Document Type: Original Article
COMSATS University Islamabad, Pakistan
Catholic University of Croatia, Croatia
Rudn University, Russia
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 non-negative 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.