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We introduce the concepts of $\mathbb{K}$-Riemann integral and radial $\mathbb{K}$-g$H$-derivative for interval-valued functions. We also give some important properties of interval-valued $\mathbb{K}$-Riemann integral, and extend interval-valued Hermite-Hadamard type inequalities in the case of $\mathbb{K}$-Riemann integral. Several examples are shown to illustrate the results.

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.

The dispersions containing two kinds of layered solids—graphene oxide (GO) and exfoliated montmorillonite (MMT) were mainly prepared, following which the binary aerogels were synthesized. The results indicate that the formation of liquid crystals (LCs) occurs at lower GO concentration and the birefringence becomes stronger when MMT is introduced into GO dispersion. Sol–gel transition forms in the binary suspensions with different mass fractions of MMT and GO. LCs with highly ordered alignment are observed in the gel and the fraction of LCs obviously increases with the increase in GO concentration. Moreover, the birefringence is observed in MMT–GO binary aerogels with the interconnected three-dimensional porous network, which is attributed to the ordered arrangement of MMT and GO nanosheets in pore walls. Among the aerogels with different MMT/GO ratios, the samples at the ratio of 10 : 1 show better adsorption capacity and removal percentage of cationic and anionic dyes.