Integral inequalities via generalized convex functions
-
2289
Downloads
-
3771
Views
Authors
Muhammad Aslam Noor
- Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan
Khalida Inayat Noor
- Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan
Farhat Safdar
- Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan
Abstract
In this paper, we introduce and investigate a new class of
generalized convex functions, called generalized log-convex
function. We establish some new Hermite-Hadamard integral
inequalities via generalized log-convex functions. Our results
represent refinement and improvement of the previously known
results. Several special cases are also discussed. The concepts and
techniques of this paper may stimulate further research in this
field.
Share and Cite
ISRP Style
Muhammad Aslam Noor, Khalida Inayat Noor, Farhat Safdar, Integral inequalities via generalized convex functions, Journal of Mathematics and Computer Science, 17 (2017), no. 4, 465-476
AMA Style
Noor Muhammad Aslam, Noor Khalida Inayat, Safdar Farhat, Integral inequalities via generalized convex functions. J Math Comput SCI-JM. (2017); 17(4):465-476
Chicago/Turabian Style
Noor, Muhammad Aslam, Noor, Khalida Inayat, Safdar, Farhat. "Integral inequalities via generalized convex functions." Journal of Mathematics and Computer Science, 17, no. 4 (2017): 465-476
Keywords
- Generalized convex functions
- generalized \(\log\)-convex functions
- Hermite-Hadamard type inequalities
MSC
References
-
[1]
M. Alomari, M. Darus, S. S. Dragomir , New inequalities of Simpson’s type for s-convex functions with applications, Res. Rep. Collect., 12 (2009), 1–18.
-
[2]
G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl., 335 (2007), 1294–1308.
-
[3]
B. C. Carlson , Special functions of applied mathematics, Academic Press, New York (1977)
-
[4]
G. Cristescu, Improved integral inequalities for products of convex functions, JIPAM. J. Inequal. Pure Appl. Math., 6 (2005), 6 pages.
-
[5]
G. Cristescu, L. Lupşa , Non-connected convexities and applications, Applied Optimization, Kluwer Academic Publishers, Dordrecht (2002)
-
[6]
M. R. Delavar, S. S. Dragomir, On \(\eta\)-convexity, Math. Inequal. Appl., 20 (2016), 203–216.
-
[7]
M. R. Delavar, F. Sajadian, Hermite-Hadamard type integral inequalities for log-\(\eta\)-convex function , Math. Comp. Sci., 1 (2016), 86–92.
-
[8]
S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University, Australia (2000)
-
[9]
M. E. Gordji, M. R. Delavar, M. De La Sen, On \(\varphi\) convex functions, J. Math. Inequal., 10 (2016), 173–183.
-
[10]
M. E. Gordji, S. S. Dragomir, M. R. Delavar, An inequality related to \(\eta\)-convex functions, II, Int. J. Nonlinear Anal. Appl., 6 (2015), 27–33.
-
[11]
D. H. Hyers, S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc., 3 (1952), 821–828.
-
[12]
C. P. Niculescu, The Hermite-Hadamard inequality for log-convex functions, Nonlinear Anal., 75 (2012), 662–669.
-
[13]
C. P. Niculescu, L. E. Persson, Convex functions and their applications, A contemporary approach, CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC, Springer, New York (2006)
-
[14]
M. A. Noor, On Hadamard integral inequalities involving two log-preinvex functions, JIPAM. J. Inequal. Pure Appl. Math., 8 (2007), 6 pages.
-
[15]
M. A. Noor, K. I. Noor, M. U. Awan, Hermite-Hadamard inequalities for relative semi-convex functions and applications, Filomat, 28 (2014), 221–230.
-
[16]
M. A. Noor, K. I. Noor, M. U. Awan, Some characterizations of harmonically log-convex functions, Proc. Jangjeon Math. Soc., 17 (2014), 51–61.
-
[17]
M. A. Noor, K. I. Noor, M. U. Awan, Generalized convexity and integral inequalities, Appl. Math. Inf. Sci., 9 (2015), 233–243.
-
[18]
M. A. Noor, K. I. Noor, S. Iftikhar, F. Safdar, Integral inequaities for relative harmonic (\(s,\eta\))-convex functions, Appl. Math. Comput. Sci., 1 (2016), 27–34.
-
[19]
M. A. Noor, K. I. Noor, F. Safdar, Generalized geometrically convex functions and inequalities, J. Inequal Appl., 2017 (2017 ), 19 pages.
-
[20]
M. A. Noor, K. I. Noor, F. Safdar, Integral inequalities via generalized (\(\alpha,m\))-convex functions, J. Nonlinear Funct. Anal., 2017 (2017 ), 13 pages.
-
[21]
J. E. Pečarić, F. Proschan, Y. L. Tong, Convex functions, partial orderings, and statistical applications, Mathematics in Science and Engineering, Academic Press, Inc., Boston, MA (1992)
-
[22]
M. Z. Sarikaya, On Hermite Hadamard inequalities for product of two log-\(\varphi\)- convex functions, Int. J. Modern Math. Sci., 6 (2013), 184–191.
-
[23]
M. Tunç, Some integral inequalities for logarithmically convex functions , J. Egyptian Math. Soc., 22 (2014), 177–181.