# A necessary and sufficient condition for the inequality of generalized weighted means

- 684 Downloads
- 2 Citations

## Abstract

We present in this paper a necessary and sufficient condition to establish the inequality between generalized weighted means which share the same sequence of numbers but differ in the weights. We first present a sufficient condition, and then obtain the more general, necessary and sufficient, condition. Our results were motivated by an inequality, involving harmonic means, found in the study of multiple importance sampling Monte Carlo technique. We present new proofs of Chebyshev’s sum inequality, Cauchy-Schwartz, and the rearrangement inequality, and derive several interesting inequalities, some of them related to the Shannon entropy, the Tsallis, and the Rényi entropy with different entropic indices, and to logsumexp mean. Those inequalities are obtained as particular cases of our general inequality, and show the potential and practical interest of our approach. We show too the relationship of our inequality with sequence majorization.

## Keywords

inequalities weighted arithmetic mean weighted harmonic mean weighted geometric mean weighted power mean weighted quasi-arithmetic mean weighted Kolmogorov mean generalized weighted mean Chebyshev’s sum inequality Cauchy-Schwartz inequality Rearrangement inequality entropy Tsallis entropy Rényi entropy logsumexp mean majorization## 1 Introduction

In the research in the multiple importance sample Monte Carlo integration problem [1, 2, 3], we were confronted with several inequalities relating harmonic means, which were either described in the literature [4, 5, 6, 7] or easy to prove from it. However, we were unable to find in the literature the following inequality (we give in an Appendix an interpretation of this inequality), which we conjectured was true.

### Conjecture 1

*For*\(\{ b_{k} \}\)

*a sequence of*

*M*

*strictly positive numbers*,

*and*\(C \ge0\),

*we have*

*where*\(\mathcal{H}\)

*stands for harmonic mean*.

## 2 Results: inequalities for generalized weighted mean

### 2.1 A new inequality for generalized weighted mean

By taking the \(\{ \alpha_{k} \}\) to be any weights obeying equation (6) we generalize Conjecture 1 to the following theorem.

### Theorem 1

*Consider a sequence of*

*M*

*strictly positive numbers*\(\{ b_{k} \}\)

*and strictly positive weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\), \(\sum_{k} \alpha_{k} =1, \sum_{k} \alpha'_{k} =1\).

*Consider that the weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\)

*obey the following condition*:

*Then the following inequality holds*:

*or*,

*equivalently*,

### Lemma 1

*Obeying condition in equation*(7)

*implies that*

### Proof

We prove now Theorem 1 under the \(\{b_{k}\}\)-increasing condition:

### Proof

*i.e.*, assume that equation (13) holds for \(M-1\), \(M\ge3\), and take as weights \(\{\frac{\alpha_{k}}{(1-\alpha_{M})}\}\) and \(\{\frac{\alpha'_{k}}{(1-\alpha '_{M})}\}\) (they add to 1 as \(\sum_{k=1}^{M-1} \alpha_{k} = 1-\alpha_{M}\), and \(\sum_{k=1}^{M-1} \alpha'_{k} = 1-\alpha'_{M}\)). These weights fulfill condition (7), and thus

*M*. □

### Corollary 1

*A sufficient condition for strict inequality in equation* (9) *is that* \(b_{1} < b_{M}\) (*i*.*e*., *the non*-*trivial case*) *and* \(\alpha_{M} \alpha'_{1}- \alpha'_{M} \alpha_{1} >0 \).

### Proof

Observe that \(\alpha_{M} \alpha'_{1}- \alpha'_{M} \alpha_{1} = 0\) would imply, from equation (11), that \(\alpha_{1} = \alpha'_{1}\) and \(\alpha_{M} = \alpha'_{M}\), and applying condition (7) with \(i=1\) we would obtain \(\alpha_{j} \ge\alpha'_{j}\) for all \(j \ge1\). Using the fact that \(\sum_{j} \alpha_{j} = \sum_{j} \alpha'_{j}=1\) we immediately arrive at \(\alpha_{j} = \alpha'_{j}\) for all *j*, thus the trivial case. If we exclude this trivial case the inequality (13) is strict. □

### Corollary 2

*Given a strictly positive sequence*\(\{ b_{k} \}\),

*and*\(k_{1} >0\), \(k_{2},C_{1}, C_{2} \ge0\),

*we have*

### Proof

Observe that Conjecture 1 is a particular case of equation (21) when \(k_{1}=k_{2}=1, C_{2}=C_{1}\).

### Corollary 3

*Given a strictly positive sequence*\(\{ b_{k} \}\),

*for any*

*γ*,

*we have*

### Proof

### Corollary 4

*Given a strictly positive sequence*\(\{ b_{k} \}\),

*and*\(\gamma_{1},\gamma _{2}\),

*such that*\(\gamma_{1}, \gamma_{2} \ge0\),

*the following inequality holds*:

### Proof

Let us see now that Theorem 1, for the weighted harmonic mean, also holds for the weighted arithmetic mean.

### Theorem 2

*Consider a sequence of*

*M*

*strictly positive numbers*\(\{ b_{k} \}\)

*and strictly positive weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\), \(\sum_{k} \alpha_{k} =1, \sum_{k} \alpha'_{k} =1\).

*If the weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\)

*obey the following condition*:

*then the following inequality holds*:

*or equivalently*

### Proof

*i.e.*with \(\{ \alpha_{k}' \}\) and \(\{ \alpha_{k} \}\) instead of \(\{ \alpha_{k} \}\) and \(\{ \alpha'_{k} \}\)) to obtain equation (35). □

### Remark 1

*C*,

### Corollary 5

*A sufficient condition for strict inequality in equation* (33) *is that* \(b_{1} < b_{M}\) (*i*.*e*., *the non trivial case*) *and* \(\alpha_{M} \alpha'_{1}- \alpha'_{M} \alpha_{1} >0\).

### Proof

From equation (35) and equation (36), we apply Corollary 1 to the sequence \(\{b_{k}^{-1}\}\) with weights \(\{ \alpha_{k}' \}\) and \(\{ \alpha_{k} \}\). As the ordering is the reverse of \(\{b_{k}\}\), and weights are switched, \(\alpha_{M}\) would take the role of \(\alpha'_{1}\) and vice versa, and the same with \(\alpha'_{M}\) and \(\alpha_{1}\). □

### Corollary 6

Chebyshev’s sum inequality

*Given the sequences*\(\{x_{1} \ge x_{2} \ge\cdots\ge x_{M} \ge0 \}, \{ y_{1} \ge y_{2} \ge\cdots\ge y_{M} \ge0 \}\)

*the following inequality holds*:

### Proof

Corollary 6 can easily be extended to the following one.

### Corollary 7

Chebyshev’s sum inequality extended

*Given the positive sequences* \(\{x_{1}, x_{2}, \ldots, x_{M} \}, \{y_{1} , y_{2}, \ldots, y_{M} \}\), *then the following holds*:

*If both sequences are sorted in the same order then*

*If in addition they are strictly positive*

*If both sequences are sorted in opposite order then*

*If in addition they are strictly positive*

### Corollary 8

Cauchy-Schwartz-Buniakowski inequality

*Given the sequences*\(\{x_{1}, x_{2}, \ldots, x_{M} \}, \{ y_{1} , y_{2}, \ldots, y_{M} \}\)

*the following inequality holds*:

### Proof

\(\mathcal{A}( \{ b_{k}^{\gamma} \}) = 1/\mathcal{H}( \{ b_{k}^{-\gamma} \})\) and reciprocally, Corollary 6, applied to sequences \(\{ b_{k}^{\gamma_{1}} \}, \{ b_{k}^{\gamma_{2}} \}, \gamma_{1}, \gamma_{1} \ge0\), together with Corollary 4, allow us to establish the following corollary.

### Corollary 9

*Given a strictly positive sequence*\(\{ b_{k} \}\),

*and*\(\gamma_{1},\gamma_{2}\)

*both positive or negative*,

*the following inequalities hold*:

Observe that we can only guarantee that equation (51) hold when both \(\gamma_{1}, \gamma_{2}\) are of the same sign. For instance, taking \(\gamma_{1}= \gamma= -\gamma_{2}\) we can easily check that equation (51) would read \(\mathcal{H}( \{ b_{k}^{\gamma} \}) \ge \mathcal{A}( \{ b_{k}^{\gamma} \})\), which is false in general (rather what is true is the inverse inequality).

Consider now \(H(\{p_{k}\})= -\sum_{k} p_{k}\log p_{k} \), the Shannon entropy of \(\{p_{k}\}\).

### Corollary 10

*For any probability distribution*\(\{p_{k}\}\)

*the following inequality holds*:

*where*,

*if for any*\(i, p_{i} =0\),

*we use the convention*\(p_{i}\log p_{i} =\lim_{p_{i} \rightarrow0} p_{i}\log p_{i} = 0, -\log p_{i} = \lim_{p_{i} \rightarrow 0} -\log p_{i} = + \infty\).

### Proof

In information theory [8], the value \(-\log p_{i}\) is considered as the information of result *i*, thus Corollary 10 says that the expected value of information is less than or equal to its average value.

### Corollary 11

*For any strictly positive probability distribution*\(\{p_{k}\}\)

*the following inequality holds*:

### Proof

*q*of probability distribution \(\{p_{k}\}\) [9, 10] is defined as

### Corollary 12

*For any probability distribution*\(\{p_{k}\}\),

*and*\(q, r\)

*both positive or both negative*,

*the following inequality holds*:

### Proof

### Corollary 13

*For any probability distribution*\(\{p_{k}\}\),

*and any*\(\beta, \gamma\ge 0\)

*the following inequality holds*:

### Proof

Let us see now that Theorem 1 extends to a weighted geometric mean.

### Theorem 3

*Consider a sequence of*

*M*

*strictly positive numbers*\(\{ b_{k} \}\)

*and strictly positive weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\), \(\sum_{k} \alpha_{k} =1, \sum_{k} \alpha'_{k} =1\).

*Consider that the weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\)

*obey the following condition*:

*Then the following inequality holds*:

### Proof

Let us see now that Theorem 1 also extends to weighted generalized (or power) mean:

### Theorem 4

*Consider a sequence of*

*M*

*strictly positive numbers*\(\{ b_{k} \}\)

*and strictly positive weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\), \(\sum_{k} \alpha_{k} =1, \sum_{k} \alpha'_{k} =1\).

*Consider that the weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\)

*obey the following condition*:

*Then the following inequality holds*:

*for any*\(p \ne0\) (

*for*\(p=0\)

*the power mean is defined as the weighted geometric mean*).

### Proof

When \(p < 0\) we just apply Theorem 1. □

Theorem 5 below extends Theorem 1 to the quasi-arithmetic or Kolmogorov generalized weighted mean.

### Theorem 5

*Let*\(f(x)\)

*be an invertible strictly positive monotonic function*,

*with inverse function*\(f^{-1}(x)\).

*Consider a sequence of*

*M*

*strictly positive numbers*\(\{ b_{k} \}\)

*and strictly positive weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\), \(\sum_{k} \alpha _{k} =1, \sum_{k} \alpha'_{k} =1\).

*Consider that the weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\)

*obey the following condition*:

*Then the following inequality holds*:

### Proof

### Corollary 14

*Given a strictly positive sequence*\(\{ b_{k} \}, 1 \le k \le M\),

*we have*

### Proof

### Corollary 15

*Given a strictly positive sequence*\(\{ b_{k} \}, 1 \le k \le M\),

*we have*

### Proof

### Theorem 6

*Be*\(\mathcal{M} (\{ b_{k} \}, \{ \alpha_{k} \})\)

*any of the means appearing in Theorems*1-5,

*of a sequence*\(\{ b_{k} \}\)

*of*

*M*

*strictly positive numbers*,

*with*\(\{ \alpha _{k} \}\), \(\{ \alpha'_{k} \}\)

*strictly positive weights obeying the condition*

*Then*,

*for any subsequence of*\(M^{\star}\)

*numbers*\(\{ b_{l} \}\)

*of*\(\{ b_{k} \}\), \(M^{\star} \le M\),

*with their corresponding normalized weights*\(\{ \alpha_{l} \}\), \(\{ \alpha '_{l} \}\),

*we have*

Finally, we consider the following theorem.

### Theorem 7

*Let*\(f(x)\)

*be an invertible strictly positive monotonic function*,

*with inverse function*\(f^{-1}(x)\).

*Consider a sequence of*

*M*

*strictly positive numbers*\(\{ b_{k} \}\)

*and functions*\(g(x), h(x), h'(x)\)

*strictly positive in the domain*\([ \min_{k} \{ b_{k} \}, \max_{k} \{ b_{k} \}]\),

*and such that in the same domain*\(g(x)\)

*is increasing*,

*and*\(h'(x)/h(x)\)

*is decreasing*.

*Then the following inequality holds*:

### Proof

Apply Theorem 5 to the sequence \(\{ g(b_{k}) \}\) with weights \(\alpha'_{k}= \frac{h'(b_{k})}{\sum_{k} h'(b_{k})} \) and \(\alpha_{k} = \frac{h(b_{k})}{\sum_{k} h(b_{k})}\). □

Observe that Corollaries 3-15 can be considered as applications of Theorem 7.

### 2.2 Relationship to majorization

### Lemma 2

*Consider the sequences of*

*M*

*strictly positive weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\), \(\sum_{k} \alpha_{k} =1, \sum_{k} \alpha'_{k} =1\).

*If for any sequence of*

*M*

*strictly positive numbers in increasing order*\(\{ b_{k} \}\)

*the following inequality holds*:

*then the following inequalities also hold*:

### Proof

*l*times, denote \(\mathbf{L} = a_{1}+\cdots+a_{l}\), \(\mathbf{L}' = a'_{1}+\cdots+a'_{l}\). Since \(a_{l+1}+\cdots+a_{M} = 1-\mathbf {L}\), \(a'_{l+1}+\cdots+a'_{M} = 1-\mathbf{L}'\) the inequality (89) gives

*i.e.*,

We can then state Theorem 8.

### Theorem 8

*Consider the sequences of*

*M*

*increasing strictly positive weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\), \(\sum_{k} \alpha _{k} =1, \sum_{k} \alpha'_{k} =1\),

*with the following condition*(

*equation*(10)):

*or equivalently*

*then the following majorization holds*:

### Proof

Observe first that equation (91) (equation (10)) is equivalent to equation (7) for sequences of *M* increasing strictly positive numbers \(\{ b_{k} \}\). We can apply then Theorem 1, and obtain the condition equation (89) in Lemma 2. It is enough then to apply Lemma 2, which guarantees that the inequalities for majorization, equation (88), are fulfilled for the decreasing sequences \(\{ \alpha_{M+1-k} \}, \{ \alpha '_{M+1-k}\}\). □

Observe that the weights in Corollary 6 are such that \(\{ \alpha_{k} \} \succ\{ \alpha'_{k}\}\), and in this way Corollary 6 can be proved by direct application of Lemma 1 in [12].

A similar theorem can be proved for decreasing weights.

### Theorem 9

*Consider the sequences of*

*M*

*decreasing strictly positive weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\), \(\sum_{k} \alpha _{k} =1, \sum_{k} \alpha'_{k} =1\),

*with the following condition*(

*equation*(10)):

*or equivalently*

*then the following majorization holds*:

### Proof

The same proof as for Theorem 8 holds. □

### Theorem 10

*Consider a convex function*\(f(x)\),

*and the sequences of*

*M*

*strictly positive weights*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \} \), \(\sum_{k} \alpha_{k} =1, \sum_{k} \alpha'_{k} =1\),

*with the following condition*(

*equation*(10)):

*or equivalently*

*then the following holds*:

*If both*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\)

*are increasing then*

*If both*\(\{ \alpha_{k} \}\)

*and*\(\{ \alpha'_{k} \}\)

*are decreasing then*

### Proof

It is enough to apply Theorems 8 and 9 together with Hardy-Littlewood-Pólya theorem [5, 13] on majorization. □

### Theorem 11

*Given the sequences of strictly positive numbers*\(\{ x_{1} , x_{2} , \ldots, x_{M} \}\), \(\{y_{1}, y_{2} , \ldots, y_{M} \}\)

*obeying the conditions*\(i \le j \Rightarrow x_{i}/y_{i} \ge x_{j}/y_{j}\),

*and*\(\sum_{k} x_{k} = \sum_{k} y_{k}\),

*then if both sequences are increasing*

*and if both sequences are decreasing*

### Proof

It is enough to normalize the sequences and apply Theorems 8 and 9, taking into account that majorization is invariant to a change in scale. □

### Theorem 12

*Given the sequences of numbers*\(\{x_{1} , x_{2} , \ldots , x_{M} \}\), \(\{y_{1}, y_{2} , \ldots, y_{M} \}\)

*obeying the condition*\(\sum_{k} x_{k} = \sum_{k} y_{k}\),

*and be*\(\{x_{1}+C , x_{2}+C , \ldots, x_{M}+C \} \), \(\{y_{1}+C, y_{2}+C , \ldots, y_{M}+C \}\)

*the sequences translated by a positive constant*

*C*

*such that*,

*for all*

*k*, \(x_{k}+C >0, y_{k}+C >0\).

*If the new sequences obey the condition*\(i \le j \Rightarrow(x_{i}+C)/(y_{i}+C) \ge(x_{j}+C)/(y_{j}+C)\),

*then if both sequences are increasing*

*and if both sequences are decreasing*

### Proof

It is enough to normalize the translated sequences and apply Theorems 8 and 9, taking into account that majorization is invariant to a change of scale and a translation. □

### Theorem 13

*Given the sequences of numbers*\(\{x_{1} , x_{2} , \ldots , x_{M} \}\), \(\{y_{1}, y_{2} , \ldots, y_{M} \}\)

*obeying the conditions*\(i \le j \Rightarrow x_{i}/y_{i} \ge x_{j}/y_{j}\), \(i \le j \Rightarrow (x_{i}-y_{i}) \ge(x_{j}-y_{j})\),

*and*\(\sum_{k} x_{k} = \sum_{k} y_{k}\),

*then if both sequences are increasing*

*and if both sequences are decreasing*

### Proof

*C*,

*C*such that, for all

*k*, \(x_{k}+C >0, y_{k}+C >0\), and we apply Theorem 12. □

We can also obtain similar results for convex functions than in Theorem 10 for Theorems 11, 12, and 13.

### 2.3 Not a necessary condition

We can see with counterexamples that the sufficient condition equation (7) appearing on all Theorems 1-5 is not a necessary condition for the inequality of the means for any strictly positive sequence \(\{b_{k}\}\) for \(M\ge3\) (although it is easy to prove it is a necessary condition for \(M=2\)).

*M*even,

We leave it to the reader to check with the other means considered in this paper.

## 3 Results: a necessary and sufficient condition

We will see in this section that the condition found in Lemma 2 is a necessary and sufficient condition.

### Theorem 14

*Consider the sequences of* *M* *numbers* \(\{ x_{k} \}\) *and* \(\{ y_{k} \}\), \(\sum_{k} x_{k} =\sum_{k} y_{k} \). *Then propositions* (1) *to* (4) *are equivalent*:

*for any sequence of*

*M*

*numbers in increasing order*\(\{ z_{k}\}\)

*the following inequality holds*:

*for any sequence of*

*M*

*numbers in increasing order*\(\{ z_{k}\}\)

*the following inequality holds*:

*the following inequalities hold*:

*the following inequalities hold*:

### Proof

*l*times, denote \(\mathbf{L}' = x_{1}+\cdots+x_{l}\), \(\mathbf{L} = y_{1}+\cdots+y_{l}\). Since \(x_{l+1}+\cdots+x_{M} = C- \mathbf{L}'\), \(y_{l+1}+\cdots+y_{M} = C-\mathbf{L}\) the inequality equation (105) gives

*i.e.*,

*l*times, and the same definitions as before for \(\mathbf{L}, \mathbf{L}'\), then equation (106) gives

*k*, \(A'_{k} - A_{k} \ge0 \), and \(\{ z_{k} \}\) is an increasing sequence.

Repeating the proof for the sequences \(\{y_{M-k+1}\}\) and \(\{x_{M-k+1}\} \) we find that (4) implies (2). □

### Remark

The case when the sequences \(\{x_{k}\}, \{y_{k}\}\) are strictly positive and \(\sum_{k} x_{k} = \sum_{k} y_{k} = 1\) is interesting and proves that condition in Lemma 2 is necessary and sufficient, and that it can be extended to harmonic, geometric, and power means, in the same way as Theorem 1 has been extended.

### Corollary 16

Rearrangement inequality

*Given the sequences of real numbers* \(\{x_{1} , x_{2} , \ldots, x_{M} \}\), \(\{y_{1}, y_{2} , \ldots, y_{M} \} \), *then the maximum of their crossed sum is obtained when both are paired in the same* (*increasing or decreasing*) *order*, *while the minimum is obtained when both are paired in inverse order*.

### Proof

## 4 Conclusions

Motivated by proving an inequality that appeared in our research in Monte Carlo multiple importance sampling (MIS), we identified first a sufficient condition for a generalized weighted means inequality where only the weights are changed. We obtained then a necessary and sufficient condition. We have given new proofs for Chebyshev’s sum, the Cauchy-Schwartz, and the rearrangement inequalities, as well as for other interesting inequalities, and we obtained results for the Shannon, the Tsallis, and the Rényi entropy and logsumexp. We also showed the relationship to majorization.

## Notes

### Acknowledgements

The authors are funded in part by grants TIN2013-47276-C6-1-R from Spanish Government and by grant number 2014-SGR-1232 from Catalan Government. The authors acknowledge the comments by David Juher to an earlier draft, and comments by anonymous reviewers that helped to improve the final version of the paper and suggested simplified proofs for Lemma 2 and Theorem 14.

## References

- 1.Veach, E, Guibas, LJ: Optimally combining sampling techniques for Monte Carlo rendering. In: Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques. SIGGRAPH ’95, pp. 419-428. ACM, New York (1995). doi: 10.1145/218380.218498 CrossRefGoogle Scholar
- 2.Havran, V, Sbert, M: Optimal combination of techniques in multiple importance sampling. In: Proceedings of the 13th ACM SIGGRAPH International Conference on Virtual-Reality Continuum and Its Applications in Industry. VRCAI ’14, pp. 141-150. ACM, New York (2014). doi: 10.1145/2670473.2670496 Google Scholar
- 3.Havran, V, Sbert, M: Optimal Combination of Techniques in Multiple Importance Sampling. Technical Report Series of DCGI CS-TR-DCGI-2014-2, Department of Computer Graphics and Interaction, Czech Technical University, FEE, (August 2014) Google Scholar
- 4.Bullen, PS: Handbook of Means and Their Inequalities. Springer, Dordrecht (2003) CrossRefzbMATHGoogle Scholar
- 5.Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1952). https://books.google.es/books?id=t1RCSP8YKt8C zbMATHGoogle Scholar
- 6.Korovkin, PP: Inequalities. Little Mathematics Library. Mir Publishers, Moscow (1975). https://archive.org/details/InequalitieslittleMathematicsLibrary zbMATHGoogle Scholar
- 7.Hoehn, L, Niven, I: Averages on the move. Math. Mag.
**58**(3), 151-156 (1985) MathSciNetCrossRefzbMATHGoogle Scholar - 8.Cover, TM, Thomas, JA: Elements of Information Theory. Wiley, New York (2006) zbMATHGoogle Scholar
- 9.Tsallis, C: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys.
**52**(1/2), 479-487 (1988) MathSciNetCrossRefzbMATHGoogle Scholar - 10.Tsallis, C: Generalized entropy-based criterion for consistent testing. Phys. Rev. E
**58**, 1442-1445 (1998) CrossRefGoogle Scholar - 11.Rényi, A: On measures of entropy and information. In: Proc. Fourth Berkeley Symp. Math. Stat. and Probability’ 60, vol. 1, pp. 547-561. University of California Press, Berkeley (1961) Google Scholar
- 12.Marjanović, MM, Kadelburg, Z: A proof of Chebyshev’s inequality. In: The Teaching of Mathematics, vol. X, 2, pp. 107-108 (2007) Google Scholar
- 13.Karamata, J: Sur une inégalité rélative aux fonctions convexes. Publ. Math. Univ. Belgrade
**1**, 145-148 (1932) zbMATHGoogle Scholar - 14.Rubinstein, RY, Kroese, DP: Simulation and the Monte Carlo Method. Wiley Series in Probability and Statistics. Wiley, New York (2008). http://books.google.com.au/books?id=1-ffZVmazvwC zbMATHGoogle Scholar
- 15.Kalos, MH, Whitlock, PA: Monte Carlo Methods: Basics. Monte Carlo Methods. Wiley, New York (1986). http://books.google.com.au/books?id=87UdAQAAMAAJ CrossRefzbMATHGoogle Scholar

## Copyright information

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.