Schwarz inequality

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Some Inequalities for Power Series of Selfadjoint Operators in Hilbert Spaces via Reverses of the Schwarz Inequality

Some Inequalities for Power Series of Selfadjoint Operators in Hilbert Spaces via Reverses of the Schwarz Inequality

Abstract. In this paper we obtain some operator inequalities for functions de…ned by power series with real coe¢ cients and, more speci…cally, with non- negative coe¢ cients. In order to obtain these inequalities some recent reverses of the Schwarz inequality for vectors in inner product spaces are utilized. Nat- ural applications for some elementary functions of interest are also provided.

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A Generalization of the Cauchy-Schwarz Inequality with Four Free Parameters and Applications

A Generalization of the Cauchy-Schwarz Inequality with Four Free Parameters and Applications

3.1.2. Sub-case 2. p = s ∈ [−2, 0] (A refinement for the Cauchy-Schwarz inequality). Suppose in (3.1) that p = s ∈ [−2, 0] and p(p + 2) = u. Consequently u ∈ [−1, 0]. By noting these assumptions we can obtain a refinement for inequality (1.1). For this purpose, first the following inequality should be considered, which is directly provable via some algebraic computations

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Generalizations of Cauchy Schwarz inequality in unitary spaces

Generalizations of Cauchy Schwarz inequality in unitary spaces

The Cauchy-Schwarz inequality is one of the most important inequalities in mathemat- ics. To date, a large number of generalizations and refinements of the inequalities (.) and (.) have been investigated in the literature (see [] and references therein, also see [–]). In this note, we will present some new generalizations of the Cauchy-Schwarz inequality (.).

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More results on a functional generalization of the Cauchy Schwarz inequality

More results on a functional generalization of the Cauchy Schwarz inequality

By using a specific functional property, some more results on a functional generalization of the Cauchy-Schwarz inequality, such as an extension of the pre-Grüss inequality and a refinement of the Cauchy-Schwarz inequality via the generalized Wagner inequality, are given for both discrete and continuous cases. MSC: 26D15; 26D20

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Reverses of the Cauchy-Bunyakovsky-Schwarz Inequality for n-tuples of Complex Numbers

Reverses of the Cauchy-Bunyakovsky-Schwarz Inequality for n-tuples of Complex Numbers

The case where the disk ¯ D (α, r) does not contain the origin, i.e., |α| > r, provides the following interesting reverse of the Cauchy-Bunyakovsky-Schwarz inequality. Theorem 2. Let a, b, p as in Theorem 1 and assume that |α| > r > 0. Then we have the inequality

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Reverses of the Schwarz Inequality in Inner Product Spaces Generalising a Klamkin-Mclenaghan Result

Reverses of the Schwarz Inequality in Inner Product Spaces Generalising a Klamkin-Mclenaghan Result

As pointed out in [4], the above results are motivated by the fact that they generalise to the case of real or complex inner product spaces some classical reverses of the Cauchy-Bunyakovsky-Schwarz inequality for positive n−tuples due to Polya- Szeg¨ o [8], Cassels [10], Shisha-Mond [9] and Greub-Rheinboldt [6].

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Refinements of the Cauchy-Bunyakovsky-Schwarz Inequality for Functions of Selfadjoint Operators in Hilbert Spaces

Refinements of the Cauchy-Bunyakovsky-Schwarz Inequality for Functions of Selfadjoint Operators in Hilbert Spaces

Corollary 4 . Let (ϕ, ψ) be a (DEC)-pair of continuous functions on [0, ∞)× [0, ∞) . If A is a selfadjoint operator on the Hilbert space (H ; h., .i) with Sp (A) ⊆ [m, M ] for some scalars m < M and if f and g are continuous on [m, M ] and with values in [0, ∞) , then we have the inequality

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A Generalization of the Cauchy Schwarz Inequality with Eight Free Parameters

A Generalization of the Cauchy Schwarz Inequality with Eight Free Parameters

The results of the recent published paper by Masjed-Jamei et al. 2009 are extended to a larger class and some of subclasses are studied in the sequel. In other words, we generalize the well known Cauchy-Schwarz and Cauchy-Bunyakovsky inequalities having eight free parameters and then introduce some of their interesting subclasses.

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A note on the Frobenius conditional number with positive definite matrices

A note on the Frobenius conditional number with positive definite matrices

In this article, we focus on the lower bounds of the Frobenius condition number. Using the generalized Schwarz inequality, we present some lower bounds for the Frobenius condition number of a positive definite matrix depending on its trace, determinant, and Frobenius norm. Also, we give some results on a kind of matrices with special structure, the positive definite matrices with centrosymmetric structure.

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Recent developments of Schwarz's type trace inequalities for operators in Hilbert spaces

Recent developments of Schwarz's type trace inequalities for operators in Hilbert spaces

For other trace inequalities see [7], [18], [41], [33], [51], [58], [74] and [80]. In this paper we survey some recent trace inequalities obtained by the author for operators in Hilbert spaces that are connected to Schwarz’s, Buzano’s and Kato’s inequalities and the reverses of Schwarz inequality known in the litera- ture as Cassels’ inequality and Shisha–Mond’s inequality. Applications for some functionals that are naturally associated to some of these inequalities and for functions of operators defined by power series are given. Examples for fundamen- tal functions such as the power, logarithmic, resolvent and exponential functions are provided as well.

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Generalized Norms Inequalities for Absolute Value Operators

Generalized Norms Inequalities for Absolute Value Operators

In this section, we generalize some unitarily invariant norms inequalities for ab- solute value operators. Our results based on several lemmas. First two lemmas contain norm inequalities of Minkowski type and generalized forms of the Cauchy- Schwarz inequality, see [4] and [2] respectively.

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Two Mappings Related To Semi-Inner Products And
Their Applications in Geometry of Normed Linear
Spaces

Two Mappings Related To Semi-Inner Products And Their Applications in Geometry of Normed Linear Spaces

generate the norm of a real normed linear space, and study properties of monotonicity and bound- edness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.

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RRRR Specifically the dot product of two vectors a

RRRR Specifically the dot product of two vectors a

equal to 1, with equality if and only if x and y are nonzero scalar multiples of each other. More generally, the Cauchy – Schwarz inequality allows one to define a notion of “the angle between the two vectors” for an arbitrary inner product, where the extendibility of such concepts from Euclidean geometry may not be intuitively clear. In particular, since the angle ∠ ∠ ∠ ∠ x 0 y is a right angle if and only if the cosine is zero, we have the following:

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Mathematical Uncertainty Relations and their Generalization for Multiple Incompatible Observables

Mathematical Uncertainty Relations and their Generalization for Multiple Incompatible Observables

Abstract: We show that the famous Heisenberg uncertainty relation for two incompatible observables can be generalized elegantly to the determinant form for N arbitrary observables. To achieve this purpose, we propose a generalization of the Cauchy-Schwarz inequality for two sets of vectors. Simple consequences of the N-ary uncertainty relation are also discussed.

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Inequalities Of Schwarz Type For n-Tuples Of Vectors In Inner Product Spaces With Applications

Inequalities Of Schwarz Type For n-Tuples Of Vectors In Inner Product Spaces With Applications

[24] K. Trenˇ cevski and R. Malˇ ceski, On a generalized n-inner product and the corresponding Cauchy-Schwarz inequality. J. Inequal. Pure Appl. Math. 7 (2006), no. 2, Article 53, 10 pp. [25] L. Tuo, Generalizations of Cauchy-Schwarz inequality in unitary spaces, submitted. [26] G.-B. Wang and J.-P. Ma, Some results on reverses of Cauchy-Schwarz inequality in inner

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Spectral Radii of Operators and High-Power Operator Inequalities

Spectral Radii of Operators and High-Power Operator Inequalities

Theorem 1. Let T ≥ 0, and S and C be arbitrary operators. Also let T S, T C, A and B be all selfadjoint operators. If n is a positive integer, then for all x, y ∈ H, y 6= x, the following are equivalent to one another and to the Cauchy-Schwarz inequality (1.1):

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Some Equivalent Forms of the Arithematic Geometric Mean Inequality in Probability: A Survey

Some Equivalent Forms of the Arithematic Geometric Mean Inequality in Probability: A Survey

The arithmetic-geometric mean inequality in short, AG inequality has been widely used in mathematics and in its applications. A large number of its equivalent forms have also been developed in several areas of mathematics. For probability and mathematical statistics, the equivalent forms of the AG inequality have not been linked together in a formal way. The purpose of this paper is to prove that the AG inequality is equivalent to some other renowned inequalities by using probabilistic arguments. Among such inequalities are those of Jensen, H ¨older, Cauchy, Minkowski, and Lyapunov, to name just a few.

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Norm inequalities for operators related to the Cauchy Schwarz and Heinz inequalities

Norm inequalities for operators related to the Cauchy Schwarz and Heinz inequalities

invariant norm, A, B, X ∈ B(H) with A, B positive, X ∈ τ · , and r > . By the convexity of f we have f ( v+  ) ≤   {f ( v+  ) + f ( v+  )} and   {f (v) + f ( v+  ) + f (   )} ≤   {f (v) + f (   )} for v ∈ [, ] \ {   } . So, norm inequality () is a refinement of inequality () obtained by Burqan [], Theorem .

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Extensions of interpolation between the arithmetic geometric mean inequality for matrices

Extensions of interpolation between the arithmetic geometric mean inequality for matrices

Our application of the methods based on the Audenaert results is presented in this paper to the operator norm and so are some interpolations for an arbitrary unitarily invariant norm. Moreover, we refine some previous inequalities as regards the Cauchy-Schwarz in- equality for the operator and Hilbert-Schmidt norms.

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Reversing the CBS-Inequality for Sequences of Vectors in Hilbert Spaces with Applications (II)

Reversing the CBS-Inequality for Sequences of Vectors in Hilbert Spaces with Applications (II)

In [6], by the use of some preliminary results obtained in [3], various reverses for the (CBS)-type inequalities (1.13) and (1.14) for sequences of vectors in Hilbert spaces were obtained. Applications for bounding the distance to a finite-dimensional subspace and in reversing the generalised triangle inequality have also been pro- vided.

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