Note on certain inequality (Q1378648)

Let \(G\), \(H\) be lattice ordered groups, \(|\cdot|\) the function defined by \(| x|=\sup\{x, 0\}-\inf\{x, 0\}\) and \(f:G^+\to H^+\) a function with increasing increments, such that \(f(0)= 0\). The author presents a proof of the inequality \[ \sum^n_{k= 1}| f(a_k)- f(a_{k- 1})|\leq f\Biggl(\sum^n_{k= 1}| a_k- a_{k-1}|\Biggr), \] where \(0= a_0,a_1,\dots, a_n\in G^+\). Furthermore, he proves that if \(f\) is a function with strictly increasing increments, then the equality holds if and only if \(a_1\leq\cdots\leq a_n\). the above result is an interesting generalization of the result of \textit{L. Maligranda} and \textit{W. Orlicz} [Monatsh. Math. 104, No. 1, 53-65 (1987; Zbl 0623.26009)] and \textit{J. E. Pečarić} and the author [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 36, 169-178 (1996)]. The author also proves a similar inequality for an increasing function \(f\) with decreasing increments and for comparable \(a_1,\dots, a_n\). Perhaps, the readers will find this article stimulating for consideration of similar problems on lattice ordered groups.