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DOI 10.1515/anly-2014-1252 | Analysis 2014; 34(2): 223–240 Sever S. Dragomir* Refinements of the Ostrowski inequality in terms of the cumulative variation and applications Abstract: Refinements of the Ostrowski inequality for functions of bounded varia- tion in terms of the cumulative variation function are given. Applications for self- adjoint operators on complex Hilbert spaces are also provided. Keywords: Ostrowski’s inequality, functions of bounded variation, cumulative variation, selfadjoint operators. AMS (2000): 26D15, 47A63 || *Corresponding Author: Sever

Abstract

We show that Tanahashi’s argument on best possibility of the grand Furuta inequality has an additional consequence.

Abstract

Some trace inequalities of Shisha-Mond type for operators in Hilbert spaces are provided. Applications in connection to Grüss inequality and for convex functions of selfadjoint operators are also given.

Abstract

One of the couple of translatable radii of an operator in the direction of another operator introduced in earlier work [PAUL, K.: Translatable radii of an operator in the direction of another operator, Scientae Mathematicae 2 (1999), 119–122] is studied in details. A necessary and sufficient condition for a unit vector f to be a stationary vector of the generalized eigenvalue problem Tf = λAf is obtained. Finally a theorem of Williams ([WILLIAMS, J. P.: Finite operators, Proc. Amer. Math. Soc. 26 (1970), 129–136]) is generalized to obtain a translatable radius of an operator in the direction of another operator.

Abstract

We consider the notion of real center of mass and total center of mass of a bounded linear operator relative to another bounded linear operator and explore their relation with cosine and total cosine of a bounded linear operator acting on a complex Hilbert space. We give another proof of the Min-max equality and then generalize it using the notion of orthogonality of bounded linear operators. We also illustrate with examples an alternative method of calculating the antieigenvalues and total antieigenvalues for finite dimensional operators.

Abstract

In this paper, we present numerous refinements of the Young inequality by the Kantorovich constant. We use these improved inequalities to establish corresponding operator inequalities on a Hilbert space and some new inequalities involving the Hilbert-Schmidt norm of matrices. We also give some refinements of the following Heron type inequality for unitarily invariant norm |||⋅||| and A, B, XMn(ℂ):

|||AνXB1ν+A1νXBν2|||(4r01)|||A12XB12|||+2(12r0)|||(1α)A12XB12+α(AX+XB2)|||,

where 14ν34,α[12,) and r 0 = min{ν, 1 – ν}.

Abstract

We consider different fractional Neumann Laplacians of order s(0,1) on domains Ωn, namely, the restricted Neumann Laplacian (-ΔΩN)Rs, the semirestricted Neumann Laplacian (-ΔΩN)Srs and the spectral Neumann Laplacian (-ΔΩN)Sps. In particular, we are interested in the attainability of Sobolev constants for these operators when Ω is a half-space.

Abstract

In this work, an operator version of Popoviciu’s inequality for positive operators on Hilbert spaces under positive linear maps for superquadratic functions is proved. Analogously, using the same technique, an operator version of Popoviciu’s inequality for convex functions is obtained. Some other related inequalities are also presented.

Abstract

Let f(z)=n=0αnzn be a function defined by power series with complex coefficients and convergent on the open disk D (0, R) ⊂ ℂ, R > 0. For any x, y ∈ ℬ, a Banach algebra, with ‖x‖, ‖y‖ < R we show among others that

f(y)f(x)yx01fa((1t)x+ty)dt
where fa(z)=n=0|αn|zn . Inequalities for the commutator such as
f(x)f(y)f(y)f(x)2fa(M)fa(M)yx,
if ‖x‖, ‖y‖ ≤ M < R, as well as some inequalities of Hermite–Hadamard type are also provided.

Abstract

Some trace inequalities of Cassels type for operators in Hilbert spaces are provided. Applications in connection to Grüss inequality and for convex functions of selfadjoint operators are also given.