Convex Functions and Their Applications

Constantin P.Niculescu
Lars-Erik Persson

1 Convex Functions on Intervals ............................. 7
1.1 ConvexFunctionsatFirstGlance ......................... 7
1.2 Young’s InequalityandItsConsequences ................... 14
1.3 SmoothnessProperties ................................... 20
1.4 AnUpperEstimateofJensen’s Inequality .................. 27
1.5 TheSubdifferential ...................................... 29
1.6 Integral Representation of Convex Functions ................ 36
1.7 Conjugate Convex Functions .............................. 40
1.8 TheIntegralFormofJensen’s Inequality ................... 44
1.9 The Hermite–Hadamard Inequality ........................ 50
1.10 ConvexityandMajorization .............................. 53
1.11 Comments.............................................. 60
2 Comparative Convexity on Intervals ....................... 65
2.1 AlgebraicVersionsofConvexity ........................... 65
2.2 TheGammaandBetaFunctions .......................... 68
2.3 GeneralitiesonMultiplicativelyConvexFunctions ........... 77
2.4 MultiplicativeConvexityofSpecialFunctions ............... 83
2.5 AnEstimateof theAM–GM Inequality.................... 85
2.6 (M,N)-ConvexFunctions ................................ 88
2.7 RelativeConvexity ...................................... 91
2.8 Comments.............................................. 97
3 Convex Functions on a Normed Linear Space ..............101
3.1 ConvexSets ............................................101
3.2 The Orthogonal Projection ...............................106
3.3 HyperplanesandSeparationTheorems .....................109
3.4 ConvexFunctions inHigherDimensions....................112
3.5 ContinuityofConvexFunctions ...........................119
3.6 PositivelyHomogeneousFunctions.........................123
3.7 TheSubdifferential ......................................128
3.8 Differentiability of Convex Functions .......................135
3.9 RecognizingConvexFunctions ............................141
3.10 TheConvexProgrammingProblem........................145
3.11 Fine Properties of Differentiability .........................152
3.12 Pr´ ekopa–Leindler Type Inequalities ........................158
3.13 Mazur–UlamSpacesandConvexity........................165
3.14 Comments..............................................171
4 Choquet’s Theory and Beyond.............................177
4.1 Steffensen–PopoviciuMeasures ............................177
4.2 TheJensen–SteffensenInequalityandMajorization ..........184
4.3 Steffensen’s Inequalities ..................................190
4.4 Choquet’sTheorem......................................192
4.5 Comments..............................................199