Holdings: Functional equations and inequalities with applications

Functional equations and inequalities with applications:

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Bibliographic Details
Main Author:	Kannappan, Pl. 1936- (Author)
Format:	Book
Language:	English
Published:	Berlin [u.a.] Springer 2009
Edition:	1. Aufl.
Series:	Springer Monographs in Mathematics
Subjects:	Inégalités (mathématiques) Équations fonctionnelles Functional equations Inequalities (Mathematics) Funktionalgleichung Funktionalungleichung
Online Access:	Inhaltsverzeichnis
Physical Description:	XXIII, 810 S. Ill., graph. darst. 235 mm x 155 mm
ISBN:	9780387894911

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245	1	0	\|a Functional equations and inequalities with applications \|c Pl. Kannappan
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264		1	\|a Berlin [u.a.] \|b Springer \|c 2009
300			\|a XXIII, 810 S. \|b Ill., graph. darst. \|c 235 mm x 155 mm
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adam_text	Contents Preface ......................................................... vii 1 Basic Equations: Cauchy and Pexider Equations .................. 1 1.1 Additive Equations ......................................... 2 1.1.1 Discontinuous Solutions ............................. 6 1.2 Algebraic Conditions — Derivation ............................ 7 1.2.1 More Algebraic Conditions .......................... 11 1.3 Additive Equation on C, K , and K+ .......................... 15 1.3.1 Additive Equation on Complex Numbers ............... 15 1.3.2 More Algebraic Conditions .......................... 17 1.4 Alternative Equations, Restricted Domains, and Conditional Cauchy Equations .......................................... 20 1.4.1 Alternative Equations ............................... 21 1.4.2 Restricted Domains and Conditional Cauchy Equations ... 23 1.4.3 Mikusinski s Functional Equation ..................... 24 1.5 The Other Cauchy Equations ................................. 26 1.5.1 Exponential Equation ............................... 27 1.5.2 Logarithmic Equation ............................... 29 1.5.3 Characterization of Exponential and Logarithmic Functions .............................. 30 1.5.4 Multiplicative Equation .............................. 33 1.6 Some Generalizations of the Cauchy Equations ................. 34 1.6.1 Jensen s Equation ................................... 34 1.6.2 Pexider s Equations ................................. 39 1.6.3 Some Generalizations ............................... 43 1.6.4 Some Special Cases ................................. 45 1.6.5 More Generalizations ............................... 52 1.7 Extensions ................................................ 53 1.7.1 Extension of the Additive Function on [0,1]............ 56 1.7.2 Quasiextensions .................................... 59 1.7.3 Extension of the Pexider Equation ..................... 60 xvi Contents 1.7.4 Extension of the Logarithmic Equation ................. 64 1.7.5 Exponential Extension ............................... 65 1.7.6 Multiplicative Extension ............................. 66 1.7.7 Extension of Derivations ............................. 67 1.7.8 Almost Everywhere Extension ........................ 67 1.8 Applications ............................................... 68 1.8.1 Economics ........................................ 68 1.8.2 Area .............................................. 72 1.8.3 Allocation Problem: Characterization of Weighted Arithmetic Means .................................. 74 1.8.4 Sum of Powers of First η Natural Numbers and Sum of Powers on Arithmetic Progressions .................... 76 1.8.5 More Sums Using the Additive Cauchy Equation ........ 81 1.8.6 Application in Combinatorics and Genetics ............. 82 2 Matrix Equations ............................................ 85 2.1 Multiplicative Equation ..................................... 91 2.2 Cosine Matrix Equation ..................................... 98 3 Trigonometric Functional Equations ............................ 105 3.1 Mixed Trigonometric Equations .............................. 107 3.2 Cosine Equation on Number Systems .......................... 113 3.3 (C) on Groups and Vector Spaces ............................. 120 3.4 Solution of (CE) on a Non-Abelian Group ...................... 124 3.4.1 Discussion ........................................ 131 3.4.2 (C) on Abstract Spaces .............................. 133 3.4.3 Jacobi s Elliptic Function Solution .................... 139 3.4.4 More Characterizations in a Single Variable ............. 145 3.4.5 More on Sine Functions on a Vector Space .............. 158 3.4.6 Sine Solution ...................................... 170 3.4.7 Characterization of the Sine .......................... 172 3.4.8 General Trigonometric Functional Equations — The Addition and Subtraction Formulas ................ 174 3.4.9 Vibration of String Equation (VS) ..................... 181 3.4.10 Wilson s Equations ................................. 186 3.4.11 Analytic Solutions .................................. 187 3.4.12 Equation (3.72) on Analytic Functions ................. 192 3.4.13 Some Generalizations ............................... 197 3.4.14 Levi-Civitá Functional Equation, Convolution Type Functional Equations, and Generalization of Cauchy-Pexider Type and d Alembert Equations ......... 202 3.4.15 Operator-Valued Solution of (C) ...................... 203 3.4.16 Solution of Equation (3.111)......................... 204 3.4.17 Inner Product Version of (3.109)...................... 205 3.4.18 A Functional Equation of d Alembert s Type ............ 207 Contents xvii 3.5 Survey — Summary of Stetkaer ............................... 209 3.5.1 Abstract ........................................... 210 3.5.2 The Abelian Solution and Related Extensions of It ....... 210 3.5.3 d Alembert s Functional Equation ..................... 215 3.5.4 Wilson s Functional Equation ......................... 216 3.5.5 A Variant of Wilson s Functional Equation .............. 217 3.5.6 Other Equations .................................... 218 Quadratic Functional Equations ................................221 4.1 General Solution and Properties .............................. 221 4.2 General Solution on a Complex Linear Space — Sesquilinear Solution ....................................... 225 4.3 Regular Solutions .......................................... 228 4.4 Generalizations and Equivalent Forms of (Q) ................... 230 4.5 Equivalence to (Q) ......................................... 232 4.6 Generalizations ............................................ 234 4.7 More Generalizations ....................................... 238 4.8 Another Form of Quadratic Function .......................... 240 4.9 Entire Functions and Quadratic Equations ...................... 241 4.10 Summary by Stetkaer [750].................................. 244 4.10.1 The Quadratic Functional Equation .................... 244 Characterization of Inner Product Spaces ........................247 5.1 Frécheťs Equation ......................................... 249 5.1.1 The Parallelepiped Law .............................. 251 5.1.2 Parallelogram Identity ............................... 253 5.1.3 More on Frécheťs Result ............................ 254 5.1.4 More Characterizations .............................. 256 5.1.5 Some Generalizations ............................... 263 5.1.5.1 Solution of a Generalization of the Fréchet Equation .......................... 263 5. ł .6 Some More Characterizations ........................ 266 5.2 Geometric Characterization .................................. 275 5.2.1 Some Generalizations ............................... 277 5.2.2 Solution of an Equation Related to Ptolemaic Inequality ................................ 283 5.2.3 Orthogonal Additivity and I.P.S ....................... 285 5.2.4 Diminnie Orthogonality ............................. 285 Stability ....................................................295 6.1 Stability of the Additive Equation ............................. 296 6.2 Stability — Multiplicative Equations ........................... 302 6.3 Stability — Logarithmic Function .............................. 303 6.4 Stability — Trigonometric Functions ........................... 305 6.4. ł Stability for Vector-Valued Functions .................. 315 xviii Contents 6.5 Stability of the Equation ƒ (x + y) + g(x - y) = h(x)k(y) ....... 318 6.6 Stability of the Sine Functional Equation ....................... 319 6.7 Stability — Alternative Cauchy Equation ....................... 319 6.8 Stability—Wave Equation ................................... 321 6.9 Stability — Polynomial Equation .............................. 322 6.10 Stability — Quadratic Equation ................................ 323 7 Characterization of Polynomials ................................329 7.1 Polynomials ............................................... 329 7.2 More Characterizations of Polynomials of Degree Two ........... 332 7.3 Generalization ............................................. 336 7.3.1 First Generalization Using Derivatives ................. 336 7.3.2 Second Generalization Without Using Any Regularity Condition ................................ 337 7.4 Another Generalization — Divided Difference ................... 337 7.5 Generalization of Divided Difference .......................... 340 7.6 Problem of W. Rudin and a Generalization ..................... 340 7.7 Generalization of Rudin s Problem ............................ 342 7.8 Frécheťs Result ............................................ 343 7.9 Polynomials in Several Variables ............................. 345 7.10 Quadratic Polynomials in Two Variables ....................... 346 7.11 Functional Equations on Groups .............................. 347 7.12 Rudin s Problem on Groups .................................. 349 7.13 Generalization ............................................. 352 8 Nondifferentiable Functions ...................................359 8.1 Weierstrass Functions ....................................... 359 8.2 Wiinderlich s Function ...................................... 361 8.3 Takagi Functions ........................................... 361 8.4 van der Waerden Type Function .............................. 362 8.5 Functional Equation Characterization of to ..................... 364 8.6 Nondifferentiability of ω .................................... 365 8.7 Riemann s Function ........................................ 367 8.8 Knopp Functions ........................................... 368 8.9 Generalization ............................................. 369 9 Characterization of Groups, Loops, and Closure Conditions ........371 9.1 Notation and Definitions .................................... 371 9.2 Closure Conditions ......................................... 372 9.3 Groups ................................................... 373 9.4 Abelian Groups ............................................ 374 9.5 Functional Equation of Identities ............................. 381 9.6 Functional Equations Arising Out of Bol, Moufang, and Extra Equations ............................................ 384 9.6.1 Bol Equation ....................................... 384 Contents xix 9.6.2 Moufang and Extra Loops ............................ 390 9.6.3 Extra Loop ........................................ 391 9.6.4 Characterizations................................... 391 9.6.5 Characterization of Moufang Loops ................... 391 9.6.6 Extra Loops ....................................... 391 9.6.7 Groups ............................................ 392 9.7 GD-groupoid .............................................. 392 9.8 More Identities ............................................ 392 9.8.1 Entropie, Bisymmetric, or Mediality Identity ............ 392 9.9 Left Inverse Property (l.i.p.), Crossed-Inverse (ci.), and Weak Inverse Property (w.i.p.) Loops .......................... 394 9.9.1 l.i.p. Loops ........................................ 394 9.9.2 ci. Loops ......................................... 395 9.9.3 w.i.p. Loops ....................................... 395 9.10 Steiner Loops .............................................. 395 9.11 Boi Loop and Power Associativity ............................ 398 9.12 More Functional Equations .................................. 398 9.13 Generalized Groupoids ...................................... 400 9.13.1 Generalized Associativity ............................ 400 9.13.2 Generalized Bisymmetry ............................. 400 10 Functional Equations from Information Theory ...................403 10.1 Introduction ............................................... 403 10.2 Notation, Basic Notions, and Preliminaries ..................... 405 10.2.1 Properties, Postulates, and Axioms .................... 406 10.2.2 Desirable Properties—Postulates ...................... 406 10.2.3 Characterization of Information Measures .............. 409 10.2.4 Shannon Entropy and Some of Its Generalizations ....... 410 10.2a Fundamental Equation of Information — Axiomatic Characterizations .........................410 1 0.2b The Fundamental Equation of Information Theory .......412 10.2c Some Generalizations of (FEI)........................ 418 10.2.5 General Solution of Equation (10.21).................. 420 1 0.2d Sum Form Functional Equation (SFE) and Its Generalizations .................................. 424 10.2d.l Representation and Characterization ..........424 10 .2e Other Measures of Information — Entropy of ТуреД НІ....................................... 432 10.2f Directed Divergence (dd) and Inaccuracy (KI) ........... 433 10.2f. 1 Generalized Directed Divergence ............. 437 10.2g Sum Form Distance Measures ........................ 441 10.2h Kullback-LeiblerType Distance Measures .............. 444 10.2i Symmetric Distance Measures ........................ 445 10.2J Some Functional Equations .......................... 447 10.2k Weighted Entropies .................................448 xx Contents 10.3 Mixed Theory of Information — Inset Measures ................. 452 10.3.1 Characterizations ................................... 454 10.3.1.1 Characterization of Inset Deviation Entropies ........................ 456 10.4 Applications ............................................... 462 10.4.1 Continuous Shannon Measure and Shannon-Wiener Inset Information ................................... 462 10.4.2 Theory of Gambles ................................. 463 10.4.3 Recursive Inset Entropies of Multiplicative Type ......... 465 11 Abel Equations and Generalizations ............................469 11.1 Solutions of Abel Equations ................................. 470 11.1.1 (AFEl)—Exponential Equation f(z + ω) = ƒ (z) f (w) .. 470 11.1.2 (AFE2)— Iteration Equation f (φ (χ)) = f (x) + 1 ....... 472 11.1.3 (AFE3) — Associative, Commutative Equations .......... 472 11.1.4 (AFE4)— Arctan Equation ........................... 473 11.1.5 (AFE5)— Trig Equation ф(х+у) = ф(х) ƒ (у)+ф(у) f (x) 473 11.1.6 (AFE6)— ψ (x + y) = g(xy) + h(x-y) ...............473 11.1.7 (АЊЂ—фОс) + ф(у) = yf(xf(y) + y f (x)) ...........479 11.1.8 System of Equations (AFE8) and (AFE8a) ..............480 11.1.9 (AFE9) and (AFE9a) ................................ 482 11.2 Generalizations and Information Measures ..................... 485 12 Regularity Conditions — Christensen Measurability ................493 12.1 Some General Results ....................................... 497 12.2 Applications ............................................... 500 12.3 Christensen Measurability ................................... 503 12.4 Functional Equations (Characterizing) from Trigonometric Functions .................................... 505 13 Difference Equations .........................................511 13.1 Cauchy Difference ......................................... 511 13.1.1 Differences that Depend on the Product ................ 516 13.1.2 Pompeiu Functional Equation and Its Generalizations .................................. 523 13.1.3 Solution of the Functional Equation (13.24a) ............ 525 13.1.4 Solution of the Functional Equation (13.24b) ............ 526 13.2 Quadratic Differences ....................................... 526 13.2.1 Differences in a Prescribed Set ........................ 528 13.3 Pexider Difference ......................................... 530 13.3.1 Some Generalizations ............................... 532 13.3.2 Measurable Solutions of the Functional Equation (13.48a) .................................. 533 Contents xxi 14 Characterization of Special Functions........................... 537 14.1 Gamma Function ........................................... 537 14.1.1 Further Properties of the Gamma Function .............. 540 14.1.2 Definitions of the Function Г(х) ...................... 541 14.2 Beta Function ............................................. 549 14.2.1 Integral Representation οι β.......................... 550 14.2.2 Other Special formulas .............................. 553 14.3 Riemann s Zeta Function .................................... 555 14.3.1 The Theta Function ................................. 556 14.4 Singular Functions ......................................... 557 14.4.1 Cantor-Lebesgue Singular Function ................... 558 14.4.2 Minkowski s Function ............................... 558 14.4.3 De Rham s Function ................................ 560 15 Miscellaneous Equations ......................................563 15.1 A General Method: Method of Determinants ................... 563 15.2 Means .................................................... 570 15.2.1 Characterizations — Arithmetic and Exponential Means ................................. 575 15.2.2 Geometric Mean and the Root Mean Power ............. 576 15.2.3 Stolarsky Mean .................................... 579 15.3 Some Comments about the Logarithmic Function ............... 581 15.4 D Alembert s Equation Revisited ............................. 588 15.4.1 Basic d Alembert Functions .......................... 589 15.4.2 D Alembert Groups: Examples ....................... 592 15.4.3 D Alembert Groups: Generalities ..................... 594 15.4.4 Solvable Finite d Alembert Groups .................... 594 15.4.5 Nonsolvable Finite d Alembert Groups ................. 595 15.5 Polynomials Revisited ...................................... 597 15.6 Inner Products Revisited ..................................... 600 16 General Inequalities ..........................................607 16.1 Cauchy Functional Inequalities ............................... 607 16.2 Subadditive and Superadditive Functions ....................... 610 16.3 Logarithmic Inequality ...................................... 612 16.4 Multiplicative Inequality .................................... 614 16.5 Convexity ................................................. 614 16.6 Trigonometric Functional Inequality ........................... 617 16.7 Cosine and Sine Functional Inequalities ........................ 624 16.8 Functional Equation Concerning the Parallelogram Identity — Quadratic Inequality ............................... 626 16.9 Inequalities for the Gamma and Beta Functions via some Classical Results ........................................... 627 16.9.1 Inequalities via Chebychev s Inequality ................ 627 16.9.2 Inequalities via Holder s Inequality .................... 632 xxii Contents 16.10 Simpson s Inequality and Applications ......................... 634 16.11 Applications for Special Means ............................... 636 16.12 Inequalities from Information Theory .......................... 638 16.12.1 Generalization ..................................... 641 16.12.2 Application to Mixed Theory of Information ............ 644 16.12.3 Continuous Solution of the Inequality (16.61)........... 648 16.13 Reproducing Scoring Systems ................................ 651 16.13.1 Solution of the Functional Inequality (16.66)............ 652 16.13.2 Symmetric Reproducing Scoring Systems .............. 654 16.13.3 A Generalization of RSS ............................. 655 16.14 More Inequalities from Information Theory .................... 656 16.15 Inequalities from Inner Product Spaces ........................ 658 16.16 Miscellaneous Inequalities ................................... 661 16.16.1 More on Convex Functions ........................... 661 16.16.2 Inequalities for Integrals ............................. 663 16.16.3 Cauchy-Schwarz-Hölder Inequalities .................. 665 17 Applications .................................................669 17.1 Binomial Expansion ........................................ 669 17.2 Scalar or Dot Product ....................................... 671 17.3 Economics ................................................ 673 17.3.1 Duopoly Model .................................... 673 17.3.1.1 Duopoly Modell.......................... 673 17.3.1.2 Duopoly Modelli ......................... 675 17.3.2 Cobb-Douglas (CD) Production Function and Quasilinearity .................................. 677 17.3.2.1 Quasilinearity ............................. 677 17.3.2.2 Determination of all Quasilinear, Linearly Homogeneous Functions .................... 678 17.4 Business Mathematics — Interest .............................. 680 17.4.1 Interest Formula .................................... 680 17.4.2 Simple Interest ..................................... 681 17.4.3 Interest Rates ...................................... 681 17.5 Physics ................................................... 682 17.5.1 Quantum Physics ................................... 682 17.5.2 Gaussian Function .................................. 684 17.5.3 Chebyshev Polynomials ............................. 688 17.5.3.1 Introduction .............................. 688 17.5.3.2 Reduction to a Difference Equation ........... 689 17.5.3.3 The Universal Solution ..................... 691 17.5.3.4 Identification of the Universal Solution ........ 692 17.5.3.5 General Remarks .......................... 693 17.6 Topology ................................................. 694 17.6.1 Integral ........................................... 697 17.6.1.1 Simpson s Rule ............................ 697 Contents xxiii 17.6.1.2 Solution of the Functional Equation (17.46)........................... 699 17.6.1.3 Solution of the Functional Equation (17.47)........................... 699 17.6.1.4 Solution of the Main Functional Equation (17.44)........................... 703 17.6.2 Determinants ...................................... 708 17.7 Digital Filtering ............................................ 710 17.8 Geometry ................................................. 711 17.9 Field Homomorphisms ...................................... 713 17.10 Pythagorean Functional Equation ............................. 715 17.11 Statistics .................................................. 717 17.11.1 Poisson Distribution ................................ 718 17.11.1.1 Characterization of (Bivariate) Poisson Distributions ....................... 718 17.11.2 Normal Distribution ................................. 719 17.11.2.1 The Equation ƒ (x)g(y) = h (ax +by)k(cx+dy) 725 17.11.2.2 The Equation f(x)g(y) = f[ Ысцх + b¡y) ... 728 ¡=і 17.11.2.3 Solution of the Functional Equation (17.72)___ 730 17.11.3 Gamma Distribution ................................ 733 17.12 Information Theory ......................................... 734 17.12.1 Bose-Einstein Entropy ............................... 734 17.12.1.1 Solution of Equations (17.81) and (17.81a) .... 735 17.12.1.2 Solution of Equation (17.81b) ............... 737 17.12.2 Sums of Powers .................................... 739 17.12.2.1 A General Result .......................... 739 17.13 Behavioural Sciences ....................................... 745 17.13.1 A Behavioural Example ............................. 746 17.13.2 Psychophysics ..................................... 748 17.13.2.1 The Conjoint Weber Law ................... 749 17.13.3 Binocular Vision ................................... 750 17.13.3.1 The Lüneburg Theory of Binocular Vision .....751 17.13.3.2 A Conjoint Representation Generalizing the Lüneburg Theory .......................... 752 17.13.4 Functional Equations Resulting from Psychophysical Invariances .......................... 753 List of Symbols ..................................................757 Bibliography ....................................................759 Author Index ....................................................795 Subject Index ....................................................803
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author	Kannappan, Pl. 1936-
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author_facet	Kannappan, Pl. 1936-
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id	DE-604.BV035674614
illustrated	Illustrated
indexdate	2024-07-09T21:43:02Z
institution	BVB
isbn	9780387894911
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-017728838
oclc_num	495198621
open_access_boolean
owner	DE-355 DE-BY-UBR DE-11 DE-898 DE-BY-UBR DE-188
owner_facet	DE-355 DE-BY-UBR DE-11 DE-898 DE-BY-UBR DE-188
physical	XXIII, 810 S. Ill., graph. darst. 235 mm x 155 mm
publishDate	2009
publishDateSearch	2009
publishDateSort	2009
publisher	Springer
record_format	marc
series2	Springer Monographs in Mathematics
spelling	Kannappan, Pl. 1936- Verfasser (DE-588)1011157209 aut Functional equations and inequalities with applications Pl. Kannappan 1. Aufl. Berlin [u.a.] Springer 2009 XXIII, 810 S. Ill., graph. darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Springer Monographs in Mathematics Inégalités (mathématiques) ram Équations fonctionnelles ram Functional equations Inequalities (Mathematics) Funktionalgleichung (DE-588)4018923-5 gnd rswk-swf Funktionalungleichung (DE-588)4155681-1 gnd rswk-swf Funktionalgleichung (DE-588)4018923-5 s Funktionalungleichung (DE-588)4155681-1 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017728838&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Kannappan, Pl. 1936- Functional equations and inequalities with applications Inégalités (mathématiques) ram Équations fonctionnelles ram Functional equations Inequalities (Mathematics) Funktionalgleichung (DE-588)4018923-5 gnd Funktionalungleichung (DE-588)4155681-1 gnd
subject_GND	(DE-588)4018923-5 (DE-588)4155681-1
title	Functional equations and inequalities with applications
title_auth	Functional equations and inequalities with applications
title_exact_search	Functional equations and inequalities with applications
title_full	Functional equations and inequalities with applications Pl. Kannappan
title_fullStr	Functional equations and inequalities with applications Pl. Kannappan
title_full_unstemmed	Functional equations and inequalities with applications Pl. Kannappan
title_short	Functional equations and inequalities with applications
title_sort	functional equations and inequalities with applications
topic	Inégalités (mathématiques) ram Équations fonctionnelles ram Functional equations Inequalities (Mathematics) Funktionalgleichung (DE-588)4018923-5 gnd Funktionalungleichung (DE-588)4155681-1 gnd
topic_facet	Inégalités (mathématiques) Équations fonctionnelles Functional equations Inequalities (Mathematics) Funktionalgleichung Funktionalungleichung
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017728838&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT kannappanpl functionalequationsandinequalitieswithapplications

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