# On Zipf-Mandelbrot entropy and $3$-convex functions

@article{Khalid2019OnZE, title={On Zipf-Mandelbrot entropy and \$3\$-convex functions}, author={Sadia Khalid and Đ. Pe{\vc}ari{\'c} and Josip E. Pe{\vc}ari{\'c}}, journal={Advances in Operator Theory}, year={2019} }

In this paper, we present some interesting results related to the bounds of Zipf–Mandelbrot entropy and the 3-convexity of the function. Further, we define linear functionals as the nonnegative differences of the obtained inequalities and we present mean value theorems for the linear functionals. Finally, we discuss the n-exponential convexity and the log-convexity of the functions associated with the linear functionals.

## One Citation

### Refinements of some Hardy–Littlewood–Pólya type inequalities via Green’s functions and Fink’s identity and related results

- Mathematics
- 2020

In this paper, first we present some interesting identities associated with Green’s functions and Fink’s identity, and further we present some interesting inequalities for r -convex functions. We…

## References

SHOWING 1-10 OF 12 REFERENCES

### Majorization, Csiszár divergence and Zipf-Mandelbrot law

- Computer Science, MathematicsJournal of inequalities and applications
- 2017

This paper shows how the Shannon entropy is connected to the theory of majorization, and gives some generalized results for majorization inequality using Csiszár f-divergence.

### 3-convex functions and generalizations of an inequality of Hardy-Littlewood-Pólya

- Mathematics
- 2013

In this paper, we present some generalizations of an inequality of Hardy-Littlewood-Polya. We give the n-exponential convexity and log-convexity of the functions associated with the linear…

### Bounds for Shannon and Zipf‐Mandelbrot entropies

- Computer Science
- 2017

Two refinements of the well‐know Jensen inequality are considered and different bounds for Shannon and Zipf‐Mandelbrot entropies are obtained and the idea used for obtaining the results may stimulate further research in this area, particularly for Zipf-Mandel Brot entropy.

### IMPROVEMENTS OF THE GIACCARDI AND THE PETROVIC INEQUALITY AND RELATED STOLARSKY TYPE MEANS

- Mathematics
- 2012

Improvements of the Giaccardi and the Petrovi´c inequality are given. The notion of nexponentially convex functions is introduced. An elegant method of producing nexponentially convex and…

### Refinements of the majorization theorems via Fink identity and related results

- Mathematics
- 2015

The well known majorization theorem ( see [5, p. 11] and [7, p. 320] ) plays an important role in our paper. By using A. M. Fink’s identity in the majorization difference, we obtain an interesting…

### Pečarić, Bounds for Shannon and Zipf–Mandelbrot entropies, Math

- Method Appl. Sci
- 2017