### Abstract

A real function is called radially Q-differentiable at the point x if, for every real number h, the finite limit d_{Q} f ( x, h ) of the ratio ( f ( x + r h ) - f ( x ) ) / r exists whenever r tends to zero through the positive rationals. We establish that, in particular, Jensen-convex functions are everywhere radially Q-differentiable. Moreover, if f is Jensen-convex, then, for each x, the mapping h {mapping} d_{Q} f ( x, h ) is subadditive, and it is an upper bound for any additive mapping A satisfying the inequality f ( x ) + A ( y - x ) {less-than or slanted equal to} f ( y ) for every y. We also characterize all set-valued mappings built up from additive solutions A of this inequality with some Jensen-convex function f.

Original language | English |
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Pages (from-to) | 99-113 |

Number of pages | 15 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 321 |

Issue number | 1 |

DOIs | |

Publication status | Published - Sep 1 2006 |

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### Keywords

- Convex function
- Cyclically monotone mapping
- Differentiability
- Subadditive function
- Subdifferential
- Subgradient

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

*321*(1), 99-113. https://doi.org/10.1016/j.jmaa.2005.07.061