Index for The Cauchy-Schwarz Master Class

Abel inequality, 208, 221
Abel, Niels Henrik, 208
Aczel, J., 287
additive bound, 4, 9, 66, 106
and Holder, 137, 264
Akerberg’s refinement, 34
al-Khazin, Abu Ja’far, 286
Alexanderson, G., 286
AM-GM inequality, 20, 21
and Kantorovich, 247
Akerberg’s refinement, 34
algorithmic proof, 245
and investments, 100
and the exponential, 91
betweeness induction, 83
general weights, 23
geometric interpretation, 33
integral analog, 113
Polya’s proof, 23
rational weights, 22
smoothing proof, 244
stability, 35
via cyclic shifts, 84
via integral analog, 114
via rearrangement, 84
via Schur concavity, 194
American Mathematical Monthly, 96,
192, 253
Andreescu, T., 251
anti-symmetric forms, 237
arithmetic mean-geometric mean
inequality, see AM-GM inequality
Arithmetica of Diophantus, 40
arrangement of spheres, 52
Artin, Emil, 47
backtracking, 137
backwards induction, 236
baseball inequality, 82
Belentepe, C., vi
Bennett, C., 288, 289
Bernoulli inequality, 31
Bessel inequality, 71, 225
betweeness, exploitation of, 74
Birkhoff’s Theorem, 207
birthday problem, 206
Bledsoe, W.W., 283
Bombieri, E., 284
box, thinking outside, 52
Brahmagupta, 286
identity, 47
Brunn–Minkowski inequality, 245, 287
Bunyakovsky, Victor Yacovlevich, 10,
190, 285
and AM-GM inequality, 115
Chebyshev contact, 76
vis-a-vis Schwarz, 11
Buzano, M.L., 286
Cai, T., vi, 246
cancellation, origins of, 210
Carleman inequality, 118
Carleson proof, 173
Polya’s proof, 27
refined via Knopp, 128
Carleson inequality, 173
Carlson inequality, 165
Cauchy inequality, 1
and quadratic forms, 14
beating, 15
by induction, 2
case of equality, 37
in an array, 16
interpolation, 48
Polya–Szego converse, 83
three term, 13
302
Page 2
Index
303
vector-scalar melange, 69
via inner product identity, 68
via monotone interpolation, 85
Cauchy’s induction argument
AM-GM inequality, 20
Cauchy inequality, 31
Jensen inequality, 101
Cauchy, Augustin-Louis, 10
Cauchy–Binet identity, 49
Cauchy–Schwarz inequality, 8
as accidental corollary, 57
cross term defects, 83
geometric proof, 58
self-generalization, 16
via Gram–Schmidt, 71
centered inequality, 115
Chebyshev
order inequality, 76
tail inequality, 86
Chebyshev, Pafnuty Lvovich, 76, 287
Chong, K.M., 244
Chung, F.R.K., 289
Clevenson, M.L., 277
closest point problem, 56
completing the square, 57
conjugate powers, 137
consistency principles, 134
convex functions, continuity of, 254
convex minorant, 98
convexity
defined, 87
differential criterion, 90
geometric characterizations, 87
strict, 89
versus J-convexity, 101
via transformation, 102
Cours d’Analyse Algebrique, 10
Cramer–Rao inequality, 18
Cronin–Kardon, C., vi
crystallographic inequality, 13
cyclic shifts, 84
cyclic sum inequality, 104
D’Angelo, J.P, 242
Davis, P.J., 287
Debeau, F., 285
Dellacherie, C., vi, 254, 290
determinant, 49
Diaconis, P., vi
Diophantus, 40, 286
Diophantus identity
and Brahmagupta, 47, 237
and complex factorization, 237
and Pythagoras, 47, 237
dissection of integrals, 106
doubly stochastic, 196
Dragomir, S.S., 241, 285
Dudley, R.M., vi
elementary symmetric function, 178
Elliot, E.B., 290
Enflo inequality, 225
Engel, A., 94, 246, 251
equality
in AM-GM inequality, 22
in Cauchy inequality, 5, 37
in Cauchy–Schwarz, 8
in Holder, 137
in Jensen inequality, 89
Erd˝os, P., 261
Euclidean distance
in R
d
, 51
triangle inequality, 53
exponential sums, 210
extend and conquer, 222
Feng, Z., 251
Fermat, Pierre de, 40
Fibonacci, 286
Fisher information, 18
Flor, P., 238
four letter identity, 49
Fuji, M., 287
gamma function, 165
Gauss, C.F., 288
general means, 120
generic improvement, 94
geometric mean
as a limit, 120
as minimum, 133
superadditivity, 100, 133
George, C., 231, 288
Gram–Schmidt process, 70
Gross–Erdmann, K.-G., 290
Gruss inequality, 119
Hadwiger–Finsler inequality, 102
Hajela, D., 289
Halmos, P., 278
Hammer, D., 231
Hardy’s inequality, 166
and Hilbert, 176
discrete, 169
geometric refinement, 177
in L
p
, 176
special instance, 177
Hardy, G.H., 197, 290
Harker–Kasper inequality, 14
harmonic mean, 126
minimax characterization, 132
harmonic series
divergence, 99, 255
Heisenberg principle, 288
Page 3
304
Index
Hewitt, E., 243
Hilbert inequality, 155
homogeneous kernel, 164
integral version, 163
max version, 163
via Toeplitz method, 165
Hilbert’s 17th problem, 46
Hilbert, David, 46, 55
HM-GM and HM-AM inequalities, 126
Holder inequality, 135, 136
case of equality, 137
converse, 139
defect estimate, 94
historical form, 151
stability, 144, 145
Holder, Otto Ludwig, 94, 135, 263, 290
homogeneity in Σ, 132
homogenization trick, 189
How to Solve It, 30
humble bound, 284
inclusion radius, 148
inner product space
Cauchy–Schwarz inequality, 8
definition, 7
integral representations, 116
interpolation
in Cauchy inequality, 48
intuition
how much, 55
refining, 53
investment inequalities, 100
isometry, 60
isoperimetric property, 19
for the cube, 34
Israel, R., 253
iteration
and discovery, 149
J-convexity, 101
Janous, Walther, 177
Jensen inequality, 87, 263
and Schur convexity, 201
case of equality, 89
for integrals, 113
geometric applications, 93
Holder’s defect estimate, 94
implies Minkowski, 150
via Cauchy’s induction, 101
Jensen, J.L.W.V., 101
Joag–Dev, K., 277
Kahane, J.-P., vi
Karayannakis, D., 286
Kedlaya, K., vi, 152, 264, 274
Knuth, D., 260
Komlos, J., 277
Korner, T., vi
Kronecker’s lemma, 177
Kubo, F., 287
Kufner, A., 290
Kuniyeda, M., 262
Lagrange identity, 39
continuous analogue, 48
simplest case, 42
Lagrange, Joseph Louis de, 40
Landau’s notation, 120
leap-forward fall-back induction, 20, 31,
36, 101
Lee, H., vi, 247, 251
light cone defined, 62
light cone inequality, 63, 245
Littlewood, J.E., 197
looking back, 25, 26
Loomis–Whitney inequality, 16
Lorentz product, 62
Lovasz, L., 278
Lozansky, E., 264
Lyusternik, L.A., 245
Maclaurin inequality, 285
Magiropoulos, M., 286
majorant principles, 284
majorization, 191
Maligranda, L., vi, 264, 288, 289
marriage lemma, 206
Marshall, A., 277
Matousek, J., vi, 285
McConnell, T.R., 277
Meng, X., vi
Mengoli, P., 99, 248
method
of halves, 122
of parameterized parameters, 164
metric space, 54
Mignotte, M., 262
Milne inequality, 50
minimal surfaces, 10
Minkowski inequality, 141
Riesz proof, 141
via Jensen, 150
Minkowski’s
conjecture, 44, 46
light cone, 62
Minkowski, Hermann, 44
Mitrinovic, D.S., 277, 283
Mobius transformation, 242
moment sequences, 149
monotonicity
and integral estimates, 118
Montgomery, H.L., 284
Motzkin, T.S., 46, 286
Muirhead condition
Page 4
Index
305
and majorization, 195
Nakhash, A., 253
names of inequalities, 11
Naor, E., 259
Needham, T., 242
Nesbitt inequality, 84, 131, 246
Neyman–Pearson lemma, 287
Niculescu, C.P., 290
Nievergelt, Y., 253
Niven, I., 260
nonnegative polynomials, 43
norm
p-norm, or
p
-norm, 140
defined, 55
normalization method, 5, 25, 26, 66
normed linear space, 55
obvious
and not, 56
Hilbert story, 56
triangle inequality, 56
Olkin, I., 277
one-trick, see 1-trick
Opic, B., 290
optimality principles, 33
order inequality, 76
order relationship
systematic exploitation, 73
to quadratic, 76
order-to-quadratic conversion, 78, 287
orthogonality, definition, 58
orthonormal, 217
orthonormal sequence, 70
other inequality of Chebyshev, 77
parameterized parameters, 164
Persson, L.E., 288, 289
pillars, three great, 87
Pitman, J., 190, 276
Plummer, M.D., 278
polarization identity, 49, 70
Polya’s
dream, 23
questions, 30
Polya, George, 30, 197, 286
Polya–Szego converse, 83
positive definite, 228
power mean continuity relations, 127
power mean curve, 124
power mean inequality, 123
simplest, 36
Prekopa–Leindler inequality, 287
principle of maximal effectiveness, 27
principle of similar sides, 139
probability model, 17
product of linear forms, 59
projection formula, 56
guessing, 58
proportionality, 50
gages of, 39
Proschan, F., 277
Ptak, V., 247
Ptolemy inequality, 69
Pythagorean theorem, 47, 51
Qian, Z., vi
quadratic form, 228
quadratic inequality, 76
qualitative inference principle, 3, 27
quasilinear representation
geometric mean, 259
Rademacher–Menchoff inequality, 217,
223
ratio monotone, 189
reflection, 60
Reznick, B., vi
Richberg, R., 272
Riesz proof of Minkowski inequality, 141
Riesz, F., 288
Riesz, M., 288
Rogers inequality, 152, 153
Rogers, L.C., 135, 152, 290
Rolle’s theorem, 102, 251
Rosset, S., 290
Rousseau, C., 264
rule of first finder, 12
Schur convexity, 191
defined, 192
Schur criterion, 193
Schur inequality, 83
Schur’s lemma, 15
Schur, Issai, 192
Schwarz inequality, 10, 11
centered, 115
pointwise proof, 115
Schwarz’s argument, 11, 63
failure, 136
in inner product space, 15
in light cone, 63
Schwarz, Hermann Amandus, 10
Selberg inequality, 225
self-generalization, 21
Holder’s inequality, 151
Cauchy inequality, 16
Cauchy–Schwarz inequality, 66
Seymour, P.D., 289
Shaman, P., vi
Sharpley, R., 288, 289
Shen, A., 231
Shepp, L., vi
Shparlinski, I.E., vi, 281
Page 5
306
Index
Siegel’s method of halves, 122
Siegel, Carl Ludwig, 122
Sigillito, V.G., 244
Simonovits, M., 277
Skillen, S., vi
slip-in trick, 117
sphere arrangement, 52
splitting trick, 106, 123, 147, 154, 227,
263, 267
defined, 226
first used, 12
grand champion, 266
stability
in Holder inequality, 145
of AM-GM inequality, 35
steepest ascent, 67
Stef˘anescu, D., 262
stress testing, 268
Stromberg, K., 243
sum of squares, 42–44, 46
superadditivity
geometric mean, 34, 100, 133
symmetry and reflection, 62
Szego, Gabor, 234
Szemeredi Regularity Lemma, 205
Tang, H., vi
telescoping, 29
thinking outside the box, 52
Three Chord Lemma, 104
tilde transformation, 193
Tiskin, A., 231
Toeplitz method, 165
Treibergs, A., 242
triangle inequality, 54
unbiased estimator, 17
van Dam, E.R., 230
van der Corput inequality, 214
van der Corput, J.G., 214
variance, 18, 116
Vaughn, H.E., 278
Vi`ete identity, 133
Vince, A., 81
Vinogradov, I.M., 281
Vitale, R., vi
von Neumann, John, 51, 286
Wagoner, S.S., 238
Walker, A.W., 192
Ward, N., vi
Watkins, W., 277
Weierstrass inequality, 190
Weitzenbock inequality, 93, 102
Weyl, H., 206, 288
Wiles, Andrew, 40
Wilf, H., vi, 235
Young inequality, 136
Zuckerman, H.S., 260
Zukav, G., 286

Mathematical Inequalities ... Getting a Good List

1-trick, 110, 144, 146, 215, 219, 227, 231

I have been struggling to get a good version of the index of the Cauchy-Schwarz Master Class on the web. What I really want is a great list of all the named inequalites and a snippet about what the CSMC says about them. This would be a ton of work, so I looked for an easy way out. Originally I just posted the PDF file from the text.

This was not very satisfactory. The problem was that I could not find any good way to add introductory material or to provide navigation, such as links back to my home page or to the back to the main Cauchy-Schwarz Master Class page. It's better now but there is still room to go.

defined, 226

It's a List of Mathematical Inequalities --- but not yet beautiful

refinement, 205, 288
first used, 12

My next idea was to use the Latex to HTML translation tools. This should have worked but didn't. Maybe you can help me. At last, I resorted to use the kind of trick that I (at least partially) understand --- Google!

When Google sees a PDF file, it converts it on the fly to an HTML file so that indexing and other actions can be done in a systematic way. Still, Google saves the converted file, and you can use it. You have to strip out the stuff at the top, and then you have to deal with the fact that what is left specifies absolute position. You can change the position co-ordinates with a Perl program, or (if you are lazy like me) you can fill the gap at with introductory material --- as I have done. NOW, we have the index. Cheers!