## Advances in Operator Theory

- Adv. Oper. Theory
- Volume 4, Number 4 (2019), 724-737.

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

Sadia Khalid, Đilda Pečarić, and Josip Pečarić

#### Abstract

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.

#### Article information

**Source**

Adv. Oper. Theory, Volume 4, Number 4 (2019), 724-737.

**Dates**

Received: 13 October 2018

Accepted: 1 February 2019

First available in Project Euclid: 15 May 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.aot/1557885622

**Digital Object Identifier**

doi:10.15352/aot.1810-1426

**Mathematical Reviews number (MathSciNet)**

MR3949971

**Zentralblatt MATH identifier**

07064101

**Subjects**

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15]

Secondary: 26A48 26A51 26D15: Inequalities for sums, series and integrals

**Keywords**

Shannon entropy Zipf-Mandelbrot entropy divided difference $n$-convex function $n$-exponential convexity logarithmic convexity

#### Citation

Khalid, Sadia; Pečarić, Đilda; Pečarić, Josip. On Zipf-Mandelbrot entropy and $3$-convex functions. Adv. Oper. Theory 4 (2019), no. 4, 724--737. doi:10.15352/aot.1810-1426. https://projecteuclid.org/euclid.aot/1557885622