Generalized results of majorization inequality via Lidstone's polynomial and newly Green functions
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Authors
Nouman Siddique
- Department of Mathematics, Govt. College University, Faisalabad 38000, Pakistan.
Naveed Latif
- General Studies Department, Jubail Industrial College, Jubail Industrial City 31961, Kingdom of Saudi Arabia.
Josip Pečarić
- Faculty of Textile Technology Zagreb, University of Zagreb, Prilaz Baruna Filipovica 28A, 10000 Zagreb, Croatia.
- RUDN University, 6 Miklukho-Maklay St, Moscow, 117198, Russia.
Abstract
Generalized results of majorization inequality are obtained by using newly Green functions defined in [N. Mahmood, R. P. Agarwal, S. I. Butt, J. Pečarić, J. Inequal. Appl., \({\bf2017}\) (2017), 17 pages]
and Lidstone's polynomial. We find
new upper bounds of Grüss and Ostrowski type.
We give further results of majorization inequality by making linear functionals constructed on convex functions \(\frac{f(x)}{x}\).
Some applications are given.
Share and Cite
ISRP Style
Nouman Siddique, Naveed Latif, Josip Pečarić, Generalized results of majorization inequality via Lidstone's polynomial and newly Green functions, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 6, 812--831
AMA Style
Siddique Nouman, Latif Naveed, Pečarić Josip, Generalized results of majorization inequality via Lidstone's polynomial and newly Green functions. J. Nonlinear Sci. Appl. (2018); 11(6):812--831
Chicago/Turabian Style
Siddique, Nouman, Latif, Naveed, Pečarić, Josip. "Generalized results of majorization inequality via Lidstone's polynomial and newly Green functions." Journal of Nonlinear Sciences and Applications, 11, no. 6 (2018): 812--831
Keywords
- Classical majorization theorem
- Fuchs's thorem
- Lidstone's interpolating polynomial
- Green Function for 'two point right focal' problem
- Čebyšev functional
- Grüss type upper bounds
- Ostrowski-type bounds
- convex function \(f(x)/x\)
- n-exponentially convex function
- mean value theorems
- Stolarsky type means
MSC
References
-
[1]
R. P. Agarwal, S. I. Bradanović, J. Pečarić , Generalizations of Sherman’s inequality by Lidstone’s interpolating polynomial, J. Inequal. Appl., 2016 (2016), 18 pages.
-
[2]
R. P. Agarwal, P. J. Y. Wong , Error Inequalities in Polynomial Interpolation and their Applications, Kluwer Academic Publishers, Dordrecht (1993)
-
[3]
G. Aras-Gazić, V. Čuljak, J. Pečarić, A. Vukelić , Generalization of Jensen’s inequality by Lidstone’s polynomial and related results, Math. Inequal. Appl., 16 (2013), 1243–1267
-
[4]
P. Cerone , On Čebyšev functional bounds , Differential & difference equations and applications, 267-277, Hindawi Publ. Corp., New York (2006)
-
[5]
P. Cerone, S. S. Dragomir, Some new Ostrowski-type bounds for the Čebyšev functional and applications, J. Math. Inequal., 8 (2014), 159–170.
-
[6]
K. R. Davidson, A. P. Donsig, Real Analysis with Real Applications, Prentice Hall, Upper Saddle River (2002)
-
[7]
L. Fuchs , A new proof of an inequality of Hardy-Littlewood-Polya, Mat. Tidsskr, B , (1947), 53–54.
-
[8]
G. H. Hardy, J. E. Littlewood, G. Pólya , Inequalities , Second ed., Cambridge University Press, London and New York (1952)
-
[9]
J. Jakšetić, J. Pečarić , Exponential convexity method, J. Convex Anal., 20 (2013), 181–197.
-
[10]
S. Karlin , Total Positivity , Stanford Univ. Press, Stanford (1968)
-
[11]
A. R. Khan, N. Latif, J. Pečarić, Exponential convexity for majorization, J. Inequal. Appl., 2012 (2012), 13 pages.
-
[12]
A. R. Khan, N. Latif, J. Pečarić, n-exponential convexity for Favard’s and Berwald’s inequalities and their applications, Adv. Inequal. Appl., 2014 (2014), 21 pages.
-
[13]
M. A. Khan, N. Latif, I. Perić, J. Pečarić , On majorization for matrices, Math. Balkanica, 27 (2013), 3–19.
-
[14]
K. A. Khan, J. Pečarić, I. Perić, Generalization of Popoviciu Type Inequalities for Symmetric Means generated by Convex Function, J. Math. Comput. Sci., 4 (2014), 1091–1113.
-
[15]
N. Mahmood, R. P. Agarwal, S. I. Butt, J. Pečarić , New Generalization of Popoviciu type inequalities via new Green functions and Montgomery identity , J. Inequal. Appl., 2017 (2017), 17 pages.
-
[16]
A. W. Marshall, I. Olkin, B. C. Arnold , Inequalities: Theory of Majorization and Its Applications, Second Ed., Springer, New York (2011)
-
[17]
J. Pečarić, F. Proschan, Y. L. Tong, , Convex functions , Partial Orderings and Statistical Applications, Academic Press, Boston (1992)
-
[18]
T. Popoviciu, Sur l’approximation des fonctions convexes d’ordre superier, Mathematica, 10 (1934), 49–54.
-
[19]
D. V. Widder, Completely convex function and Lidstone series, Trans. Amer. Math. Soc., 51 (1942), 387–398.