### Abstract

In this paper our aim is to show that if a probability density function is geometrically concave (convex), then the corresponding cumulative distribution function and the survival function are geometrically concave (convex) too, under some assumptions. The proofs are based on the so-called monotone form of l'Hospital's rule and permit us to extend our results to the case of the concavity (convexity) with respect to Hölder means. To illustrate the applications of the main results, we discuss in details the geometrical concavity of the probability density function, cumulative distribution function and survival function of some common continuous univariate distributions. Moreover, at the end of the paper, we present a simple alternative proof to Schweizer's problem related to the Mulholland's generalization of Minkowski's inequality.

Original language | English |
---|---|

Pages (from-to) | 182-196 |

Number of pages | 15 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 363 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1 2010 |

### Fingerprint

### Keywords

- Convexity (concavity) with respect to Hölder means
- Geometrically concave (convex) functions
- Log-concave (log-convex) functions
- Monotone form of l'Hospital's rule
- Mulholland's inequality
- Statistical distributions

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Geometrically concave univariate distributions.** / Baricz, A.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 363, no. 1, pp. 182-196. https://doi.org/10.1016/j.jmaa.2009.08.029

}

TY - JOUR

T1 - Geometrically concave univariate distributions

AU - Baricz, A.

PY - 2010/3/1

Y1 - 2010/3/1

N2 - In this paper our aim is to show that if a probability density function is geometrically concave (convex), then the corresponding cumulative distribution function and the survival function are geometrically concave (convex) too, under some assumptions. The proofs are based on the so-called monotone form of l'Hospital's rule and permit us to extend our results to the case of the concavity (convexity) with respect to Hölder means. To illustrate the applications of the main results, we discuss in details the geometrical concavity of the probability density function, cumulative distribution function and survival function of some common continuous univariate distributions. Moreover, at the end of the paper, we present a simple alternative proof to Schweizer's problem related to the Mulholland's generalization of Minkowski's inequality.

AB - In this paper our aim is to show that if a probability density function is geometrically concave (convex), then the corresponding cumulative distribution function and the survival function are geometrically concave (convex) too, under some assumptions. The proofs are based on the so-called monotone form of l'Hospital's rule and permit us to extend our results to the case of the concavity (convexity) with respect to Hölder means. To illustrate the applications of the main results, we discuss in details the geometrical concavity of the probability density function, cumulative distribution function and survival function of some common continuous univariate distributions. Moreover, at the end of the paper, we present a simple alternative proof to Schweizer's problem related to the Mulholland's generalization of Minkowski's inequality.

KW - Convexity (concavity) with respect to Hölder means

KW - Geometrically concave (convex) functions

KW - Log-concave (log-convex) functions

KW - Monotone form of l'Hospital's rule

KW - Mulholland's inequality

KW - Statistical distributions

UR - http://www.scopus.com/inward/record.url?scp=70350735988&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350735988&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2009.08.029

DO - 10.1016/j.jmaa.2009.08.029

M3 - Article

VL - 363

SP - 182

EP - 196

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -