# On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures

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## Abstract

**:**

## 1. Introduction

## 2. $(p,\beta ,\lambda )$-Fisher–Rényi Complexity and the Extended Stam Inequality

#### 2.1. Rényi Entropy, Extended Fisher Information and Rényi–Fisher Complexity

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

#### 2.2. Shift and Scale Invariance, Bounding from Below and Minimizing Distributions

**Proposition**

**1.**

**Proof.**

**Proposition**

**2**

**Proof.**

**Proposition**

**3.**

**Proof.**

#### 2.3. Some Explicitly Known Minimizing Distributions

**Definition**

**4**

#### 2.3.1. The Case $\beta =\lambda $

#### 2.3.2. Stretched Deformed Gaussian: The Symmetric Case

#### 2.3.3. Dealing with the Usual Fisher Information

#### 2.3.4. The Symmetrical of the Usual Fisher Information

## 3. Extended Optimal Stam Inequality: A Step Further

#### 3.1. Differential-Escort Distribution: A Brief Overview

**Definition**

**5**

**Property**

**1.**

**Proposition**

**4.**

#### 3.2. Enlarging the Validity Domain of the Extended Stam Inequality

- Consider a point $(\beta ,\lambda )\in {\tilde{\mathcal{D}}}_{p}$ and find an index $\alpha \in {\mathbb{R}}_{+}^{*}$ such that ${\mathfrak{A}}_{\alpha}(\beta ,\lambda )\in {\mathcal{D}}_{p}$, which is a point of the intersection between ${\mathcal{D}}_{p}$ and the line joining $(0,1)$ and $(\beta ,\lambda )$.
- Apply Proposition 2 for the point $(p,{\mathfrak{A}}_{\alpha}(\beta ,\lambda ))$, leading to the minimizing distribution ${\rho}_{p,{\mathfrak{A}}_{\alpha}(\beta ,\lambda )}$ and its corresponding bound.
- Then, remarking that ${\mathfrak{A}}_{{\alpha}^{-1}}\circ {\mathfrak{A}}_{\alpha}(\beta ,\lambda )=(\beta ,\lambda )$, the minimizer of the extended complexity writes ${\rho}_{p,\beta ,\lambda}={\mathfrak{E}}_{\alpha}\left[{\rho}_{p,{\mathfrak{A}}_{\alpha}(\beta ,\lambda )}\right]$ and the corresponding bound can be computed from this minimizer or noting that ${K}_{p,\beta ,\lambda}={\alpha}^{2}{K}_{p,{\mathfrak{A}}_{\alpha}(\beta ,\lambda )}$.

**Definition**

**6**

**Proposition**

**5**

**Proof.**

## 4. Applications to Quantum Physics

#### 4.1. Brief Review on the Quantum Systems with Radial Potential

#### 4.2. $(p,\beta ,\lambda )$-Fisher–Rényi Complexity and the Hydrogenic System

#### 4.3. $(p,\beta ,\lambda )$-Fisher–Rényi Complexity and the Harmonic System

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Proof of Proposition 2

#### Appendix A.1. The Case λ ≠ 1

#### Appendix A.1.1. The Sub-Case λ < 1

#### Appendix A.1.2. The Sub-Case λ > 1

#### Appendix A.2. The Case λ = 1

## Appendix B. Proof of Proposition 3

## Appendix C. Proof of Proposition 5

#### Appendix C.1. The (p,β,λ)-Fisher–Rényi Complexity is Lowerbounded over ${\tilde{\mathcal{D}}}_{p}$

#### Appendix C.2. Explicit Expression for the Minimizers.

#### Appendix C.2.1. The Case 1 − p*β < λ < 1

#### Appendix C.2.2. The Case λ > 1

#### Appendix C.2.3. The Case λ = 1

#### Appendix C.3. Symmetry through the Involution ${\mathfrak{T}}_{p}$.

#### Appendix C.4. Explicit Expression of the Lower Bound.

## References

- Sen, K.D. Statistical Complexity. Application in Electronic Structure; Springer: New York, NY, USA, 2011. [Google Scholar]
- López-Ruiz, R.; Mancini, H.L.; Calbet, X. A statistical measure of complexity. Phys. Lett. A
**1995**, 209, 321–326. [Google Scholar] [CrossRef] - López-Ruiz, R. Shannon information, LMC complexity and Rényi entropies: A straightforward approach. Biophys. Chem.
**2005**, 115, 215–218. [Google Scholar] [CrossRef] [PubMed] - Chatzisavvas, K.C.; Moustakidis, C.C.; Panos, C.P. Information entropy, information distances, and complexity in atoms. J. Chem. Phys.
**2005**, 123, 174111. [Google Scholar] [CrossRef] [PubMed] - Sen, K.D.; Panos, C.P.; Chatzisavvas, K.C.; Moustakidis, C.C. Net Fisher information measure versus ionization potential and dipole polarizability in atoms. Phys. Lett. A
**2007**, 364, 286–290. [Google Scholar] [CrossRef] [Green Version] - Bialynicki-Birula, I.; Rudnicki, Ł. Entropic uncertainty relations in quantum physics. In Statistical Complexity. Application in Electronic Structure; Sen, K.D., Ed.; Springer: Berlin, Germay, 2010. [Google Scholar]
- Dehesa, J.S.; López-Rosa, S.; Manzano, D. Entropy and complexity analyses of D-dimensional quantum systems. In Statistical Complexities: Application to Electronic Structure; Sen, K.D., Ed.; Springer: Berlin, Germany, 2010. [Google Scholar]
- Huang, Y. Entanglement detection: Complexity and Shannon entropic criteria. IEEE Trans. Inf. Theor.
**2013**, 59, 6774–6778. [Google Scholar] [CrossRef] - Ebeling, W.; Molgedey, L.; Kurths, J.; Schwarz, U. Entropy, complexity, predictability and data analysis of time series and letter sequences. In Theory of Disaster; Springer: Berlin, Germany, 2000. [Google Scholar]
- Angulo, J.C.; Antolín, J. Atomic complexity measures in position and momentum spaces. J. Chem. Phys.
**2008**, 128, 164109. [Google Scholar] [CrossRef] [PubMed] - Rosso, O.A.; Ospina, R.; Frery, A.C. Classification and verification of handwritten signatures with time causal information theory quantifiers. PLoS ONE
**2016**, 11, e0166868. [Google Scholar] [CrossRef] [PubMed] - Toranzo, I.V.; Sánchez-Moreno, P.; Rudnicki, Ł.; Dehesa, J.S. One-parameter Fisher-Rényi complexity: Notion and hydrogenic applications. Entropy
**2017**, 19, 16. [Google Scholar] [CrossRef] - Angulo, J.C.; Romera, E.; Dehesa, J.S. Inverse atomic densities and inequalities among density functionals. J. Math. Phys.
**2000**, 41, 7906–7917. [Google Scholar] [CrossRef] - Dehesa, J.S.; López-Rosa, S.; Martínez-Finkelshtein, A.; Yáñez, R.J. Information theory of D-dimensional hydrogenic systems: Application to circular and Rydberg states. Int. J. Quantum Chem.
**2010**, 110, 1529–1548. [Google Scholar] [CrossRef] - López-Rosa, S.; Esquievel, R.O.; Angulo, J.C.; Antolín, J.; Dehesa, J.S.; Flores-Gallegos, N. Fisher information study in position and momentum spaces for elementary chemical reactions. J. Chem. Theor. Comput.
**2010**, 6, 145–154. [Google Scholar] [CrossRef] [PubMed] - Romera, E.; Sánchez-Moreno, P.; Dehesa, J.S. Uncertainty relation for Fisher information of D-dimensional single-particle systems with central potentials. J. Math. Phys.
**2006**, 47, 103504. [Google Scholar] [CrossRef] - Sánchez-Moreno, P.; Zozor, S.; Dehesa, J.S. Upper bounds on Shannon and Rényi entropies for central potential. J. Math. Phys.
**2011**, 52, 022105. [Google Scholar] [CrossRef] - Zozor, S.; Portesi, M.; Sánchez-Moreno, P.; Dehesa, J.S. Position-momentum uncertainty relation based on moments of arbitrary order. Phys. Rev. A
**2011**, 83, 052107. [Google Scholar] [CrossRef] - Martin, M.T.; Plastino, A.R.; Plastino, A. Tsallis-like information measures and the analysis of complex signals. Phys. A Stat. Mech. Appl.
**2000**, 275, 262–271. [Google Scholar] [CrossRef] - Portesi, M.; Plastino, A. Generalized entropy as measure of quantum uncertainty. Phys. A Stat. Mech. Appl.
**1996**, 225, 412–430. [Google Scholar] [CrossRef] - Massen, S.E.; Panos, C.P. Universal property of the information entropy in atoms, nuclei and atomic clusters. Phys. Lett. A
**1998**, 246, 530–533. [Google Scholar] [CrossRef] - Guerrero, A.; Sanchez-Moreno, P.; Dehesa, J.S. Upper bounds on quantum uncertainty products and complexity measures. Phys. Rev. A
**2011**, 84, 042105. [Google Scholar] [CrossRef] - Dehesa, J.S.; Sánchez-Moreno, P.; Yáñez, R.J. Crámer-Rao information plane of orthogonal hypergeometric polynomials. J. Comput. Appl. Math.
**2006**, 186, 523–541. [Google Scholar] [CrossRef] - Antolín, J.; Angulo, J.C. Complexity analysis of ionization processes and isoelectronic series. Int. J. Quantum Chem.
**2009**, 109, 586–593. [Google Scholar] [CrossRef] - Angulo, J.C.; Antolín, J.; Sen, K.D. Fisher-Shannon plane and statistical complexity of atoms. Phys. Lett. A
**2008**, 372, 670–674. [Google Scholar] [CrossRef] - Romera, E.; Dehesa, J.S. The Fisher-Shannon information plane, an electron correlation tool. J. Chem. Phys.
**2004**, 120, 8906–8912. [Google Scholar] [CrossRef] [PubMed] - Puertas-Centeno, D.; Toranzo, I.V.; Dehesa, J.S. The biparametric Fisher-Rényi complexity measure and its application to the multidimensional blackbody radiation. J. Stat. Mech. Theor. Exp.
**2017**, 2017, 043408. [Google Scholar] [CrossRef] - Sobrino-Coll, N.; Puertas-Centeno, D.; Toranzo, I.V.; Dehesa, J.S. Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems. J. Stat. Mech. Theor. Exp.
**2017**, 2017, 083102. [Google Scholar] [CrossRef] - Puertas-Centeno, D.; Toranzo, I.V.; Dehesa, J.S. Biparametric complexities and the generalized Planck radiation law. arXiv, 2017; arXiv:1704.08452v. [Google Scholar]
- Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 623–656. [Google Scholar] [CrossRef] - Fisher, R.A. On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. A
**1922**, 222, 309–368. [Google Scholar] - Rudnicki, Ł.; Toranzo, I.V.; Sánchez-Moreno, P.; Dehesa., J.S. Monotone measures of statistical complexity. Phys. Lett. A
**2016**, 380, 377–380. [Google Scholar] - Rényi, A. On measures of entropy and information. In Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, USA, 20 June–30 July 1960; pp. 547–561. [Google Scholar]
- Lutwak, E.; Yang, D.; Zhang, G. Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information. IEEE Trans. Inf. Theor.
**2005**, 51, 473–478. [Google Scholar] [CrossRef] - Bercher, J.F. On a (β,q)-generalized Fisher information and inequalities invoving q-Gaussian distributions. J. Math. Phys.
**2012**, 53, 063303. [Google Scholar] [CrossRef] [Green Version] - Lutwak, E.; Lv, S.; Yang, D.; Zhang, G. Extension of Fisher information and Stam’s inequality. IEEE Trans. Inf. Theor.
**2012**, 58, 1319–1327. [Google Scholar] [CrossRef] - Stam, A.J. Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inf. Control
**1959**, 2, 101–112. [Google Scholar] [CrossRef] - Cover, T.M.; Thomas, J.A. Elements of Information Theory, 2nd ed.; John Wiley & Sons: Hoboken, NJ, USA, 2006. [Google Scholar]
- Kay, S.M. Fundamentals for Statistical Signal Processing: Estimation Theory; Prentice Hall: Upper Saddle River, NJ, USA, 1993. [Google Scholar]
- Lehmann, E.L.; Casella, G. Theory of Point Estimation, 2nd ed.; Springer: New York, NY, USA, 1998. [Google Scholar]
- Bourret, R. A note on an information theoretic form of the uncertainty principle. Inf. Control
**1958**, 1, 398–401. [Google Scholar] [CrossRef] - Leipnik, R. Entropy and the uncertainty principle. Inf. Control
**1959**, 2, 64–79. [Google Scholar] [CrossRef] - Vignat, C.; Bercher, J.F. Analysis of signals in the Fisher-Shannon information plane. Phys. Lett. A
**2003**, 312, 27–33. [Google Scholar] [CrossRef] - Sañudo, J.; López-Ruiz, R. Statistical complexity and Fisher-Shannon information in the H-atom. Phys. Lett. A
**2008**, 372, 5283–5286. [Google Scholar] - Dehesa, J.S.; López-Rosa, S.; Manzano, D. Configuration complexities of hydrogenic atoms. Eur. Phys. J. D
**2009**, 55, 539–548. [Google Scholar] [CrossRef] - López-Ruiz, R.; Sañudo, J.; Romera, E.; Calbet, X. Statistical complexity and Fisher-Shannon information: Application. In Statistical Complexity. Application in Electronic Structure; Springer: New York, NY, USA, 2012. [Google Scholar]
- Manzano, D. Statistical measures of complexity for quantum systems with continuous variables. Phys. A Stat. Mech. Appl.
**2012**, 391, 6238–6244. [Google Scholar] [CrossRef] - Gell-Mann, M.; Tsallis, C. (Eds.) Nonextensive Entropy: Interdisciplinary Applications; Oxford University Press: Oxford, UK, 2004. [Google Scholar]
- Tsallis, C. Introduction to Nonextensive Statistical Mechanics—Approaching a Complex World; Springer: New York, NY, USA, 2009. [Google Scholar]
- Puertas-Centeno, D.; Rudnicki, L.; Dehesa, J.S. LMC-Rényi complexity monotones, heavy tailed distributions and stretched-escort deformation. 2017; in preparation. [Google Scholar]
- Agueh, M. Sharp Gagliardo-Nirenberg inequalities and mass transport theory. J. Dyn. Differ. Equ.
**2006**, 18, 1069–1093. [Google Scholar] [CrossRef] - Agueh, M. Sharp Gagliardo-Nirenberg inequalities via p-Laplacian type equations. Nonlinear Differ. Equ. Appl.
**2008**, 15, 457–472. [Google Scholar] - Costa, J.A.; Hero, A.O., III; Vignat, C. On solutions to multivariate maximum α-entropy problems. In Proceedings of the 4th International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition, Lisbon, Portugal, 7–9 July 2003; pp. 211–226. [Google Scholar]
- Johnson, O.; Vignat, C. Some results concerning maximum Rényi entropy distributions. Ann. Inst. Henri Poincare B Probab. Stat.
**2007**, 43, 339–351. [Google Scholar] [CrossRef] - Nanda, A.K.; Maiti, S.S. Rényi information measure for a used item. Inf. Sci.
**2007**, 177, 4161–4175. [Google Scholar] [CrossRef] - Panter, P.F.; Dite, W. Quantization distortion in pulse-count modulation with nonuniform spacing of levels. Proc. IRE
**1951**, 39, 44–48. [Google Scholar] [CrossRef] - Loyd, S.P. Least squares quantization in PCM. IEEE Trans. Inf. Theor.
**1982**, 28, 129–137. [Google Scholar] [CrossRef] - Gersho, A.; Gray, R.M. Vector Quantization and Signal Compression; Kluwer: Boston, MA, USA, 1992. [Google Scholar]
- Campbell, L.L. A coding theorem and Rényi’s entropy. Inf. Control
**1965**, 8, 423–429. [Google Scholar] [CrossRef] - Humblet, P.A. Generalization of the Huffman coding to minimize the probability of buffer overflow. IEEE Trans. Inf. Theor.
**1981**, 27, 230–232. [Google Scholar] [CrossRef] - Baer, M.B. Source coding for quasiarithmetic penalties. IEEE Trans. Inf. Theor.
**2006**, 52, 4380–4393. [Google Scholar] [CrossRef] - Bercher, J.F. Source coding with escort distributions and Rényi entropy bounds. Phys. Lett. A
**2009**, 373, 3235–3238. [Google Scholar] [CrossRef] [Green Version] - Bobkov, S.G.; Chistyakov, G.P. Entropy Power Inequality for the Rényi Entropy. IEEE Trans. Inf. Theor.
**2015**, 61, 708–714. [Google Scholar] [CrossRef] - Pardo, L. Statistical Inference Based on Divergence Measures; Chapman & Hall: Boca Raton, FL, USA, 2006. [Google Scholar]
- Harte, D. Multifractals: Theory and Applications, 1st ed.; Chapman & Hall/CRC: Boca Raton, FL, USA, 2001. [Google Scholar]
- Jizba, P.; Arimitsu, T. The world according to Rényi: Thermodynamics of multifractal systems. Ann. Phys.
**2004**, 312, 17–59. [Google Scholar] [CrossRef] - Beck, C.; Schögl, F. Thermodynamics of Chaotic Systems: An Introduction; Cambridge University Press: Cambridge, UK, 1993. [Google Scholar]
- Bialynicki-Birula, I. Formulation of the uncertainty relations in terms of the Rényi entropies. Phys. Rev. A
**2006**, 74, 052101. [Google Scholar] [CrossRef] - Zozor, S.; Vignat, C. On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles. Phys. A Stat. Mech. Appl.
**2007**, 375, 499–517. [Google Scholar] [CrossRef] - Zozor, S.; Vignat, C. Forme entropique du principe d’incertitude et cas d’égalité asymptotique. In Proceedings of the Colloque GRETSI, Troyes, France, 11–14 Septembre 2007. (In French). [Google Scholar]
- Zozor, S.; Portesi, M.; Vignat, C. Some extensions to the uncertainty principle. Phys. A Stat. Mech. Appl.
**2008**, 387, 4800–4808. [Google Scholar] [CrossRef] - Zozor, S.; Bosyk, G.M.; Portesi, M. General entropy-like uncertainty relations in finite dimensions. J. Phys. A
**2014**, 47, 495302. [Google Scholar] [CrossRef] - Jizba, P.; Dunningham, J.A.; Joo, J. Role of information theoretic uncertainty relations in quantum theory. Ann. Phys.
**2015**, 355, 87–115. [Google Scholar] [CrossRef] - Jizba, P.; Ma, Y.; Hayes, A.; Dunningham, J.A. One-parameter class of uncertainty relations based on entropy power. Phys. Rev. E
**2016**, 93, 060104. [Google Scholar] [CrossRef] [PubMed] - Hammad, P. Mesure d’ordre α de l’information au sens de Fisher. Rev. Stat. Appl.
**1978**, 26, 73–84. (In French) [Google Scholar] - Pennini, F.; Plastino, A.R.; Plastino, A. Rényi entropies and Fisher information as measures of nonextensivity in a Tsallis setting. Phys. A Stat. Mech. Appl.
**1998**, 258, 446–457. [Google Scholar] [CrossRef] - Chimento, L.P.; Pennini, F.; Plastino, A. Naudts-like duality and the extreme Fisher information principle. Phys. Rev. E
**2000**, 62, 7462–7465. [Google Scholar] [CrossRef] - Casas, M.; Chimento, L.; Pennini, F.; Plastino, A.; Plastino, A.R. Fisher information in a Tsallis non-extensive environment. Chaos Solitons Fractals
**2002**, 13, 451–459. [Google Scholar] [CrossRef] - Pennini, F.; Plastino, A.; Ferri, G.L. Semiclassical information from deformed and escort information measures. Phys. A Stat. Mech. Appl.
**2007**, 383, 782–796. [Google Scholar] [CrossRef] - Bercher, J.F. On generalized Cramér-Rao inequalities, generalized Fisher information and characterizations of generalized q-Gaussian distributions. J. Phys. A
**2012**, 45, 255303. [Google Scholar] [CrossRef] [Green Version] - Bercher, J.F. Some properties of generalized Fisher information in the context of nonextensive thermostatistics. Phys. A Stat. Mech. Appl.
**2013**, 392, 3140–3154. [Google Scholar] [CrossRef] [Green Version] - Bercher, J.F. On escort distributions, q-gaussians and Fisher information. In Proceedings of the 30th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Chamonix, France, 4–9 July 2010; pp. 208–215. [Google Scholar]
- Devroye, L. Non-Uniform Random Variate Generation; Springer: New York, NY, USA, 1986. [Google Scholar]
- Korbel, J. Rescaling the nonadditivity parameter in Tsallis thermostatistics. Phys. Lett. A
**2017**, 381, 2588–2592. [Google Scholar] [CrossRef] - Olver, F.W.J.; Lozier, D.W.; Boisvert, R.F.; Clark, C.W. NIST Handbook of Mathematical Functions; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Dover: New York, NY, USA, 1970. [Google Scholar]
- Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series, and Products, 7th ed.; Academic Press: San Diego, CA, USA, 2007. [Google Scholar]
- Prudnikov, A.P.; Brychkov, Y.A.; Marichev, O.I. Integrals and Series, Volume 3: More Special Functions; Gordon and Breach: New York, NY, USA, 1990. [Google Scholar]
- Nieto, M.M. Hydrogen atom and relativistic pi-mesic atom in N-space dimensions. Am. J. Phys.
**1979**, 47, 1067–1072. [Google Scholar] [CrossRef] - Yáñez, R.J.; van Assche, W.; Dehesa, J.S. Position and momentum information entropies of the D-dimensional harmonic oscillator and hydrogen atoms. Phys. Rev. A
**1994**, 50, 3065–3079. [Google Scholar] [CrossRef] [PubMed] - Avery, J.S. Hyperspherical Harmonics and Generalized Sturmians; Kluwer Academic: Dordrecht, The Netherlands, 2002. [Google Scholar]
- Yáñez, R.J.; van Assche, W.; González-Férez, R.; Sánchez-Dehesa, J. Entropic integrals of hyperspherical harmonics and spatial entropy of D-dimensional central potential. J. Math. Phys.
**1999**, 40, 5675–5686. [Google Scholar] [CrossRef] - Louck, J.D.; Shaffer, W.H. Generalized orbital angular momentum of the n-fold degenerate quantum-mechanical oscillator. Part I. The twofold degenerate oscillator. J. Mol. Spectrosc.
**1960**, 4, 285–297. [Google Scholar] - Louck, J.D.; Shaffer, W.H. Generalized orbital angular momentum of the n-fold degenerate quantum-mechanical oscillator. Part II. The n-fold degenerate oscillator. J. Mol. Spectrosc.
**1960**, 4, 298–333. [Google Scholar] - Nirenberg, L. On elliptical partial differential equations. Annali della Scuola Normale Superiore di Pisa
**1959**, 13, 115–169. [Google Scholar] - Gelfand, I.M.; Fomin, S.V. Calculus of Variations; Prentice Hall: Englewood Cliff, NJ, USA, 1963. [Google Scholar]
- Van Brunt, B. The Calculus of Variations; Springer: New York, NY, USA, 2004. [Google Scholar]

**Figure 1.**(

**a**) the domain ${\mathcal{D}}_{p}$ for a given p is represented by the gray area (here $p>2$). The thick line belongs to ${\mathcal{D}}_{p}$. The dashed line represents ${\mathcal{L}}_{p}$, corresponding to the Lutwak situation of Section 2.3.1, where the relation holds and the minimizers are explicitly known (stretched deformed Gaussian distributions), whereas ${\overline{\mathcal{L}}}_{p}$ corresponds to Section 2.3.2 (${\mathcal{B}}_{p}$ and ${\overline{\mathcal{B}}}_{p}$ obtained by the Gagliardo–Nirenberg inequality are their restrictions to ${\mathcal{D}}_{p}$); (

**b**) same situation for $p=2$, with the domains ${\mathcal{A}}_{2}$ and ${\overline{\mathcal{A}}}_{2}$ (dashed lines) that correspond to the situations of Section 2.3.3 and Section 2.3.4, respectively, (${\mathcal{L}}_{2}$ and ${\overline{\mathcal{L}}}_{2}$ are not represented for the clarity of the figure).

**Figure 2.**Given a p, the domain in gray represents ${\tilde{\mathcal{D}}}_{p}$, where we know that the $(p,\beta ,\lambda )$-Fisher–Rényi complexity is optimally lower bounded and where the minimizers can be deduced from proposition 2. (

**a**) the domain in dark gray represents ${\mathcal{D}}_{p}$, which is obviously included in ${\tilde{\mathcal{D}}}_{p}$; the dot is a particular point $(\beta ,\lambda )\in {\mathcal{D}}_{p}$ and the dotted line represents its transform by $\mathfrak{A}$; (

**b**) the domain in dark gray represents $\mathfrak{A}\left({\mathcal{L}}_{p}\right)\subset {\tilde{\mathcal{D}}}_{p}$, which obviously contains ${\mathcal{L}}_{p}$ represented by the dashed line; (

**c**) same as (

**b**) with ${\overline{\mathcal{L}}}_{p}$ and $\mathfrak{A}\left({\overline{\mathcal{L}}}_{p}\right)\subset {\tilde{\mathcal{D}}}_{p}$. This illustrates that ${\tilde{\mathcal{D}}}_{p}=\mathfrak{A}\left({\mathcal{L}}_{p}\right)\cup \mathfrak{A}\left({\overline{\mathcal{L}}}_{p}\right)$.

**Figure 3.**Fisher information ${F}_{p,\beta}$ (left graph), Rényi entropy power ${N}_{\lambda}$ (center graph), and $(p,\beta ,\lambda )$-Fisher–Rényi complexity ${C}_{p,\beta ,\lambda}$ (right graph) of the radial hydrogenic distribution in position space with dimensions $d=3(\circ ),\phantom{\rule{0.222222em}{0ex}}12(*)$ versus the quantum numbers n and l. The complexity parameters are $p=2,\phantom{\rule{0.166667em}{0ex}}\beta =1,\phantom{\rule{0.166667em}{0ex}}\lambda =7$.

**Figure 4.**$(p,\beta ,\lambda )$-Fisher–Rényi complexity (normalized to its lower bound), ${C}_{p,\beta ,\lambda}$, with $(p,\lambda ,\beta )=(2,0.8,7),(2,1,1),(5,2,7)$ for the radial hydrogenic distribution in the position space with dimensions $d=3(\circ )$ and $12(*)$.

**Figure 5.**Fisher information ${F}_{p,\beta}$ (left graph), Rényi entropy power ${N}_{\lambda}$ (center graph), and $(p,\beta ,\lambda )$-Fisher–Rényi complexity ${C}_{p,\beta ,\lambda}$ (right graph) of the radial hydrogenic distribution in momentum space with dimensions $d=3(\circ ),\phantom{\rule{0.222222em}{0ex}}12(*)$ versus the quantum numbers n and l. The complexity parameters are $p=2,\phantom{\rule{0.166667em}{0ex}}\beta =1,\phantom{\rule{0.166667em}{0ex}}\lambda =7$.

**Figure 6.**$(p,\beta ,\lambda )$-Fisher–Rényi complexity (normalized to its lower bound), ${C}_{p,\beta ,\lambda}$, with $(p,\lambda ,\beta )=(2,0.8,7),\phantom{\rule{0.222222em}{0ex}}(2,1,1),\phantom{\rule{0.222222em}{0ex}}(5,2,7)$ for the radial hydrogenic distribution in the momentum space with dimensions $d=3(\circ )$ and $12(*)$.

**Figure 7.**Fisher information ${F}_{p,\beta}$ (left graph), Rényi entropy power ${N}_{\lambda}$ (center graph), and $(p,\beta ,\lambda )$-Fisher–Rényi complexity ${C}_{p,\beta ,\lambda}$ (right graph) versus n and l for the radial harmonic system in position space with dimensions $d=3(\circ ),\phantom{\rule{4pt}{0ex}}12(*)$. The informational parameters are $p=2,\phantom{\rule{0.166667em}{0ex}}\beta =1,\phantom{\rule{0.166667em}{0ex}}\lambda =7$.

**Figure 8.**$(p,\beta ,\lambda )$-Fisher–Rényi complexity (normalized to its lower bound) ${C}_{p,\beta ,\lambda}$ with $(p,\lambda ,\beta )=(2,0.8,7),\phantom{\rule{0.166667em}{0ex}}(2,1,1),\phantom{\rule{0.166667em}{0ex}}(5,2,7)$ for the oscillator system in the position space with dimensions $d=3(\circ ),12(*)$.

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Zozor, S.; Puertas-Centeno, D.; Dehesa, J.S.
On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures. *Entropy* **2017**, *19*, 493.
https://doi.org/10.3390/e19090493

**AMA Style**

Zozor S, Puertas-Centeno D, Dehesa JS.
On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures. *Entropy*. 2017; 19(9):493.
https://doi.org/10.3390/e19090493

**Chicago/Turabian Style**

Zozor, Steeve, David Puertas-Centeno, and Jesús S. Dehesa.
2017. "On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures" *Entropy* 19, no. 9: 493.
https://doi.org/10.3390/e19090493