JOZSEF´ SANDOR´ THEORY OF MEANS AND THEIR INEQUALITIES This book is dedicated to the memory of my parents: ILKA SANDOR´ (1928-2010) JOZSEF´ SANDOR´ (1911-1981) Contents Preface 9 1 Classical means 13 1.1 On the arithmetic mean { geometric mean inequality for n numbers . 13 1.2 A refinement of arithmetic mean { geometric mean inequality . 16 1.3 On Bernoulli's inequality . 19 1.4 A refinement of the harmonic-geometric inequality . 22 1.5 On certain conjectures on classical means . 25 1.6 On weighted arithmetic-geometric inequality . 30 1.7 A note on the inequality of means . 32 1.8 On an inequality of Sierpinski . 37 1.9 On certain inequalities for means of many arguments . 40 1.10 A note on log-convexity of power means . 45 1.11 A note on the ' and functions . 50 1.12 Generalizations of Lehman's inequality . 53 2 Logarithmic, identric and related means 65 2.1 On the identric and logarithmic means . 65 2.2 A note on some inequalities for means . 79 2.3 Refinements of certain inequalities for means . 84 2.4 On certain identities for means . 89 5 2.5 Inequalities for means . 99 2.6 Inequalities for means of two arguments . 106 2.7 An application of Gauss' quadrature formula . 117 2.8 On certain subhomogeneous means . 120 2.9 Monotonicity and convexity properties of means . 130 2.10 Logarithmic convexity of the means It and Lt . 141 2.11 On certain logarithmic inequalities . 145 2.12 A note on the logarithmic mean . 152 2.13 A basic logarithmic inequality, and the logarithmic mean . 154 2.14 On certain inequalities for means in two variables . 159 2.15 On a logarithmic inequality . 163 2.16 Series expansions related to the logarithmic mean . 166 2.17 On some inequalities for the identric, logarithmic and related means . 171 2.18 New refinements of two inequalities for means . 183 2.19 On certain entropy inequalities . 189 3 Integral inequalities and means 195 3.1 Some integral inequalities . 195 3.2 Some simple integral inequalities . 201 3.3 Generalization of the Hadamard integral inequalities . 212 3.4 Applications of the Cauchy-Bouniakowsky inequality in the theory of means . 214 3.5 On some exponential means . 231 3.6 A property of an exponential mean . 237 3.7 Some new inequalities for means and convex functions . 239 3.8 Inequalities for general integral means . 245 3.9 On upper Hermite-Hadamard inequalities for geometric-convex and log-convex functions . 252 3.10 On certain identities for means, III . 261 6 3.11 On some exponential means, II . 279 3.12 On global bounds for generalized Jensen's inequality . 290 4 Means and their Ky Fan type inequalities 301 4.1 On an inequality of Ky Fan . 301 4.2 A refinement of the Ky Fan inequality . 311 4.3 A converse of Ky Fan's inequality . 317 4.4 On certain new Ky Fan type inequalities for means . 319 4.5 An extension of Ky Fan's inequalities . 327 4.6 Notes on certain inequalities by H¨older, Lewent and Ky Fan . 330 4.7 On certain new means and their Ky Fan type inequalities . 334 4.8 On common generalizations of some inequalities . 344 5 Stolarsky and Gini means 349 5.1 On reverse Stolarsky type inequalities for means . 349 5.2 Inequalities for certain means in two arguments . 357 5.3 A note on certain inequalities for bivariate means . 366 5.4 A note on logarithmically completely monotonic ratios of certain mean values . 370 5.5 On means generated by two positive functions . 378 5.6 On certain weighted means . 386 5.7 Stolarsky and Gini means . 396 5.8 A note on the Gini means . 415 5.9 Inequalities for the ratios of certain bivariate means . 420 5.10 Inequalities involving logarithmic mean of arbitrary order . 440 6 Sequential means 447 6.1 On some inequalities for means . 447 6.2 On inequalities for means by sequential method . 454 7 6.3 On certain inequalities for means, III . 463 6.4 On two means by Seiffert . 473 6.5 The Schwab-Borchardt mean . 477 6.6 Refinements of the Mitrinovi´c-Adamovi´cinequality with application . 494 6.7 A note on bounds for the Neuman-S´andormean using power and identric means . 499 6.8 A note on a bound of the combination of arithmetic and harmonic means for the Seiffert’s mean . 504 6.9 The Huygens and Wilker-type inequalities as inequalities for means of two arguments . 509 6.10 On Huygens' inequalities . 520 6.11 New refinements of the Huygens and Wilker hyperbolic and trigonometric inequalities . 532 6.12 On two new means of two variables . 538 Final references 551 Author Index 565 8 Preface The aim of this book is to present short notes or articles, as well as studies on some topics of the Theory of means and their inequalities. This is mainly a subfield of Mathematical analysis, but one can find here also applications in various other fields as Number theory, Numerical analysis, Trigonometry, Networks, Information theory, etc. The material is divided into six chapters: Classical means; Logarithmic, identric and related means; Integral inequalities and means; Means and their Ky Fan type inequalities; Stolarsky and Gini means; and Sequential means. Chapter 1 deals essentially with the classical means, including the arithmetic, geometric, harmonic means of two or more numbers and the relations connecting them. One can find here more new proofs of the classical arithmetic mean-geometric mean-harmonic mean inequality, as well as their weighted version. The famous Sierpinski inequality connect- ing these means, or applications of the simple Bernoulli inequality offer strong refinements of these results, including the important Popoviciu and Rado type inequalities. One of the subjects is the log-convexity prop- erties of the power means, which is applied so frequently in mathematics. The chapter contains also applications of some results for certain arith- metic functions, the theory of Euler's constant e, or electrical network theory. The largest chapter of the book is Chapter 2 on the identric, logarith- mic and related means. These special means play a fundamental role in the study of many general means, including certain exponential, integral 9 means, or Stolarsky and Gini means, etc. There are considered various identities or inequalities involving these means, along with strong refine- ments of results known in the literature. Monotonicity, convexity and subhomogeneity properties are also studied. Series representations, and their applications are also considered. Applications for certain interesting logarithmic inequalities, having importance in other fields of mathemat- ics, or in the entropy theory are also provided. Chapter 3 contains many important integral inequalities, as the Cauchy-Bouniakowski integral inequality, the Hadamard (or Hermite- Hadamard) integral inequalities, the Jensen integral inequality, etc. Re- finements, as well as generalizations or extensions of these inequalities are considered. As the inequalities offer in fact results for the integral mean of a function, many consequences or applications for special means are obtained. The classical monotonicity notion and its extension of sec- tion 1 give in a surprising manner interesting and nice results for means. The Jensen functionals considered in the last section, offer very general results with many applications. Chapter 4 studies the famous Ky Fan type inequalities. As this in- equality contains in a limiting case the arithmetic mean-geometric mean inequality, these results are connected with Chapter 1. However, here one obtains a more detailed and complicated study of these inequalities, involving many new means. Section 1 presents one of the author's early results, rediscovered later by other authors, namely the equivalence of Ky Fan's inequality with Henrici's inequality. Extensions, converses, re- finements are also provided. The related Wang-Wang inequality is also considered. Ky Fan type inequalities will be considered also in Chapter 5, on the Stolarsky and Gini means. These general means are studied extensively in sections 7 and 9 of this chapter. Particular cases, as the generalized logarithmic means, and a particular Gini mean, which is also a weighted geometric mean, are included, too. This last mean is strongly related, by 10 an identity, with the identric mean of Chapter 2, and has many connec- tions with other means, so plays a central role in many results. Finally, Chapter 6 deals with means, called by the author as \sequen- tial means", which may be viewed essentially as the common limit of certain recurrent sequences. These are the famous arithmetic-geometric mean of Gauss, the Schwab-Borchardt mean, etc. Here the classical loga- rithmic mean is considered also as a such mean, and many other famous particular means, as the Seiffert means, or the Neuman-S´andormean, are studied. Two new means of the author are introduced in the last section. These means have today a growing literature, and their importance is recognized by specialists of the field. All sections in all chapters are based on the author's original papers, published in various national and international journals. We have in- cluded a \Final references" section with the titles of the most important publications of the author in the theory of means. The book is concluded with an author index, focused on articles (and not pages). A citation of type I.2(3) shows that the respective author is cited three times in section 2 of chapter 1. Finally, we wish to mention the importance of this domain of mathe- matics. Usually, researchers encounter in their studies the need of certain bounds, which essentially may be reduced to a relation between means.