Generalization and reverses of the left Fejer inequality for convex functions
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Authors
S. S. Dragomir
- Mathematics, School of Engineering & Science, Victoria University, P. O. Box 14428, Melbourne City, MC 8001, Australia.
- chool of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa.
Abstract
In this paper we establish a generalization of the left Fej´er inequality for general Lebesgue integral on measurable spaces
as well as various upper bounds for the difference
\[\frac{1}{\int^b_a g(x)dx} \int^b_ah(x)g(x)dx-h\left(\frac{a+b}{2}\right),\]
where \(h : [a, b] \rightarrow \mathbb{R}\) is a convex function and \(g : [a, b] \rightarrow [0,\infty)\) is an integrable weight. Applications for discrete means and
Hermite-Hadamard type inequalities are also provided.
Share and Cite
ISRP Style
S. S. Dragomir, Generalization and reverses of the left Fejer inequality for convex functions, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 6, 3231--3244
AMA Style
Dragomir S. S., Generalization and reverses of the left Fejer inequality for convex functions. J. Nonlinear Sci. Appl. (2017); 10(6):3231--3244
Chicago/Turabian Style
Dragomir, S. S.. "Generalization and reverses of the left Fejer inequality for convex functions." Journal of Nonlinear Sciences and Applications, 10, no. 6 (2017): 3231--3244
Keywords
- Convex functions
- integral inequalities
- Jensen’s type inequalities
- Fejér type inequalities
- Lebesgue integral
- Hermite-Hadamard type inequalities
- special means.
MSC
References
-
[1]
A. G. Azpeitia, Convex functions and the Hadamard inequality, Rev. Colombiana Mat., 28 (1994), 7–12.
-
[2]
S. S. Dragomir, C. E. M. Pearce, Selected topics on HermiteHadamard inequalities and applications, RGMIA Monographs, Victoria University, Melbourne City, Australia (2000)
-
[3]
L. Fejér, Über die Fourierreihen, II, (Hungarian) Math. Natur. Ungar. Akad Wiss., 24 (1906), 369–390.
-
[4]
A. Guessab, G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Theory, 115 (2002), 260–288.
-
[5]
E. Kikianty, S. S. Dragomir, Hermite-Hadamard’s inequality and the p-HH-norm on the Cartesian product of two copies of a normed space, Math. Inequal. Appl., 13 (2010), 1–32.
-
[6]
M. Merkle, Remarks on Ostrowski’s and Hadamard’s inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 10 (1999), 113–117.
-
[7]
C. E. M. Pearce, A. M. Rubinov, P-functions, quasi-convex functions, and Hadamard-type inequalities, J. Math. Anal. Appl., 240 (1999), 92–104.
-
[8]
J. Pečarić, A. Vukelić , Hadamard and Dragomir-Agarwal inequalities, the Euler formulae and convex functions, Functional equations, inequalities and applications, Kluwer Acad. Publ., Dordrecht, (2003), 105–137.
-
[9]
G. Toader, Superadditivity and Hermite-Hadamard’s inequalities, Studia Univ. Babe-Bolyai Math., 39 (1994), 27–32.
-
[10]
G.-S. Yang, M.-C. Hong, A note on Hadamard’s inequality, Tamkang J. Math., 28 (1997), 33–37.
-
[11]
G.-S. Yang, K.-L. Tseng, On certain integral inequalities related to Hermite-Hadamard inequalities, J. Math. Anal. Appl., 239 (1999), 180–187.