicsLorestan UniversityP. O. Box 465, Khoramabad, Iran.
2 - Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran
3 - Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran

Received: 2024-09-04 Accepted : 2025-03-17 Published : 2025-12-01

Keywords: Accretive matrix, Matrix mean, Sector matrix, Matrix monotone function, Sector preserving function.,

Abstract :

Abstract

 

In recent decades, matrix mean inequalities for positive matrices have been studied and investigated, and it is important to understand how these inequalities are applied to piecewise matrices. In this paper, we have obtained several mean inequalities for sector matrices, some of which are extensions of the results obtained by Bedrani et al. For example, for a positive unitary linear mapping , sector matrices    and , a real-valued function preserving  , and arbitrary means , we have proved the following important inequality:

,

This inequality for positive matrices and harmonic and geometric means is as follows:
.

Among the other results of the article, with the above conditions, we can mention the following useful inequality:

.

 

In the following, as applications of these inequalities, we have stated and proved the smoothness condition, determinant, and singular values for piecewise matrices. Finally, we have given examples that express the importance of the inequalities proved in the article.

References:

[1] Bedrani, Y., Kittaneh, F., Sababheh, M. (2021). From positive to accretive matrices. Positivity, 25(4), 1601–1629.
[2] Bedrani, Y., Kittaneh, F., Sababheh, M. (2021). Numerical radii of accretive matrices. Linear and Multilinear Algebra, 69(5), 957–970.
[3] Bhatia, R. (1997). Matrix Analysis. Springer-Verlag, New York.
[4] Bhatia, R. (2007). Positive Definite Matrices. Princeton University Press, Princeton.
[5] Durury, S., Lin, M. (2014). Singular value inequalities for matrices with numerical ranges in a sector. Operators and Matrices, 8, 1143–1148.
[6] Ghazanfari, A., Malekinejad, S. (2021). Heron means and Pólya inequality for sector matrices. Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, 64, 329–339.
[7] Ghazanfari, A. G., Malekinejad, S. (2023). Operator monotone functions on accretive matrices. Positivity, 27(5), 59.
[8] Horn, R. A., Johnson, C. R. (2013). Matrix Analysis. Cambridge University Press, Cambridge.
[9] Lin, M. (2015). Extension of a result of Hanynsworth and Hartfiel. Archiv der Mathematik, 104, 93–100.
[10] Lin, M. (2016). Some inequalities for sector matrices. Operators and Matrices, 10(4), 915–921.
[11] Malekinejad, S., Khosravi, M., Sheikhhosseini, A. (2022). Mean inequalities for sector matrices involving positive linear maps. Positivity, 26(44), 1–17.
[12] Tan, F., Che, H. (2019). Inequalities for sector matrices and positive linear maps. The Electronic Journal of Linear Algebra, 35, 418–423.
[13] Zhan, F. (2015). A matrix decomposition and its applications. Linear and Multilinear Algebra, 63(10), 2033–2042