# Entropy Optimization, Generalized Logarithms, and Duality Relations

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## Abstract

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_{q}entropies

## 1. Introduction

## 2. ${\mathit{S}}_{\mathit{q}}$ Entropies, q-Logarithms, and q-Exponential Maximum-Entropy Probability Distributions

## 3. Generalized Entropies and Logarithms

#### 3.1. The Duality Condition Satisfied by the ${S}_{q}$ Thermostatistics

#### 3.2. The Simplest Duality Relation

#### 3.3. More General Duality Relations

#### 3.4. Duality Relations: The Inverse Problem

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Plot of the Mittag-Leffler function ${E}_{a,b}\left(x\right)$, for $b=1$ and illustrative values of the parameter a; ${E}_{1,1}\left(x\right)={e}^{x}$.

**Figure 2.**Plot of the inverse Mittag-Leffler function, ${E}_{a,b}^{(-1)}\left(x\right)$, for $b=1$ and specific values of the parameter a; ${E}_{1,1}^{(-1)}\left(x\right)=\mathrm{ln}x$.

**Figure 3.**Plotof the function $\mathcal{C}\left(x\right)$ corresponding to $J\left(x\right)={E}_{a}\left(x\right)$, for different values of the parameter a. The function $\mathcal{C}\left(x\right)$ appears in the definition of a trace-form entropic measure (51), and is given by Equation (52). For $a=1$, one has ${E}_{1}\left(x\right)=\mathrm{exp}\left(x\right)$ and $C\left(x\right)=-x\mathrm{ln}x$.

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**MDPI and ACS Style**

Plastino, A.R.; Tsallis, C.; Wedemann, R.S.; Haubold, H.J.
Entropy Optimization, Generalized Logarithms, and Duality Relations. *Entropy* **2022**, *24*, 1723.
https://doi.org/10.3390/e24121723

**AMA Style**

Plastino AR, Tsallis C, Wedemann RS, Haubold HJ.
Entropy Optimization, Generalized Logarithms, and Duality Relations. *Entropy*. 2022; 24(12):1723.
https://doi.org/10.3390/e24121723

**Chicago/Turabian Style**

Plastino, Angel R., Constantino Tsallis, Roseli S. Wedemann, and Hans J. Haubold.
2022. "Entropy Optimization, Generalized Logarithms, and Duality Relations" *Entropy* 24, no. 12: 1723.
https://doi.org/10.3390/e24121723