Title

SCHUR-GEOMETRIC CONVEXITY
OF STOLARSKY'S EXTENDED MEAN VALUES

Authors

R.C.L. Janardhana, K.M. Nagaraja, V. Lokesha
Research and Development Center
Bharathiar University
Coimbatore, 641046, INDIA
Department of Mathematics
Govt. First Grade College
Rajajinagar, Bengaluru, 560010, Karanataka, INDIA
Department of Mathematics
JSS Academy of Technical Education
Uttarahalli, Kengeri Main Road, Bangalore-60, Karanataka, INDIA
PG Department of Mathematics
VSK University, Jnana Sagara Campus
Vinayaka Nagara, Bellary-04, Karanataka, INDIA

Abstract

In the recent years, the Schur convexity and Schur-geometrically convexity of Stolarsky's mean values have attracted the attention of a considerable number of mathematicians and researchers. In this paper, the Schur-geometric convexity of Stolarsky's extended type mean values are discussed.

History

Received: December 8, 2016
Revised: February 19, 2017
Published: April 21, 2017

AMS Classification, Key Words

AMS Subject Classification: 26D10, 26D15
Key Words and Phrases: schur-geometric, convexity, concavity, harmonic mean, contra harmonic mean

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Bibliography

1

P.S. Bullen, Handbook of means and their inequalities,Kluwer Acad. Publ., Dordrecht, (2003).

2

Y.M. Chu, X.M. Zhang and G.D. Wang, The Schur geometrical convexityof the extended mean values, J. Convex. Anal., 15 4 (2008), 707-718.

3

Y.M. Chu and X.M. Zhang, Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave, J. Math. Kyoto Univ. 48 1 (2008), 229-238.

4

Y.M. Chu and Y.P. Lv, The Schur harmonic convexity of the Hamysymmetric function and its applications, J. Inequal. Appl. (2009), 10, doi: https://doi.org/10.1155/2009/838529.

5

G.H. Hardy, J.E. Littlewood and G. Pólya, Some simple inequalities satisfied by convex functions, Messenger Math. 58 (1929), 145-152.

6

E.B. Leach and M.C. Sholander, Extended mean values II, J. Math. Anal. Appl., 92 (1983), 207-223.

7

D.M. Li and H.N. Shi, Schur convexity and Schur-geometrically concavity of generalized exponent mean, J. Math. Inequal. 3 2 (2009), 217-225.

8

V. Lokesha, S. Padmanabhan, K.M. Nagaraja and Y. Simsek, Relation between Greek means and various other means, General Mathematics, 17 3 (2009), 3-13.

9

V. Lokesha, K.M. Nagaraja, B.N. Kumar and S. Padmanabhan, Oscillatory type mean in Greek means, Int. e-Journal ofEngg. Maths Theory and Applications, 9 3 (2010), 18-26.

10

A.W. Marshall and I. Olkin, Inequalities: Theorey of Majorization and Its Applications, New York, Academic Press (1979).

11

K.M. Nagaraja, K. Murali and V. Lokesha, Schur convexity and concavity of Gnan mean, Proceedings of the Jangjeon Math. Society 17 3 (2014), 355-367.

12

K.M. Nagaraja and S.K. Sahu, Schur Geometric convexity of Gnan mean for two variables, Journal of the International Mathematical Virtual Institute 3 (2013), 39-59.

13

K.M. Nagaraja, P.S.K. Reddy and K. Sridevi, Schur Harmonic convexity of Gnan mean for two variables, Journal of the International Mathematical Virtual Institute 3 (2013), 61-80.

14

K.M. Nagaraja, P.S.K. Reddy and B. Naveenkumar, Refinement of Inequality involving ratio of Means for four Positive Arguments, Bulletin of International Mathematical Virtual Institute 3 (2013), 135-138.

15

K.M. Nagaraja, V. Lokesha and S. Padmanabhan, A simple proof on strengthening and extension of inequalities, Advn. Stud. Contemp. Math., 17 1 (2008), 97-103.

16

F. Qi, J. Sandor and S.S. Dragomir, Notes on the Schur-convexity of the extended mean values, Taiwanese J. Math., 9 3 (2005), 411-420.

17

F. Qi, A note on Schur-convexity of extended mean values, Rocky Mountain J. Math. 35 5 (2005), 1787-1793.

18

H.N. Shi, S.H. Wu and F. Qi, An alternative note on the Schur-convexity of the extended mean values, Math. Inequal. Appl. 9 2 (2006), 219-224.

19

S. Simic, An extension of Stolarsky means, Novi Sad J. Math. 38 3 (2008), 81-89.

20

S. Simic, On weighted Stolarsky means, Sarajevo Journal of Mathematics 7 19 (2011), 3-9.

21

K.B. Stolarsky, Generalizations of the Logarithmic Mean, Math. Mag. 48 (1975), 87-92.

22

K.B. Stolarsky, The power and generalized Logarithmic Means, Amer.Math. Monthly 87 (1980), 545-548.

23

Zh.H. Yang, On the homogeneous functions with two parametersand its monotonicity, J. Inequal. Pure Appl. Math. 6 (2005).

24

X.M. Zhang, Geometrically Convex Functions, Hefei, An. hui University Press (2004) (Chinese).

25

Zhen-Hang Yang, Necessary and Suffcient Condition for Schur Convexity of the Two-Parameter Symmetric Homogeneous Means, Applied Mathematical Sciences 5 64 (2011), 3183 - 3190.

How to Cite?

DOI: 10.12732/ijpam.v114i1.7 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 114
Issue: 1
Pages: 77 - 90

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