On Zipf-Mandelbrot entropy and 3-convex functions. (English) Zbl 1468.26006

Khalid, Sadia; Pečarić, Đilda; Pečarić, Josip

Adv. Oper. Theory 4, No. 4, 724-737 (2019).

The authors prove inequalities for the Zipf-Mandelbrot entropy, given in [M. A. Khan et al., Math. Methods Appl. Sci. 40, No. 18, 7316–7322 (2017; Zbl 1381.26027)], and inequalities for a differentiable function satisfying some condition, which is related to the Zipf-Mandelbrot entropy. Furthermore, the authors prove the mean value theorem for a differentiable and 3-convex function which was defined by J. E. Pečarić et al. [Convex functions, partial orderings, and statistical applications. Boston, MA etc.: Academic Press (1992; Zbl 0749.26004)] and whose proof is similar to the proof of Theorem 2.8 in [S. Khalid et al., Glas. Mat., III. Ser. 48, No. 2, 335–356 (2013; Zbl 1303.26021)]. Finally, \(n\)-exponential convexity, given in [J. Pečarić and J. Perić, An. Univ. Craiova, Ser. Mat. Inf. 39, No. 1, 65–75 (2012; Zbl 1274.26069)] and log-convextity, given by J. E. Pečarić et al. [Convex functions, partial orderings, and statistical applications. Boston, MA etc.: Academic Press (1992; Zbl 0749.26004)] are investigated.
The results are useful for beginners on inequalities related to convex functions.

Reviewer: Choonkil Park (Seoul)

MSC:

26A51	Convexity of real functions in one variable, generalizations
26A24	Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A48	Monotonic functions, generalizations
26D15	Inequalities for sums, series and integrals

Keywords:

Shannon entropy; Zipf-Mandelbrot entropy; divided difference; \(n\)-convex function; Cauchy means; \(n\)-exponential convexity; logarithmic convexity

Citations:

Zbl 1381.26027; Zbl 0749.26004; Zbl 1303.26021; Zbl 1274.26069