MEAN VALUE
THEOREMS
AND
FUNCTIONAL
EQUATIONS
THEOREMS
AND
FUNCTIONAL
EQUATIONS
P. K. Sahoo
T. Riedel
T. Riedel
University of Louisville
Louisville, KY, USA
Louisville, KY, USA
World Scientific
Singapore »New Jersey • London • Hong Kong
Contents
Preface vii
Chapter 1 Additive and Biadditive Functions 1
1.1 Continuous Additive Functions 1
1.2 Discontinuous Additive Functions 6
1.3 Other Criteria for Linearity 11
1.4 Additive Functions on the Real and Complex Plane 12
1.5 Biadditive Functions 18
1.6 Some Open Problems 22
1.2 Discontinuous Additive Functions 6
1.3 Other Criteria for Linearity 11
1.4 Additive Functions on the Real and Complex Plane 12
1.5 Biadditive Functions 18
1.6 Some Open Problems 22
Chapter 2 Lagrange's Mean Value Theorem and Related
Functional Equations 25
Functional Equations 25
2.1 Lagrange's Mean Value Theorem 25
2.2 Applications of the MVT 30
2.3 Associated Functional Equations 40
2.4 the MVT for Divided Differences 58
2.5 Limiting Behavior of Mean Values 64
2.6 Cauchy's MVT and Functional Equations 77
2.7 Some Open Problems 80
2.2 Applications of the MVT 30
2.3 Associated Functional Equations 40
2.4 the MVT for Divided Differences 58
2.5 Limiting Behavior of Mean Values 64
2.6 Cauchy's MVT and Functional Equations 77
2.7 Some Open Problems 80
Chapter 3 Pompeiu's Mean Value Theorem and Associated
Functional Equations 83
Functional Equations 83
3.1 Pompeiu's Mean Value Theorem 83
3.2 Stamate Type Equations 85
3.3 An Equation of Kuczma 92
3.2 Stamate Type Equations 85
3.3 An Equation of Kuczma 92
XI
xii
Contents
3.4 Equations Motivated by Simpson's Rule 98
3.5 Some Generalizations 108
3.6 Some Open Problems 125
3.5 Some Generalizations 108
3.6 Some Open Problems 125
Chapter 4 Two-dimensional Mean Value Theorems and
Functional Equations 127
Functional Equations 127
4.1 MVTs for Functions in Two Variables 127
4.2 Mean Value Type Functional Equations 129
4.3 Generalized Mean Value Type Equations 135
4.4 Cauchy's MVT for Functions in Two Variables 144
4.5 Some Open Problems 144
4.2 Mean Value Type Functional Equations 129
4.3 Generalized Mean Value Type Equations 135
4.4 Cauchy's MVT for Functions in Two Variables 144
4.5 Some Open Problems 144
Chapter 5 Some Generalizations of Lagrange's Mean Value
Theorem 147
Theorem 147
5.1 MVTs for Real Functions 147
5.2 MVTs for Real Valued Functions on the Plane 156
5.3 MVTs for Vector Valued Functions on the Reals 160
5.4 MVTs for Vector Valued Functions on the Plane 163
5.5 MVTs for Functions on the Complex Plane 168
5.6 A Conjecture of Furi and Martelli 180
5.2 MVTs for Real Valued Functions on the Plane 156
5.3 MVTs for Vector Valued Functions on the Reals 160
5.4 MVTs for Vector Valued Functions on the Plane 163
5.5 MVTs for Functions on the Complex Plane 168
5.6 A Conjecture of Furi and Martelli 180
Chapter 6 Mean Value Theorems for Some Generalized
Derivatives 181
Derivatives 181
6.1 Symmetric Differentiation of Real Functions 181
6.2 A Quasi-Mean Value Theorem 188
6.3 An Application 192
6.4 Generalizations of MVTs . . . \ 193
6.5 _ Dini Derivatives of Real Functions 195
6.6 MVTs for Nondifferentiable Functions 201
6.2 A Quasi-Mean Value Theorem 188
6.3 An Application 192
6.4 Generalizations of MVTs . . . \ 193
6.5 _ Dini Derivatives of Real Functions 195
6.6 MVTs for Nondifferentiable Functions 201
Chapter 7 Some Integral Mean Value Theorems and Related
Topics 207
Topics 207
7.1 The Integral MVT and Generalizations 207
7.2 Integral Representation of Means 218
7.3 Coincidence of Mean Values 224
7.4 Some Open Problems 227
Bibliography 233
Index 2427.2 Integral Representation of Means 218
7.3 Coincidence of Mean Values 224
7.4 Some Open Problems 227
Bibliography 233
