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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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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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** (.).

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

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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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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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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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.

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.

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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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.

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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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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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.

[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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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):

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.

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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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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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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