A similar analysis to Theorem 4. If T 1 and T 2 commute, then. The A-G-H weighted mean inequality, i. Firstly, we show the second inequality. Assume the inequality holds for some k. Then, by virtue of. Then, using.
Some means inequalities for positive operators in Hilbert spaces
The authors also would like to thank the reviewers very much for their useful suggestions. Competing interests. All the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
National Center for Biotechnology Information , U. Journal of Inequalities and Applications. J Inequal Appl. Published online Jan Jin Liang and Guanghua Shi.
Author information Article notes Copyright and License information Disclaimer. Jin Liang, Email: nc. Corresponding author. Received Sep 10; Accepted Dec Abstract In this paper, we obtain two refinements of the ordering relations among Heinz means with different parameters via the Taylor series of some hyperbolic functions and by the way, we derive new generalizations of Heinz operator inequalities.
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Keywords: means inequalities, positive linear operators, Hilbert space. Introduction Since Heinz proved a series of useful norm inequalities, which are closely related to the Cordes inequality and the Furuta inequality, in , many researchers have devoted themselves to sharping the Heinz inequalities and extending the Heinz norm inequalities to more general cases with the help of a Bernstein type inequality for nonselfadjoint operators, the convexity of norm functions, the Jensen functional and its properties, the Hermite-Hadamard inequality, and so on. Theorem 2. Theorem 3.
Subjects Primary: 47A Operator inequalities 47A Numerical range, numerical radius 47A Norms inequalities, more than one norm, etc. Secondary: 30B Power series including lacunary series. Dragomir, S.
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More by S. More by Y. Read more about accessing full-text Buy article. Abstract Article info and citation First page Abstract In this paper we obtain some operator inequalities for functions defined by power series with complex coefficients and, more specifically, with nonnegative coefficients. Article information Source Afr. Export citation.
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If , then for any. Now, consider the logarithmic function. We have the following. Then one has the inequalities.
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Dragomir SS: Some Ostrowski's type vector inequalities for functions of selfadjoint operators in Hilbert spaces. Dragomir SS: Comparison between functions of selfadjoint operators in Hilbert spaces and integral means. Dragomir SS: Approximating n-time differentiable functions of selfadjoint operators in Hilbert spaces by two point Taylor's type expansion.
Element, Zagreb; Linear Algebra and Its Applications , 2—3 — Houston Journal of Mathematics , 19 3 — Utilitas Mathematica , — Houston Journal of Mathematics , 30 1 — Linear and Multilinear Algebra , 58 7—8 — Dragomir SS: Some trapezoidal vector inequalities for continuous functions of delfadjoint operators in Hilbert spaces.
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enter site Reprints and Permissions. Search all SpringerOpen articles Search. Abstract On utilizing the spectral representation of self-adjoint operators in Hilbert spaces, some inequalities for the composite operator , where and for various classes of continuous functions are given. Introduction Let be a self-adjoint operator on the complex Hilbert space with the spectrum included in the interval for some real numbers and let be its spectral family.
Some Vector Inequalities The following representation in terms of the spectral family is of interest in itself.