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