Hermite-Hadamard inequality in the geometry of Banach spaces

Background......Page 21
Historical consideration......Page 23
Characterization of convexity......Page 24
Connection to special means......Page 26
Some generalizations of the Hermite-Hadamard inequality......Page 30
Motivation......Page 34
Outline of the thesis......Page 36
Banach spaces......Page 41
Dual space and reflexivity......Page 45
Geometrical properties of Banach spaces......Page 46
Banach sequence spaces......Page 49
Bochner function spaces......Page 52
Hermite-Hadamard inequality in normed spaces......Page 55
The p-HH-norm......Page 56
Completeness and reflexivity......Page 63
Convexity and smoothness......Page 65
Ostrowski type inequality involving the p-HH-norms......Page 71
Ostrowski inequality......Page 72
Ostrowski inequality for absolutely continuous functions on linear spaces......Page 75
Application for semi-inner products......Page 80
Inequalities involving the p-HH-norm and the p-norm......Page 85
Ostrowski inequality for convex functions on linear spaces......Page 89
Application for semi-inner products......Page 90
Inequalities involving the p-HH-norm and the p-norm......Page 92
Comparison analysis......Page 97
Grüss type inequality involving the p-HH-norms......Page 101
Grüss inequality and Cebyšev functional......Page 102
Inequalities involving the p-HH-norms......Page 104
New bounds for the generalized Cebyšev functional D......Page 109
Application to the Cebyšev functional......Page 112
Cebyšev functional for convex functions......Page 113
Application to the p-HH-norms......Page 115
Notions of orthogonality in normed spaces......Page 119
Birkhoff\'s orthogonality......Page 121
Carlsson type orthogonality......Page 123
Relations between main orthogonalities......Page 127
Hermite-Hadamard type orthogonality......Page 128
Existence and main properties......Page 132
Uniqueness......Page 135
Characterizations of inner product spaces......Page 142
Alternative proofs for special cases......Page 153
Existence......Page 154
Characterizations of inner product spaces......Page 160
The p-HH-norms on Cartesian powers and sequence spaces......Page 165
Extending the p-HH-norms......Page 167
Equivalency of the p-norms and the p-HH-norms......Page 169
The 2-HH-norm......Page 177
Convexity and smoothness......Page 179
Strict convexity and uniform convexity......Page 180
Smoothness......Page 181
The hp spaces......Page 183
Summary......Page 193
Main achievements......Page 196
Future work......Page 197
Lebesgue space......Page 201
Linear operator......Page 203
References......Page 1