### Abstract

We present several inequalities for f_{a}=Γ(a,x)/Γ(a,0) (a>0, x≥0) where Γ(a,x) is the incomplete gamma function. One of our theorems states that the inequalities. f_{a} (S_{p}(x_{1},...,x_{n})) ≤ f_{a}(x_{1})...f_{a}(x_{n}) ≤ f_{a}(S_{q}(x_{1},...,x_{n})) (p,q>0) hold for all nonnegative real numbers x_{1},...,x_{n} (n≥2) if and only if p≤min(a,1) and q≥max(a,1). Here, St(x_{1},...,x_{n}) denotes the power sum of order t. This extends and complements a result published by Ismail and Laforgia in 2006.

Original language | English |
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Pages (from-to) | 167-178 |

Number of pages | 12 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 385 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2012 |

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### Keywords

- Arithmetic, geometric, and harmonic means
- Completely monotonic
- Concave
- Convex
- Functional inequalities
- Grünbaum-type inequality
- Incomplete gamma function
- Power means
- Power sums
- Subadditive
- Turán-type inequality

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics