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

#### 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
Accepted: 1 February 2019
First available in Project Euclid: 15 May 2019

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

Digital Object Identifier
doi:10.15352/aot.1810-1426

Mathematical Reviews number (MathSciNet)
MR3949971

Zentralblatt MATH identifier
07064101

#### 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

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