Soft Computing

, Volume 22, Issue 9, pp 2843–2849 | Cite as

Nonlinear integrals and Hadamard-type inequalities

  • Sadegh Abbaszadeh
  • Ali Ebadian


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
$$\begin{aligned} \int _{[0,1]}^{\oplus } f\big ((1- t)a+ tb\big ) \odot \mathrm {d}m \leqslant g^{-1}\left( \frac{1}{2}\right) \odot \big (f(a)\oplus f(b)\big ), \end{aligned}$$
for which by choosing the convex and increasing function \(g(x)= x\), 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.


Pseudo-operation g-integral Hadamard inequality Convex function 



This work was supported by Iranian National Science Foundation: [Grant Number 95004084].

Compliance with ethical standards

Conflict of interest

The authors declares that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsPayame Noor UniversityTehranIran

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