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Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2013, Article ID 698906, 1 page http://dx.doi.org/10.1155/2013/698906 Editorial Means and Their Inequalities Mowaffaq Hajja, 1 Peter S. Bullen, 2 Janusz Matkowski, 3 Edward Neuman, 4 and Slavko Simic 5 1 Yarmouk University, Irbid, Jordan 2 University of British Columbia, 2329 West Mall, Vancouver, BC, Canada V6T 1Z4 3 Faculty of Mathematics, Computer Science and Econometrics, University of Zielona G´ ora, Szafrana 4a, 65-516 Zielona G´ ora, Poland 4 Southern Illinois University, 1220 Lincoln Drive, Carbondale, IL 62901, USA 5 Serbian Academy of Sciences and Arts, Belgrade, Serbia Correspondence should be addressed to Mowaffaq Hajja; [email protected] Received 24 June 2013; Accepted 24 June 2013 Copyright © 2013 Mowaffaq Hajja et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e theory of means has its roots in the work of the Py- thagoreans who introduced the harmonic, geometric, and arithmetic means with reference to their theories of music and arithmetic. Later, Pappus introduced seven other means and gave the well-known elegant geometric proof of the celebrated inequalities among the harmonic, geometric, and arithmetic means. Nowadays, the families and types of means that are being investigated by researchers and the variety of questions that are being asked about them are beyond the scope of any single survey, with the voluminous book Handbook of Means and eir Inequalities by P. S. Bullen being the best such reference in this direction. e theory of means has grown to occupy a prominent place in mathematics with hundreds of papers on the subject appearing every year. e strong relations and interactions of the theory of means with the theories of inequalities, functional equations, and probability and statistics add greatly to its importance. Continuous versions of some means and inequalities among them tie it with real analysis and the theory of integration. e fact that centers of triangles and simplices can be viewed as means of points in the Euclidean spaces makes the subject of interest to geometers. Positivity and copositivity tests in the theory of forms naturally give rise to questions on internality tests of means arising from forms, making this aspect of the subject of interest to algebraists as well. Extensions of Gauss’s outstand- ing discoveries that relate the evaluation of certain elliptic integrals to iterations of the arithmetic and geometric means that led to the beautiful arithmeticogeometric mean resulted in so many interesting results and lines of research. A quick look at the table of contents of the book Pi and the AGM by J. M. Borwein and P. B. Borwein shows how extensive this line of research is and also shows that the subject is related to almost everything. e theory of means has applications in so many other diverse fields. Quoting from the preface of the aforemen- tioned book of P. S. Bullen, these include electrostatics, heat conduction, chemistry, and even medicine. is issue contains several papers that pertain to some of the the aforementioned subjects. One of the papers is an exposition of certain elementary aspects of the subject, together with several open problems that are within the comprehension of a graduate student. It is hoped that such questions will lead to contributions from experts and amateurs alike. Mowaffaq Hajja Peter S. Bullen Janusz Matkowski Edward Neuman Slavko Simic

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Page 1: Editorial Means and Their Inequalitiesdownloads.hindawi.com/journals/ijmms/2013/698906.pdf · Means and Their Inequalities MowaffaqHajja, 1 PeterS.Bullen, 2 JanuszMatkowski, 3 EdwardNeuman,

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2013, Article ID 698906, 1 pagehttp://dx.doi.org/10.1155/2013/698906

EditorialMeans and Their Inequalities

Mowaffaq Hajja,1 Peter S. Bullen,2 Janusz Matkowski,3

Edward Neuman,4 and Slavko Simic5

1 Yarmouk University, Irbid, Jordan2University of British Columbia, 2329 West Mall, Vancouver, BC, Canada V6T 1Z43 Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora, Szafrana 4a,65-516 Zielona Gora, Poland

4 Southern Illinois University, 1220 Lincoln Drive, Carbondale, IL 62901, USA5 Serbian Academy of Sciences and Arts, Belgrade, Serbia

Correspondence should be addressed to Mowaffaq Hajja; [email protected]

Received 24 June 2013; Accepted 24 June 2013

Copyright © 2013 Mowaffaq Hajja et al.This is an open access article distributed under the Creative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The theory of means has its roots in the work of the Py-thagoreans who introduced the harmonic, geometric, andarithmetic means with reference to their theories of musicand arithmetic. Later, Pappus introduced seven other meansand gave the well-known elegant geometric proof of thecelebrated inequalities among the harmonic, geometric, andarithmetic means.

Nowadays, the families and types of means that are beinginvestigated by researchers and the variety of questions thatare being asked about them are beyond the scope of any singlesurvey, with the voluminous book Handbook of Means andTheir Inequalities by P. S. Bullen being the best such referencein this direction. The theory of means has grown to occupy aprominent place in mathematics with hundreds of papers onthe subject appearing every year.

The strong relations and interactions of the theory ofmeans with the theories of inequalities, functional equations,and probability and statistics add greatly to its importance.

Continuous versions of some means and inequalitiesamong them tie it with real analysis and the theory ofintegration. The fact that centers of triangles and simplicescan be viewed as means of points in the Euclidean spacesmakes the subject of interest to geometers.

Positivity and copositivity tests in the theory of formsnaturally give rise to questions on internality tests of means

arising from forms, making this aspect of the subject ofinterest to algebraists as well. Extensions of Gauss’s outstand-ing discoveries that relate the evaluation of certain ellipticintegrals to iterations of the arithmetic and geometric meansthat led to the beautiful arithmeticogeometric mean resultedin so many interesting results and lines of research. A quicklook at the table of contents of the book Pi and the AGM byJ. M. Borwein and P. B. Borwein shows how extensive thisline of research is and also shows that the subject is related toalmost everything.

The theory of means has applications in so many otherdiverse fields. Quoting from the preface of the aforemen-tioned book of P. S. Bullen, these include electrostatics, heatconduction, chemistry, and even medicine.

This issue contains several papers that pertain to some ofthe the aforementioned subjects.

One of the papers is an exposition of certain elementaryaspects of the subject, together with several open problemsthat are within the comprehension of a graduate student. Itis hoped that such questions will lead to contributions fromexperts and amateurs alike.

Mowaffaq HajjaPeter S. Bullen

Janusz MatkowskiEdward Neuman

Slavko Simic

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