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A general formulation of the Jensen–Mercer operator inequality for operator convex functions, continuous fields of operators and unital fields of positive linear mappings is given. As consequences, a global upper bound for Jensen’s operator functional and some properties of the quasi-arithmetic operator means and quasi-arithmetic operator means of Mercer’s type are obtained.

We establish an operator extension of the following generalization of Bohr’s inequality, due to M.P. Vasi´c and D.J. Kečkić: ½Sn i=1 zi½r£ (Sni=1 a1i (1-r))r-1 Sni=1 ai ½zi½r (r<1, ziÎC, ai>0.1£i£n).We also present some inequalities related to our noncommutative generalization of Bohr’s inequality.

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In this paper, we prove Jensen’, s operator inequality for an h-convex function and we point out the results for classes of continuous fields of operators. Also, some generalizations of Jensen’, s operator inequality and some properties of the h-convex function are given.

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.

Minimum and maximum operators are two wellknown t-norm and s-norm used frequently in fuzzy systems. In this paper, two different types of fuzzy inequalities are simultaneously studied where the convex combination of minimum and maximum operators is applied as the fuzzy relational composition. Some basic properties and theoretical aspects of the problem are derived and four necessary and sufficient conditions are presented. Moreover, an algorithm is proposed to solve the problem and an example is described to illustrate the algorithm.

We prove that if positive invertible operators A and B satisfy an operator inequality (Bs/2A(s-t)/2BtA(s-t)/2Bs/2) 1/2s³B for some t>s>0, Then(1) If t³3s-2³0, then logB³logA, and if t³s+2 is additionally assumed, then B³A.(2) If 0<s<1/2, then logB³logA, and if t³s+2 is additionally assumed, then B³A.It is an interesting application of the Furuta inequality. Furthermore we consider some related results.