We give the Choi-Davis-Jensen type inequality without using convexity. Applying our main results,... more We give the Choi-Davis-Jensen type inequality without using convexity. Applying our main results, we also give new inequalities improving previous known results. In particular, we show some inequalities for relative operator entropies and quantum mechanical entropies.
In this paper we deal with Sherman's inequality and its complementary inequalities for operator c... more In this paper we deal with Sherman's inequality and its complementary inequalities for operator convex functions, whose arguments are the bounded self-adjoint operators from C *-algebra on a Hilbert space and positive linear mappings between C *-algebras. We introduce the so called Sherman's operator and study its properties. Applying an extended idea of convexity to operator functions of several variables, we obtain multidimensional Sherman's operator inequality. We define multidimensional Sherman's operator and study its properties. At the end, we observe applications to some operator inequalities related to connections, solidarities, and multidimensional weighted geometric mean.
Utilizing the Mond-Pečarić method and the properties of operator means, we present some Ando's ty... more Utilizing the Mond-Pečarić method and the properties of operator means, we present some Ando's type inequalities. Some inequalities in the existing literature are also generalized by our results.
In Chapter 1 we give a very brief and quick review of the basic facts about a Hilbert space and (... more In Chapter 1 we give a very brief and quick review of the basic facts about a Hilbert space and (bounded linear) operators on a Hilbert space, which will recur throughout the book. In Chapter 2 we tell the history of the Kantorovich inequality, and describe how the Kantorovich inequality develops in the field of operator inequalities. In such context, the method for convex functions established by Mond and Pe\v{;c};ari\'{;c}; (commonly known as "the Mond- Pecaric method") has outlined a more complete picture of that inequality in the field of operator inequalities. We discuss ratio and difference type converses of operator versions of Jensen's inequality. These constants in terms of spectra of given self- adjoint operators have many interesting properties and are connected with a closed relation, and play an essential role in the remainder of this book. In Chapter 3 we explain fundamental operator inequalities related to the Furuta inequality. The base point is the Lowner-Heinz inequality. It induces weighted geometric means, which serves as an excellent technical tool. The chaotic order log A >= log B is conceptually important in the late discussion. In Chapter 4 we study the order preserving operator inequality in another direction which differs from the Furuta inequality. We investigate the Kantorovich type inequalities related to the operator ordering and the chaotic one. In Chapter 5 as applications of the Mond-Pecaric method for convex functions, we discuss inequalities involving the operator norm. Among others, we show a converse of the Araki-Cordes inequality, the norm inequality of seve\-ral geometric means and a complement of the Ando-Hiai inequality. Also, we discuss Holder's inequality and its converses in connection with the operator geometric mean. In Chapter 6 we define the geometric mean of n- operators due to Ando-Li-Mathias and Lowson-Lim. We present an alternative proof of the power convergence of the symmetrization procedure on the weighted geometric mean due to Lawson and Lim. We show a converse of the weighted arithmetic-geometric mean inequality of n-operators. In Chapter 7 we give some differential-geometrical structure of operators. The space of positive invertible operators of a unital C*-algebra has the natural structure of a reductive homogenous manifold with a Finsler metric. Then a pair of points A and B can be joined by a unique geodesic A #_t B for t in [0, 1] and we consider estimates of the upper bounds for the difference between the geodesic and extended interpolation paths in terms of the spectra of positive operators. In Chapter 8 we give some properties of Mercer's type inequalities. A variant of Jensen's operator inequality for convex functions, which is a generalization of Mercer's result, is proved. We show a monotonicity property for Mercer's power means for operators, and a comparison theorem for quasi-arithmetic means for operators. In Chapter 9 a general formulation of Jensen's operator inequality for some non-unital fields of positive linear mappings is given. Next, we consider different types of converse inequalities. We discuss the ordering among power functions in a general setting. We get the order among power means and some comparison theorems for quasi- arithmetic means. We also give a refined calculation of bounds in converses of Jensen's operator inequality. In Chapter 10 we give Jensen's operator inequality without operator convexity. We observe this inequality for n-tuples of self-adjoint operators, unital n-tuples of positive linear mappings and real valued convex functions with conditions on the operators bounds. In the present context, we also give an extension and a refinement of Jensen's operator inequality. As an application we get the order among quasi-arithmetic operator means. In Chapter 11 we observe some operator versions of Bohr's inequality. Using a general result involving matrix ordering, we derive several inequalities of Bohr's type. Furthermore, we present an approach to Bohr's inequality based on a generalization of the parallelogram theorem with absolute values of operators. Finally, applying Jensen's operator inequality we get a generalization of Bohr's inequality.
Ovi nastavni materijali prate nastavni plan i program kolegija Matematika II koji se održava u lj... more Ovi nastavni materijali prate nastavni plan i program kolegija Matematika II koji se održava u ljetnom semestru prve godine preddiplomskog studija Fakulteta strojarsta i brodogradnje Sveucilista u Zagrebu. Sadržaj je podijeljen u cetiri cjeline: Redovi potencija, Tehnike integriranja, Diferencijalne jednadžbe, Realne funkcije vise varijabli, Visestruki integrali. U svakoj cjelini ukratko je objasnjena teorija te je rijeseno dosta primjera i zadataka kako bi studenti i samostalno mogli savladati navedeno gradivo. Materijali sadrže mnogo slika i animacija.
For a selfadjoint operator A on a Hilbert space H and a normalized positive linear map Φ, a quasi... more For a selfadjoint operator A on a Hilbert space H and a normalized positive linear map Φ, a quasi-arithmetic mean is defined by ϕ −1 (Φ(ϕ(A))) for a strictly monotone function ϕ. In this paper, we shall show an order relation among quasi-arithmetic means for convex functions through positive linear maps and its complementary problems, in which we use the Mond-Pečarić method for convex functions.
Potaknuti clancima o novim nastavnim planovima i programima u srednjim skolama i fakultetima, te ... more Potaknuti clancima o novim nastavnim planovima i programima u srednjim skolama i fakultetima, te o novim temama seminarskih i diplomskih radova na sveucilistima u Hrvatskoj (vidjeti [1], [6], [7], [4]), u ovom clanku imamo namjeru govoriti o tome kako se danas izvodi nastava matematike na prvoj godini dodiplomskog studija FSB-u u skladu s Bolonjskom deklaracijom, te kako se provjeravaju ishodi ucenja.
We survey several significant results on the Bohr inequality and presented its generalizations in... more 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.
The authors discuss on the constant K>0 for the operator inequality 0≤B p ≤KA p if 0≤B≤A with ... more The authors discuss on the constant K>0 for the operator inequality 0≤B p ≤KA p if 0≤B≤A with 0<mI≤A≤MI for some scalars m, M and p≥1. It is called Kantorovich type operator inequality. They obtain a more precise constant K than M. Fujii, E. Kamei and Y. Seo [Sci. Math. 3, 263–272 (2000; Zbl 0967.47007)] and M. Hashimoto and T. Yamazaki [Sci. Math. 3, 127–136 (2000; Zbl 0969.47012)].
ABSTRACT The authors continue their investigations from [Linear Algebra Appl. 318, No. 1–3, 87–10... more ABSTRACT The authors continue their investigations from [Linear Algebra Appl. 318, No. 1–3, 87–107 (2000; Zbl 0971.47014)]. They show further general complementary inequalities to operator inequalities on a positive linear map associated with two operator means on Hilbert spaces. Connections with the work of J. S. Aujla and H. L. Vasudeva [Math. Jap. 41, No. 2, 383–388 (1995; Zbl 0819.47021)], J. I. Fuji [Math. Jap. 41, No. 3, 531–535 (1995; Zbl 0826.47016)] on Hadamard products and operator means, and B. Mond, J. E. Pečarić, J. Šunde and S. Varošanec [Linear Algebra Appl. 264, 117–126 (1997; Zbl 0963.47011)] on mixed operator means are established.
Let A, Φ and ω be k-touples of positive invertible operators A j with Sp(A j )⊆[m,M] for some sca... more Let A, Φ and ω be k-touples of positive invertible operators A j with Sp(A j )⊆[m,M] for some scalars 0<m<M, normalized positive linear maps Φ j , and positive numbers ω j with ∑ j=1 k ω j =1, respectively. For A, Φ, ω, and a real number r≠0, let M k [r] (A,Φ,ω):=(∑ j=1 k ω j Φ j (A j r )) 1/r . The authors examine the constants α 1 ,α 2 , β 1 and β 2 such that α 2 M k [s] (A,Φ,ω)≤M k [r] (A,Φ,ω)≤α 1 M k [s] (A,Φ,ω) and β 2 I≤M k [s] (A,Φ,ω)-M k [r] (A,Φ,ω)≤β 1 I hold if r≤s, r,s≠0.
The authors discuss on the constant K>0 for the operator inequality 0≤B p ≤KA p if 0≤B≤A with ... more The authors discuss on the constant K>0 for the operator inequality 0≤B p ≤KA p if 0≤B≤A with 0<mI≤A≤MI for some scalars m, M and p≥1. It is called Kantorovich type operator inequality. They obtain a more precise constant K than M. Fujii, E. Kamei and Y. Seo [Sci. Math. 3, 263–272 (2000; Zbl 0967.47007)] and M. Hashimoto and T. Yamazaki [Sci. Math. 3, 127–136 (2000; Zbl 0969.47012)].
ABSTRACT The authors continue their investigations from [Linear Algebra Appl. 318, No. 1–3, 87–10... more ABSTRACT The authors continue their investigations from [Linear Algebra Appl. 318, No. 1–3, 87–107 (2000; Zbl 0971.47014)]. They show further general complementary inequalities to operator inequalities on a positive linear map associated with two operator means on Hilbert spaces. Connections with the work of J. S. Aujla and H. L. Vasudeva [Math. Jap. 41, No. 2, 383–388 (1995; Zbl 0819.47021)], J. I. Fuji [Math. Jap. 41, No. 3, 531–535 (1995; Zbl 0826.47016)] on Hadamard products and operator means, and B. Mond, J. E. Pečarić, J. Šunde and S. Varošanec [Linear Algebra Appl. 264, 117–126 (1997; Zbl 0963.47011)] on mixed operator means are established.
We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tu... more We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tuples of positive linear mappings, and real-valued continuous convex functions with conditions on the operators' bounds. We also study operator quasiarithmetic means under the same conditions.
We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tu... more We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tuples of positive linear mappings, and real-valued continuous convex functions with conditions on the operators' bounds. We also study operator quasiarithmetic means under the same conditions.
As a continuation of our previous research [J. Mićić, J. Pečarić and Y. Seo, Converses of Jensen'... more As a continuation of our previous research [J. Mićić, J. Pečarić and Y. Seo, Converses of Jensen's operator inequality, accepted to Oper. Matrices 4 (2010), 3, 385-403], we discuss order among quasi-arithmetic means of positive operators with fields of positive linear mappings (φt)t∈T such that T φt(1) dµ(t) = k1 for some positive scalar k.
We give new results on Levinson's operator inequality and its converse for normalized positive li... more We give new results on Levinson's operator inequality and its converse for normalized positive linear mappings and some large class of '3-convex functions at a point c' .
We give the Choi-Davis-Jensen type inequality without using convexity. Applying our main results,... more We give the Choi-Davis-Jensen type inequality without using convexity. Applying our main results, we also give new inequalities improving previous known results. In particular, we show some inequalities for relative operator entropies and quantum mechanical entropies.
In this paper we deal with Sherman's inequality and its complementary inequalities for operator c... more In this paper we deal with Sherman's inequality and its complementary inequalities for operator convex functions, whose arguments are the bounded self-adjoint operators from C *-algebra on a Hilbert space and positive linear mappings between C *-algebras. We introduce the so called Sherman's operator and study its properties. Applying an extended idea of convexity to operator functions of several variables, we obtain multidimensional Sherman's operator inequality. We define multidimensional Sherman's operator and study its properties. At the end, we observe applications to some operator inequalities related to connections, solidarities, and multidimensional weighted geometric mean.
Utilizing the Mond-Pečarić method and the properties of operator means, we present some Ando's ty... more Utilizing the Mond-Pečarić method and the properties of operator means, we present some Ando's type inequalities. Some inequalities in the existing literature are also generalized by our results.
In Chapter 1 we give a very brief and quick review of the basic facts about a Hilbert space and (... more In Chapter 1 we give a very brief and quick review of the basic facts about a Hilbert space and (bounded linear) operators on a Hilbert space, which will recur throughout the book. In Chapter 2 we tell the history of the Kantorovich inequality, and describe how the Kantorovich inequality develops in the field of operator inequalities. In such context, the method for convex functions established by Mond and Pe\v{;c};ari\'{;c}; (commonly known as "the Mond- Pecaric method") has outlined a more complete picture of that inequality in the field of operator inequalities. We discuss ratio and difference type converses of operator versions of Jensen's inequality. These constants in terms of spectra of given self- adjoint operators have many interesting properties and are connected with a closed relation, and play an essential role in the remainder of this book. In Chapter 3 we explain fundamental operator inequalities related to the Furuta inequality. The base point is the Lowner-Heinz inequality. It induces weighted geometric means, which serves as an excellent technical tool. The chaotic order log A >= log B is conceptually important in the late discussion. In Chapter 4 we study the order preserving operator inequality in another direction which differs from the Furuta inequality. We investigate the Kantorovich type inequalities related to the operator ordering and the chaotic one. In Chapter 5 as applications of the Mond-Pecaric method for convex functions, we discuss inequalities involving the operator norm. Among others, we show a converse of the Araki-Cordes inequality, the norm inequality of seve\-ral geometric means and a complement of the Ando-Hiai inequality. Also, we discuss Holder's inequality and its converses in connection with the operator geometric mean. In Chapter 6 we define the geometric mean of n- operators due to Ando-Li-Mathias and Lowson-Lim. We present an alternative proof of the power convergence of the symmetrization procedure on the weighted geometric mean due to Lawson and Lim. We show a converse of the weighted arithmetic-geometric mean inequality of n-operators. In Chapter 7 we give some differential-geometrical structure of operators. The space of positive invertible operators of a unital C*-algebra has the natural structure of a reductive homogenous manifold with a Finsler metric. Then a pair of points A and B can be joined by a unique geodesic A #_t B for t in [0, 1] and we consider estimates of the upper bounds for the difference between the geodesic and extended interpolation paths in terms of the spectra of positive operators. In Chapter 8 we give some properties of Mercer's type inequalities. A variant of Jensen's operator inequality for convex functions, which is a generalization of Mercer's result, is proved. We show a monotonicity property for Mercer's power means for operators, and a comparison theorem for quasi-arithmetic means for operators. In Chapter 9 a general formulation of Jensen's operator inequality for some non-unital fields of positive linear mappings is given. Next, we consider different types of converse inequalities. We discuss the ordering among power functions in a general setting. We get the order among power means and some comparison theorems for quasi- arithmetic means. We also give a refined calculation of bounds in converses of Jensen's operator inequality. In Chapter 10 we give Jensen's operator inequality without operator convexity. We observe this inequality for n-tuples of self-adjoint operators, unital n-tuples of positive linear mappings and real valued convex functions with conditions on the operators bounds. In the present context, we also give an extension and a refinement of Jensen's operator inequality. As an application we get the order among quasi-arithmetic operator means. In Chapter 11 we observe some operator versions of Bohr's inequality. Using a general result involving matrix ordering, we derive several inequalities of Bohr's type. Furthermore, we present an approach to Bohr's inequality based on a generalization of the parallelogram theorem with absolute values of operators. Finally, applying Jensen's operator inequality we get a generalization of Bohr's inequality.
Ovi nastavni materijali prate nastavni plan i program kolegija Matematika II koji se održava u lj... more Ovi nastavni materijali prate nastavni plan i program kolegija Matematika II koji se održava u ljetnom semestru prve godine preddiplomskog studija Fakulteta strojarsta i brodogradnje Sveucilista u Zagrebu. Sadržaj je podijeljen u cetiri cjeline: Redovi potencija, Tehnike integriranja, Diferencijalne jednadžbe, Realne funkcije vise varijabli, Visestruki integrali. U svakoj cjelini ukratko je objasnjena teorija te je rijeseno dosta primjera i zadataka kako bi studenti i samostalno mogli savladati navedeno gradivo. Materijali sadrže mnogo slika i animacija.
For a selfadjoint operator A on a Hilbert space H and a normalized positive linear map Φ, a quasi... more For a selfadjoint operator A on a Hilbert space H and a normalized positive linear map Φ, a quasi-arithmetic mean is defined by ϕ −1 (Φ(ϕ(A))) for a strictly monotone function ϕ. In this paper, we shall show an order relation among quasi-arithmetic means for convex functions through positive linear maps and its complementary problems, in which we use the Mond-Pečarić method for convex functions.
Potaknuti clancima o novim nastavnim planovima i programima u srednjim skolama i fakultetima, te ... more Potaknuti clancima o novim nastavnim planovima i programima u srednjim skolama i fakultetima, te o novim temama seminarskih i diplomskih radova na sveucilistima u Hrvatskoj (vidjeti [1], [6], [7], [4]), u ovom clanku imamo namjeru govoriti o tome kako se danas izvodi nastava matematike na prvoj godini dodiplomskog studija FSB-u u skladu s Bolonjskom deklaracijom, te kako se provjeravaju ishodi ucenja.
We survey several significant results on the Bohr inequality and presented its generalizations in... more 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.
The authors discuss on the constant K>0 for the operator inequality 0≤B p ≤KA p if 0≤B≤A with ... more The authors discuss on the constant K>0 for the operator inequality 0≤B p ≤KA p if 0≤B≤A with 0<mI≤A≤MI for some scalars m, M and p≥1. It is called Kantorovich type operator inequality. They obtain a more precise constant K than M. Fujii, E. Kamei and Y. Seo [Sci. Math. 3, 263–272 (2000; Zbl 0967.47007)] and M. Hashimoto and T. Yamazaki [Sci. Math. 3, 127–136 (2000; Zbl 0969.47012)].
ABSTRACT The authors continue their investigations from [Linear Algebra Appl. 318, No. 1–3, 87–10... more ABSTRACT The authors continue their investigations from [Linear Algebra Appl. 318, No. 1–3, 87–107 (2000; Zbl 0971.47014)]. They show further general complementary inequalities to operator inequalities on a positive linear map associated with two operator means on Hilbert spaces. Connections with the work of J. S. Aujla and H. L. Vasudeva [Math. Jap. 41, No. 2, 383–388 (1995; Zbl 0819.47021)], J. I. Fuji [Math. Jap. 41, No. 3, 531–535 (1995; Zbl 0826.47016)] on Hadamard products and operator means, and B. Mond, J. E. Pečarić, J. Šunde and S. Varošanec [Linear Algebra Appl. 264, 117–126 (1997; Zbl 0963.47011)] on mixed operator means are established.
Let A, Φ and ω be k-touples of positive invertible operators A j with Sp(A j )⊆[m,M] for some sca... more Let A, Φ and ω be k-touples of positive invertible operators A j with Sp(A j )⊆[m,M] for some scalars 0<m<M, normalized positive linear maps Φ j , and positive numbers ω j with ∑ j=1 k ω j =1, respectively. For A, Φ, ω, and a real number r≠0, let M k [r] (A,Φ,ω):=(∑ j=1 k ω j Φ j (A j r )) 1/r . The authors examine the constants α 1 ,α 2 , β 1 and β 2 such that α 2 M k [s] (A,Φ,ω)≤M k [r] (A,Φ,ω)≤α 1 M k [s] (A,Φ,ω) and β 2 I≤M k [s] (A,Φ,ω)-M k [r] (A,Φ,ω)≤β 1 I hold if r≤s, r,s≠0.
The authors discuss on the constant K>0 for the operator inequality 0≤B p ≤KA p if 0≤B≤A with ... more The authors discuss on the constant K>0 for the operator inequality 0≤B p ≤KA p if 0≤B≤A with 0<mI≤A≤MI for some scalars m, M and p≥1. It is called Kantorovich type operator inequality. They obtain a more precise constant K than M. Fujii, E. Kamei and Y. Seo [Sci. Math. 3, 263–272 (2000; Zbl 0967.47007)] and M. Hashimoto and T. Yamazaki [Sci. Math. 3, 127–136 (2000; Zbl 0969.47012)].
ABSTRACT The authors continue their investigations from [Linear Algebra Appl. 318, No. 1–3, 87–10... more ABSTRACT The authors continue their investigations from [Linear Algebra Appl. 318, No. 1–3, 87–107 (2000; Zbl 0971.47014)]. They show further general complementary inequalities to operator inequalities on a positive linear map associated with two operator means on Hilbert spaces. Connections with the work of J. S. Aujla and H. L. Vasudeva [Math. Jap. 41, No. 2, 383–388 (1995; Zbl 0819.47021)], J. I. Fuji [Math. Jap. 41, No. 3, 531–535 (1995; Zbl 0826.47016)] on Hadamard products and operator means, and B. Mond, J. E. Pečarić, J. Šunde and S. Varošanec [Linear Algebra Appl. 264, 117–126 (1997; Zbl 0963.47011)] on mixed operator means are established.
We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tu... more We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tuples of positive linear mappings, and real-valued continuous convex functions with conditions on the operators' bounds. We also study operator quasiarithmetic means under the same conditions.
We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tu... more We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tuples of positive linear mappings, and real-valued continuous convex functions with conditions on the operators' bounds. We also study operator quasiarithmetic means under the same conditions.
As a continuation of our previous research [J. Mićić, J. Pečarić and Y. Seo, Converses of Jensen'... more As a continuation of our previous research [J. Mićić, J. Pečarić and Y. Seo, Converses of Jensen's operator inequality, accepted to Oper. Matrices 4 (2010), 3, 385-403], we discuss order among quasi-arithmetic means of positive operators with fields of positive linear mappings (φt)t∈T such that T φt(1) dµ(t) = k1 for some positive scalar k.
We give new results on Levinson's operator inequality and its converse for normalized positive li... more We give new results on Levinson's operator inequality and its converse for normalized positive linear mappings and some large class of '3-convex functions at a point c' .
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