Jadranka Mićić

Complementary inequalities to inequalities of Jensen and Ando based on the Mond–Pečarić method

Abstract Applying the Mond–Pecaric method to unital positive linear maps, we shall show several complementary inequalities to Jensens inequalities on positive linear maps and consequently obtain complementary inequalities to Andos inequalities associated with operator means. We shall apply them to obtain complementary estimates for the results by Ando, Aujla–Vasudeva and Fujii on Hadamard product and operator means.

The Inequalities for Quasiarithmetic Means

Overview and refinements of the results are given for discrete, integral, functional and operator variants of inequalities for quasiarithmetic means. The general results are applied to further refinements of the power means. Jensens inequalities have been systematically presented, from the older variants, to the most recent ones for the operators without operator convexity.

Function order of positive operators based on the Mond–Pečarić method

We shall show function order preserving operator inequalities under general setting, based on Kantorovich type inequalities for convex functions due to Mond–Pecaric: Let A and B be positive operators on a Hilbert space H satisfying MI⩾B⩾mI>0. Let f(t) be a continuous convex function on [m,M]. If g(t) is a continuous increasing convex function on [m,M]∪Sp(A), then for a given α>0 A⩾B⩾0impliesαg(A)+βI⩾f(B), where β=maxm⩽t⩽M{f(m)+(f(M)−f(m))(t−m)/(M−m)−αg(t)}. As applications, we shall extend Kantorovich type operator inequalities by Furuta, Yamazaki and Yanagida, and present operator inequalities on the usual order and the chaotic order via Ky Fan–Furuta constant. Among others, we show the following inequality: If A⩾B>0 and MI⩾B⩾mI>0, then Mp−1mq−1Aq⩾(q−1)q−1qq(Mp−mp)q(M−m)(mMp−Mmp)q−1Aq⩾Bp holds for all p>1 and q>1 such that qmp−1⩽Mp−mpM−m⩽qMp−1.

Refined converses of Jensen’s inequality for operators

In this paper converses of a generalized Jensen’s inequality for a continuous field of self-adjoint operators, a unital field of positive linear mappings and real-valued continuous convex functions are studied. New refined converses are presented by using the Mond-Pečarić method improvement. Obtained results are applied to refine selected inequalities with power functions.MSC:47A63, 47A64.

Operator inequalities of Jensen type

Abstract We present some generalized Jensen type operator inequalities involving sequences of self-adjoint operators. Among other things, we prove that if f : [0;1) → ℝ is a continuous convex function with f(0) ≤ 0, then for all operators Ci such that (i=1 , ... , n) for some scalar M ≥ 0, where and

Inequalities among quasi-arithmetic means for continuous field of operators

In this paper we study inequalities among quasi-arithmetic means for a continuous field of self-adjoint operators, a field of positive linear mappings and continuous strictly monotone functions which induce means. We present inequalities with operator convexity and without operator convexity of appropriate functions. Also, we present a general formulation of converse inequalities in each of these cases. Furthermore, we obtain refined inequalities without operator convexity. As applications, we obtain inequalities among power means.

Inequalities of Furuta and Mond-Pečarić on the Hadamard product.

A case management system includes a case database storing case records in association with respective internal case identifiers. Operation includes automatically generating memorable case identifiers and providing them to users for use in identifying respective case records, the memorable case identifiers being generated by encoding the internal case identifiers with respective user identifiers as respective sequences of words (e.g., 3-word sequences) of a natural language of the users according to an encoding function. Case records are retrieved from the case database and provided to the users based on memorable case identifiers received from the users, by decoding received memorable case identifiers into respective internal case identifiers and accessing the case database using the respective internal case identifiers from the decoding.

Some complementary inequalities to Jensen's operator inequality

In this paper, we study some complementary inequalities to Jensen’s inequality for self-adjoint operators, unital positive linear mappings, and real valued twice differentiable functions. New improved complementary inequalities are presented by using an improvement of the Mond-Pečarić method. These results are applied to obtain some inequalities with quasi-arithmetic means.

Bohr’s Inequality Revisited

We survey several significant results on the Bohr inequality and presented its generalizations in some new approaches. These are some Bohr-type inequalities of Hilbert space operators related to the matrix order and the Jensen inequality. An eigenvalue extension of Bohr’s inequality is discussed as well.

Recent Research on Jensen's Inequality for Oparators

Let f be an operator convex function defined on an interval I. Ch. Davis [1] proved1 a Schwarz inequality f (φ(x)) ≤ φ ( f (x)) (2) where φ : A → B(K) is a unital completely positive linear mapping from a C∗-algebra A to linear operators on a Hilbert space K, and x is a self-adjoint element in A with spectrum in I. Subsequently M. D. Choi [2] noted that it is enough to assume that φ is unital and positive. In fact, the restriction of φ to the commutative C∗-algebra generated by x is automatically completely positive by a theorem of Stinespring.