### Abstract

The recent generalizations of the Boltzmann-Gibbs statistics mathematically rely on the deformed logarithmic and exponential functions defined through some deformation parameters. In the present work, we investigate whether a deformed logarithmic/exponential map is a bijection from R+ /R (set of positive real numbers/all real numbers) to R/ R+, as their undeformed counterparts. We show that their inverse map exists only in some subsets of the aforementioned (co)domains. Furthermore, we present conditions which a generalized deformed function has to satisfy, so that the most important properties of the ordinary functions are preserved. The fulfillment of these conditions permits us to determine the validity interval of the deformation parameters. We finally apply our analysis to Tsallis q -deformed functions and discuss the interval of concavity of the Ŕnyi entropy.

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

Article number | 103301 |

Journal | Journal of Mathematical Physics |

Volume | 50 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2009 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*50*(10), [103301]. https://doi.org/10.1063/1.3227657

**A note on the definition of deformed exponential and logarithm functions.** / Oikonomou, Thomas; Bagci, G. Baris.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 50, no. 10, 103301. https://doi.org/10.1063/1.3227657

}

TY - JOUR

T1 - A note on the definition of deformed exponential and logarithm functions

AU - Oikonomou, Thomas

AU - Bagci, G. Baris

PY - 2009

Y1 - 2009

N2 - The recent generalizations of the Boltzmann-Gibbs statistics mathematically rely on the deformed logarithmic and exponential functions defined through some deformation parameters. In the present work, we investigate whether a deformed logarithmic/exponential map is a bijection from R+ /R (set of positive real numbers/all real numbers) to R/ R+, as their undeformed counterparts. We show that their inverse map exists only in some subsets of the aforementioned (co)domains. Furthermore, we present conditions which a generalized deformed function has to satisfy, so that the most important properties of the ordinary functions are preserved. The fulfillment of these conditions permits us to determine the validity interval of the deformation parameters. We finally apply our analysis to Tsallis q -deformed functions and discuss the interval of concavity of the Ŕnyi entropy.

AB - The recent generalizations of the Boltzmann-Gibbs statistics mathematically rely on the deformed logarithmic and exponential functions defined through some deformation parameters. In the present work, we investigate whether a deformed logarithmic/exponential map is a bijection from R+ /R (set of positive real numbers/all real numbers) to R/ R+, as their undeformed counterparts. We show that their inverse map exists only in some subsets of the aforementioned (co)domains. Furthermore, we present conditions which a generalized deformed function has to satisfy, so that the most important properties of the ordinary functions are preserved. The fulfillment of these conditions permits us to determine the validity interval of the deformation parameters. We finally apply our analysis to Tsallis q -deformed functions and discuss the interval of concavity of the Ŕnyi entropy.

UR - http://www.scopus.com/inward/record.url?scp=70350725947&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350725947&partnerID=8YFLogxK

U2 - 10.1063/1.3227657

DO - 10.1063/1.3227657

M3 - Article

AN - SCOPUS:70350725947

VL - 50

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 10

M1 - 103301

ER -