Novel versions of Hölder's-Like and related inequalities with newly defined $L_P$ space, and their applications over fuzzy domain

It is widely recognized that fuzzy number theory relies on the characteristic function. However, within the fuzzy realm, the characteristic function transforms into a membership function contingent upon the interval [0,1]. This implies that real numbers and intervals represent exceptional cases of fuzzy numbers. By considering this approach, this paper introduces a new $L_P$ space and novel refinements for integral variations of Hölder's inequality which is known as Hölder's-like inequality over fuzzy domain. Numerous prevailing inequalities associated with Hölder's-like inequality can be enhanced through the newly acquired inequalities, as demonstrated through an application. By using newly defined special means, some new versions of integral inequalities have obtained where differentiable mappings are real-valued convex-like (or convex fuzzy) mappings Lastly, nontrivial numerical examples are also included to validate the accuracy of the presented inequalities as they vary with the parameter $ꙍ$ .