### Abstract

Motivated by the well-known implications among t-convexity properties of real functions, analogous relations among the upper and lower M-convexity properties of real functions are established. More precisely, having an n-tuple (M_{1}, ..., M_{n}) of continuous two-variable means, the notion of the descendant of these means (which is also an n-tuple (N_{1}, ..., N_{n}) of two-variable means) is introduced. In particular, when all the means M_{i} are weighted arithmetic, then the components of their descendants are also weighted arithmetic means. More general statements are obtained in terms of the generalized quasi-arithmetic or Matkowski means. The main results then state that if a function f is M_{i}-convex for all i∈{1, ..., n}, then it is also N_{i}-convex for all i∈{1, ..., n}. Several consequences are discussed.

Original language | English |
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Pages (from-to) | 1852-1874 |

Number of pages | 23 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 434 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 15 2016 |

### Keywords

- Convexity with respect to a mean
- Descendant of means
- Divided differences
- Fixed point theorems

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics