<abstract> <p>To create various kinds of inequalities, the idea of convexity is essential. Convexity and integral inequality hence have a significant link. This study's goals are to introduce a new class of generalized convex fuzzy-interval-valued functions (convex 𝘍𝘐𝘝𝘍s) which are known as $ \left(\mathfrak{p}, \mathfrak{J}\right) $-convex 𝘍𝘐𝘝𝘍s and to establish Jensen, Schur and Hermite-Hadamard type inequalities for $ \left(\mathfrak{p}, \mathfrak{J}\right) $-convex 𝘍𝘐𝘝𝘍s using fuzzy order relation. The Kulisch-Miranker order relation, which is based on interval space, is used to define this fuzzy order relation level-wise. Additionally, we have demonstrated that, as special examples, our conclusions encompass a sizable class of both new and well-known inequalities for $ \left(\mathfrak{p}, \mathfrak{J}\right) $-convex 𝘍𝘐𝘝𝘍s. We offer helpful examples that demonstrate the theory created in this study's application. These findings and various methods might point the way in new directions for modeling, interval-valued functions and fuzzy optimization issues.</p> </abstract>