a carleman-knopp type inequality for pseudo-integral

‎pseudo-analysis has applications in several fields‎, ‎including game theory and optimization problems‎. ‎pseudo-analysis is a generalized form of ordinary classical analysis that has two main operations‎. ‎in fact‎, ‎these two operations‎, ‎which are called pseudo-multiplication $otimes$ and pseudo-addition $oplus$‎, ‎are the basis of the formation of a semi-ring on the interval [c,d] of $ [-infty,infty]$‎. ‎the pseudo-operations $otimes$ and $oplus$ on [c,d] produce three types of semi-ring‎. ‎first‎, ‎the semi-ring $([c,d],sup,otimes)$ or $([c,d],inf,otimes)$ in which $otimes$ is generated‎, ‎the second‎, ‎a semi-ring where $otimes$ and $oplus$ are defined by the continuous and strictly monotone function $psi$‎, ‎the third‎, ‎a semi-ring in which both pseudo-operations $otimes$ and $oplus$ are idempotent‎. ‎in this article‎, ‎we intend to state and prove some of the most recent generalizations of carleman-knopp’s type inequalities via pseudo-integrals‎.