We prove an OPERATOR arithmetic-harmonic MEAN type inequality in Krein space setting, by using some block matrix techniques of indefinite type. We also give an example which shows that the OPERATOR arithmetic-geometric-harmonic MEAN inequality for two invertible selfadjoint OPERATORs on Krein spaces is not valid, in general.

Please click on PDF to view the abstract.

Picture fuzzy set is the generalization of fuzzy set and intuitionistic fuzzy set. It is useful for handling uncertainty by considering the positive membership, neutral membership and negative membership degrees independently for each element of a universal set. The main objective of this article is to develop some picture fuzzy MEAN OPERATORs, including Picture Fuzzy Harmonic MEAN (PFHM), Picture Fuzzy Weighted Harmonic MEAN (PFWHM), Picture Fuzzy Arithmetic MEAN (PFAM), Picture Fuzzy Weighted Arithmetic MEAN (PFWAM), Picture Fuzzy Geometric MEAN (PFGM) and Picture Fuzzy Weighted Geometric MEAN (PFWGM), to aggregate the picture fuzzy sets. Moreover, we discuss some relevant properties of these OPERATORs. Furthermore, we apply these MEAN OPERATORs to make decisions with practical examples. Finally, to show the efficiency and the validity of the proposed OPERATORs, we compare our results with the results of existing methods and concluded from the comparison that our proposed methods are more effective and reliable.

In this paper we derive some improvements of MEANs inequalities for Hilbert space OPERATORs. More precisely, we obtain refinements and reverses of the arithmetic-geometric OPERATOR MEAN inequality. As an application, we also deduce an improved variant for the refined arithmetic-Heinz MEAN in- equality. We also present some eigenvalue inequalities for differences of certain OPERATOR MEANs.