Abstract
The functional defined as the difference between the right-hand and the left-hand side of the Hardy-Littlewood maximal inequality is studied and its properties, such as exponential and logarithmic convexity, are explored. Furthermore, related analogues of the Lagrange and Cauchy mean value theorems are derived. Finally, using this functional, a new family of the Cauchy-type means is generated. These means are shown to be monotone.
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