Nonlinear integrals and Hadamard-type inequalities

The Hadamard integral inequality for nonlinear integrals has been proved by some researchers, but the obtained inequalities do not look like the classical Hadamard inequality. In this paper, we provide a refinement of the Hadamard integral inequality for g-integrals as ∫[0,1]⊕f((1-t)a+tb)⊙dm⩽g-112⊙(f(a)⊕f(b)),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{[0,1]}^{\oplus } f\big ((1- t)a+ tb\big ) \odot \mathrm {d}m \leqslant g^{-1}\left( \frac{1}{2}\right) \odot \big (f(a)\oplus f(b)\big ), \end{aligned}$$\end{document}for which by choosing the convex and increasing function g(x)=x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(x)= x$$\end{document}, we get the classical Hadamard inequality. Consequently, we establish some novel integral inequalities, the Hadamard-type integral inequalities for a pseudo-multiplication of n convex (concave) functions, in the framework of g-integrals.