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Abstract In this paper, we introduce the concepts of map T $\mathcal {T}$ and interval-valued T $\mathcal {T}$ -convex, and give some basic properties. Further, we extend fractional Hermite–Hadamard inequalities in the case of T $\mathcal {T}$ -convex and Ostrowski type inequalities for interval-valued functions. Several examples are presented to illustrate the results.