## Annals of Functional Analysis

- Ann. Funct. Anal.
- Volume 8, Number 1 (2017), 142-151.

### An Inequality for expectation of means of positive random variables

Paolo Gibilisco and Frank Hansen

#### Abstract

Suppose that $X$, $Y$ are positive random variables and $m$ is a numerical (commutative) mean. We prove that the inequality $\mathrm{E}\left(m\right(X,Y\left)\right)\le m\left(\mathrm{E}\right(X),\mathrm{E}(Y\left)\right)$ holds if and only if the mean is generated by a concave function. With due changes we also prove that the same inequality holds for all operator means in the Kubo–Ando setting. The case of the harmonic mean was proved by C. R. Rao and B. L. S. Prakasa Rao.

#### Article information

**Source**

Ann. Funct. Anal., Volume 8, Number 1 (2017), 142-151.

**Dates**

Received: 9 June 2016

Accepted: 1 August 2016

First available in Project Euclid: 12 November 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.afa/1478919628

**Digital Object Identifier**

doi:10.1215/20088752-3750087

**Mathematical Reviews number (MathSciNet)**

MR3572337

**Zentralblatt MATH identifier**

1354.26053

**Subjects**

Primary: 26E60: Means [See also 47A64]

Secondary: 47A64: Operator means, shorted operators, etc. 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

**Keywords**

numerical means operator means concavity random matrices

#### Citation

Gibilisco, Paolo; Hansen, Frank. An Inequality for expectation of means of positive random variables. Ann. Funct. Anal. 8 (2017), no. 1, 142--151. doi:10.1215/20088752-3750087. https://projecteuclid.org/euclid.afa/1478919628