## Tbilisi Mathematical Journal

### On some Hermite-Hadamard-type integral inequalities for co-ordinated $\boldsymbol{(\alpha, \mbox{QC})}$- and $\boldsymbol{(\alpha, \mbox{CJ})}$-convex functions

#### Abstract

In the article, the authors introduce the new concepts “co-ordinated $(\alpha, \mbox{QC})$-, $(\alpha,\mbox{JQC})$-, $(\alpha, \mbox{CJ})$- and $(\alpha, \mbox{J})$-convex functions”, establish some Hermite-Hadamard's type integral inequalities for the co-ordinated $(\alpha, \mbox{QC})$-, $(\alpha,\mbox{JQC})$-, $(\alpha, \mbox{CJ})$- and $(\alpha, \mbox{J})$-convex functions.

#### Article information

Source
Tbilisi Math. J., Volume 8, Issue 2 (2015), 75-86.

Dates
Accepted: 24 May 2015
First available in Project Euclid: 12 June 2018

https://projecteuclid.org/euclid.tbilisi/1528769008

Digital Object Identifier
doi:10.1515/tmj-2015-0012

Mathematical Reviews number (MathSciNet)
MR3360644

#### Citation

Xi, Bo-Yan; Sun, Jian; Bai, Shu-Ping. On some Hermite-Hadamard-type integral inequalities for co-ordinated $\boldsymbol{(\alpha, \mbox{QC})}$- and $\boldsymbol{(\alpha, \mbox{CJ})}$-convex functions. Tbilisi Math. J. 8 (2015), no. 2, 75--86. doi:10.1515/tmj-2015-0012. https://projecteuclid.org/euclid.tbilisi/1528769008

#### References

• R.-F. Bai, F. Qi, and B.-Y. Xi, Hermite-Hadamard type inequalities for the $m$- and $(\alpha,m)$-logarithmically convex functions, Filomat 27 (2013), no. 1, 1–7.
• S.-P. Bai, S.-H. Wang, and F. Qi, Some Hermite-Hadamard type inequalities for $n$-time differentiable $(\alpha,m)$-convex functions, J. Inequal. Appl. 2012, 2012:267; Available online at.
• S. S. Dragomir, On the Hadamard's inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese J. Math. 5 (2001), no. 4, 775–788.
• S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University (2000). Available online at http://www.staff.vu.edu.au/RGMIA/monographs/hermite${_{}$hadamard.html}.
• S. S. Dragomir and C. E. M. Pearce, Quasi-convex functions and Hadamard's inequality, Bull. Austr. Math. Soc. 57 (1998), no 3, 377–385.
• S. S. Dragomir, J. Pečarić, and L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math. 21 (1995), no. 3, 335–341.
• M. A. Latif and S. S. Dragomir, On some new inequalities for differentiable co-ordinated convex functions, J. Inequal. Appl. 2012, 2012:28; Available online at.
• J. Pečarić, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press (1992), Inc.
• F. Qi and B.-Y. Xi, Some integral inequalities of Simpson type for GA-$\varepsilon$-convex functions, Georgian Math. J. 20 (2013), no. 4, 775–788; Available online at.
• B.-Y. Xi, R.-F. Bai, and F. Qi, Hermite-Hadamard type inequalities for the $m$- and $(\alpha,m)$-geometrically convex functions, Aequationes Math. 84 (2012), no. 3, 261–269; Available online at.
• B.-Y. Xi, J. Hua, and F. Qi, Hermite-Hadamard type inequalities for extended $s$-convex functions on the co-ordinates in a rectangle, Journal of Applied Analysis 20 (2014), no.1, 29-39; Available online at.
• B.-Y. Xi and F. Qi, Integral inequalities of Simpson type for logarithmically convex functions, Advanced Studies in Contemporary Mathematics 23 (2013), no. 4, 559–566.
• B.-Y. Xi and F. Qi, Some integral inequalities of Hermite-Hadamard type for convex functions with applications to means, J. Funct. Spaces Appl. 2012 (2012), Article ID 980438, 14 pages; Available online at.
• B.-Y. Xi and F. Qi, Some Hermite-Hadamard type inequalities for differentiable convex functions and applications, Hacettepe Journal of Mathematics and Statistics 42 (2013), no. 3, 243–257..
• B.-Y. Xi and F. Qi, Hermite-Hadamard type inequalities for geometrically $r$-convex functions, Studia Scientiarum Mathematicarum Hungarica, 51 (2014), no.4, 530-546; Available online at.
• B.-Y. Xi, S.-H. Wang, and F. Qi, Some inequalities for $(h,m)$-convex functions, Journal of Inequalities and Applications 2014, 2014:100, 12 pages; Available online at.
• B.-Y. Xi, S.-H. Wang, and F. Qi, Properties and inequalities for the $h$- and $(h,m)$-logarithmically convex functions, Creative Mathematics and Informatics 23 (2014), no. 1, 123-130.