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Zbl 1231.47012
Dragomir, S.S.
Ostrowski's type inequalities for continuous functions of selfadjoint operators on Hilbert spaces: a survey of recent results. (English)
[J] Ann. Funct. Anal. AFA 2, No. 1, 139-205, electronic only (2011). ISSN 2008-8752/e

This is a survey paper which introduces several inequalities related to the Ostrowski inequality. Let $f:[a,b] \to \mathbb{R}$ be a differentiable function on $(a,b)$ with the property that $|f'(t)|\leq M$ for all $t\in (a,b)$. Then the Ostrowski inequality is known as $$ \left| f(x) -\frac{1}{b-a}\int_{a}^{b}f(t)\,dt\right| \leq \left[ \frac{1}{4}+\left(\frac{x-\frac{a+b}{2}}{b-a}\right)^{2}\right](b-a)M $$ for all $x\in [a,b]$. Moreover, it is known that the constant $\frac{1}{4}$ is optimal. The author himself has contributed several results to the study of the above inequality, and reviews them in the present survey paper. The contents of this paper is as follows. (1) Introduction. (2) Continuous functions of selfadjoint operators. (3) The spectral representation theorem. (4) Ostrowski type inequalities for Hölder continuous functions. (5) Other Ostrowski inequalities for continuous functions. (6) More Ostrowski type inequalities. (7) Some vector inequalities for monotonic functions. (8) Ostrowski type vector inequalities. (9) Bounds for the difference between functions and integral means. (10) Ostrowski's type inequalities for $n$-time differentiable functions.
[Takeaki Yamazaki (Kawagoe)]

MSC 2000:: *47A63 Operator inequalities, etc.
47-02 Research monographs (operator theory)
26D15 Inequalities for sums, series and integrals of real functions

Keywords: Ostrowski's inequality; selfadjoint operators; positive operators; functions of selfadjoint operators.

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