Lecture notes. Fundamental inequalities: techniques and applications
|
|
- James Lindsey
- 5 years ago
- Views:
Transcription
1 Lecture notes Fundmentl inequlities: techniques nd pplictions Mnh Hong Duong Mthemtics Institute, University of Wrwick Emil: Februry 8, 207
2 2 Abstrct Inequlities re ubiquitous in Mthemtics (nd in rel life). For exmple, in optimiztion theory (prticulrly in liner progrmming) inequlities re used to described constrints. In nlysis inequlities re frequently used to derive priori estimtes, to control the errors nd to obtin the order of convergence, just to nme few. Of prticulr importnce inequlities re Cuchy-Schwrz inequlity, Jensen s inequlity for convex functions nd Fenchel s inequlities for dulity. These inequlities re simple nd flexible to be pplicble in vrious settings such s in liner lgebr, convex nlysis nd probbility theory. The im of this mini-course is to introduce to undergrdute students these inequlities together with useful techniques nd some pplictions. In the first section, through vriety of selected problems, students will be fmilir with mny techniques frequently used. The second section discusses their pplictions in mtrix inequlities/nlysis, estimting integrls nd reltive entropy. No dvnced mthemtics is required. This course is tught for Wrwick s tem for Interntionl Mthemtics Competition for University Students, 24th IMC 207. The competition is plnned for students completing their first, second, third or fourth yer of university eduction.
3 Contents Fundmentl inequlities: Cuchy-Schwrz inequlity, Jensen s inequlity for convex functions nd Fenchel s dul inequlity 4. Cuchy-Schwrz inequlity Convex functions Jensen s inequlity Convex conjugte nd Fenchel s inequlity Some techniques to prove inequlities
4 Fundmentl inequlities: Cuchy-Schwrz inequlity, Jensen s inequlity for convex functions nd Fenchel s dul inequlity In this section, we review three bsic inequlities tht re Cuchy-Schwrz inequlity, Jensen s inequlity for convex functions nd Fenchel s inequlity for dulity. For simplicity of presenttion, we only consider simplest underlying spces such s R n or finite set. These inequlities, however, cn be stted in much more complex situtions. Mny techniques for proving inequlities re presented vi selected exmples nd exercises in Section Cuchy-Schwrz inequlity Let u, v be two vectors of n inner product spce (over R, for simplicity ). The Cuchy- Schwrz inequlity sttes tht u, v u v. Proof. If v = 0, the inequlity is obvious true. Let v 0. For ny λ R, we hve 0 u + λv 2 = u 2 + 2λ u, v + λ 2 v 2. Consider this s qudrtic function of λ. Therefore we hve u, v 2 u 2 v 2, which implies the Cuchy-Schwrz inequlity..2 Convex functions Definition. (Convex function). Let X R n be convex set. A function f : X R is cll convex if for ll x, x 2 X nd t [0, ] f(tx + ( t)x 2 ) tf(x ) + ( t)f(x 2 ). A function f : X R is cll strictly convex if for ll x, x 2 X nd t (0, ) f(tx + ( t)x 2 ) < tf(x ) + ( t)f(x 2 ). Exmple.. Exmples of convex functions: ffine functions: f(x) = x + b, for, b R n, exponentil functions: f(x) = e x for ny R,
5 5 ( n Eucliden-norm: f(x) = x = x 2 i ) 2. Verifying convexity of differentible function Note tht in Definition., the function f does not require to be differentible. The following criteri cn be used to verify the convexity of f when it is differentible. () If f : X R is differentible, then it is convex if nd only if f(x) f(y) + f(y) (x y) for ll x, y X. (2) If f : X R is twice differentible, then it is convex if nd only if its Hessin 2 f(x) is semi-positive definite for ll x X. Some importnt properties of convex functions Lemm.2. Below re some importnt properties of convex functions. () If f nd g re convex functions, then so re m(x) = mx{f(x), g(x)} nd s(x) = f(x) + g(x). (2) If f nd g re convex functions nd g is non-decresing, then h(x) = g(f(x)) is convex. Proof. These properties cn be directly proved by verifying the definition.. Jensen s inequlity Theorem. (Jensen s inequlity). Let f be convex function, 0 α i ; i =,..., n such tht α i =. Then for ll x,..., x n, we hve ( ) f α i x i α i f(x i ). () Proof. We prove by induction. The cses n =, 2 re obvious. Suppose tht the sttement is true for n =,..., K. Suppose tht α,..., α K re non-negtive numbers such tht K α i =. We need to prove tht ( K ) f α i x i K α i f(x i ).
6 6 Since K α i =, t lest one of the coefficients α i must be strictly positive. Assume tht α > 0. Then by the conducting ssumptions, we obtin ( K ) ( f α i x i = f α x + ( α ) where we hve used the fct tht K K i=2 α ) i x i α ( K α ) i α f(x ) + ( α )f x i α α f(x ) + ( α ) = K α i f(x i ), i=2 α i α =. K i=2 i=2 α i α f(x i ) Remrk.4 (Jensen s inequlity- probbilistic form). Jensen s inequlity cn lso be stted using probbilistic form. Let (Ω, A, µ) be probbility spce. If g is rel-vlued function tht is µ integrble nd if f is convex function on the rel line, then ( ) f g dµ f g dµ. Ω Ω Exmple.2 (Exmples of Jensen s inequlity). ) For ll rel numbers x,..., x n, it holds ( ) 2 x i n x 2 i. Proof. Since f(x) = x 2 is convex, we hve ( n ) 2 ( x i = f n ) x i n f(x i) = n x 2 i, which is the desired sttement. 2) Arithmetic-Geometric (AM-GM) Inequlity. Let (x i ) i n nd (λ i ) i n be rel number stisfying x i 0, λ i 0, λ i =.
7 7 Then, with the convention 0 0 =, λ i x i n x λ i i. (2) In prticulr, tking λ =... = λ n = yields n x i n /n x... x n. Proof. By tking the logrithmic both sides, (2) is equivlent to ( λ i ln(x i ) ln λ i x i ). This is exctly Jensen s inequlity pplying to the convex function f(x) = ln(x)..4 Convex conjugte nd Fenchel s inequlity The convex conjugte of function f : R d R is, f : R d R, defined by f (y) = sup x R d {x y f(y)}. Fenchel s inequlity: for x, y R d, we hve f(x) + f (y) x y. Exmple.. Exmples of Fenchel s inequlity ) f(x) = x 2, then f (y) = sup{x y x 2 } = y 2. Fenchel s inequlity reds 2 2 x R d 2 ( x 2 + y 2 ) x y. 2) f(x) = p x p where p >. Then f (y) = sup{x y p x p } = q y q where + =. p q x R d Fenchel s inequlity becomes: for x, y R d nd p, q > such tht + =, we hve p q p x p + q y q x y.
8 8.5 Some techniques to prove inequlities In prctises, three inequlities introduced in previous sections often do not pper in stndrd forms. It is crucil to recognize them. In this section, through exercises we will lern some techniques to prove inequlities. Exercise. Let i, b i R, b i > 0 for i =,..., n. Prove tht 2 i b i ( n ) 2 i n b i. Proof. By the Cuchy-Schrz inequlity we hve ( ) 2 ( i ) 2 ( 2 )( i = bi i b i ). bi b i Exercise 2 (Problem 6, IMC 205). Prove tht n= Proof. By the AM-GM Inequlity we hve which implies tht Dividing both sides by n(n + ) yields Hence by summing up over n we obtin n(n + ) < 2. 2(n + ) = n + (n + ) > 2 n(n + ), n= 2(n + ) 2 n(n + ) >. ( < 2 n ). n(n + ) n + n(n + ) < 2 n= ( ) n = 2. n + Exercise. Let A, B, C be three ngles of tringle. Prove tht sin A + sin B + sin C 2.
9 9 Proof. Consider the function f(x) = sin x. Since f (x) = sin 2 (x) 0, f is concve. Therefore, sin A + sin B + sin C ( A + B + C ) sin = sin π = 2. Exercise 4 (Problem, IMC 206). Let n be positive integer nd,..., n ; b,..., b n be rel number such tht i + b i > 0 for i =,..., n. Prove tht ( n ) 2 i b i b 2 i b i b i i i + b i. () ( i + b i ) Proof. We notice the similr form of both sides of (). For A, B R we hve Applying (4) for A = i, B = b i, we get LHS = i b i b 2 i i + b i = AB B 2 A + B = B 2B2 A + B ( ) b i 2b2 i = i + b i b i 2b 2 i i + b i. (4) Similrly now pplying (4) for A = b i, B = n b i, we obtin ( n ) 2 2 b i RHS = b i n. i + b i Therefore () is equivlent to ( n b i By Cuchy-Schwrz inequlity we hve ( ) 2 ( b i = which implies (5) s desired. ) 2 n i + b i b i ) 2 ( i + b i i + b i b 2 i i + b i (5) b 2 ) i i + b i Exercise 5 (Problem, IMC 200). Let 0 < < b. Prove tht b (x 2 + )e x2 dx e 2 e b2 ( i + b i ),
10 0 Proof. By the AM-GM Inequlity x 2 + 2x > 0 for ny 0 < x b, we hve b (x 2 + )e x2 dx b 2xe x2 dx = e x2 b = b (x 2 + )e x2 dx. Exercise 6 (Problem 6, IMC 200). Let n be n integer nd let f n (x) = sin x sin(2x)... sin(2 n x). Prove tht f n (x) 2 f n (π/). Proof. Let g(x) = sin x sin(2x) /2. We hve 2 ( g(x) = sin x sin(2x) /2 4 = 4 sin x sin x sin x ) 2 cos x 2 sin 2 x + cos 2 x ( ) 2 4 = = g(π/). 4 2 Note we hve use the AM-GM inequlity tht sin 2 x + cos 2 x = sin 2 x + sin 2 x + sin 2 x + ( cos x) sin x sin x sin x cos x. Therefore, let K = 2 [ ( /2) n ], we hve f n (x) = sin x sin(2x)... sin(2 n x) ) ) /2 ( ) /4 = ( sin x sin(2x) /2 ( sin(2x) sin(4x) /2 sin(4x) sin(8x) /2 ) K ( ) K/2... ( sin(2 n x sin(2 n x) /2 sin(2 n x) = g(x) g(2x) /2... g(2 n x) K ( sin(2 n x) g(π/)g(x) g(2π/) /2... g(2 n π/) K = f n (π/) / sin(2 n π/) K/2 ( 2 ) K/2 = f n (π/) fn (π/) 2. This is the desired inequlity. ) K/2 Exercise 7 (IMO 995). Let, b, c be positive rel numbers such tht bc =. Prove tht (b + c) + b (c + ) + c ( + b) 2.
11 Proof. Let x =, y =, z =. Then x, y, z re positive rel numbers nd xyz =. We b c hve (b + c) = ) = x2 y + z. Similrly b (c + ) = x ( y + z y2 z + x, c ( + b) = z2 x + y. By Cuchy-Schrz inequlity (see lso Exercise ) nd the Arithmetic-Geometric Inequlity we hve x 2 y + z + y2 z + x + z2 x + y (x + y + z)2 2(x + y + z) = x + y + z 2 xyz 2 = 2. Exercise 8 (Problem, The 26th Annul Vojtech Jrnik Interntionl Competition 206). Let, b, c be positive rel number such tht + b + c =. Prove tht ( + bc Proof. By the AM-GM inequlity we hve )( b c)( + c + ) 728 b + bc = + bc + bc + bc 4 4 b c, nd Therefore, ( + bc ( + b + c ) 27 = bc, i.e., bc 27. )( b c)( + c + ) ( 4 b = 64 bc 4 b c )(4 4 b c )( = 728. ) 4 c b Exercise 9. Let x, y R, y > 0. Prove tht e x + y(ln y ) x y. Proof. Let f(x) = e x. Then for y > 0, we hve f (y) = sup x R {x y e x } = y(ln y ). By Fenchel s inequlity we hve f(x) + f (y) = e x + y(ln y ) x y s desired.
12 2 Exercise 0 (IMO 200). Let, b, c be positive rel numbers. Prove tht 2 + 8bc + b b2 + 8c + c c2 + 8b. Proof. Since the expression on the LHS does not chnge when we replce (, b, c) by (k, kb, kc) for rbitrry k R, we cn ssume tht + b + c =. Since x x is convex for x > 0, pplying Jensen s inequlity we obtin 2 + 8bc + Next we show tht b b2 + 8c + c c2 + 8b (2 + 8bc) + b(b 2 + 8c) + c(c 2 + 8b) = + b + c + 24bc. (6) +b +c +24bc = (+b+c) = +b +c +( 2 b+ 2 c+b 2 c+b 2 +c 2 +c 2 b)+6bc, (7) which is equivlent to show tht ( 2 b + 2 c + b 2 c + b 2 + c 2 + c 2 b) 6bc. This is indeed true becuse of the AM-GM inequlity ( 2 b + 2 c + b 2 c + b 2 + c 2 + c 2 b) b 2 c b 2 b 2 c c 2 c 2 b = 6bc. The desired inequlity follows from (6) nd (7).
Convex Sets and Functions
B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationBest Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationIntroduction to the Calculus of Variations
Introduction to the Clculus of Vritions Jim Fischer Mrch 20, 1999 Abstrct This is self-contined pper which introduces fundmentl problem in the clculus of vritions, the problem of finding extreme vlues
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More informationThe Banach algebra of functions of bounded variation and the pointwise Helly selection theorem
The Bnch lgebr of functions of bounded vrition nd the pointwise Helly selection theorem Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics, University of Toronto Jnury, 015 1 BV [, b] Let < b. For f
More informationPresentation Problems 5
Presenttion Problems 5 21-355 A For these problems, ssume ll sets re subsets of R unless otherwise specified. 1. Let P nd Q be prtitions of [, b] such tht P Q. Then U(f, P ) U(f, Q) nd L(f, P ) L(f, Q).
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationUNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE
UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More informationg i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f
1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where
More information11 An introduction to Riemann Integration
11 An introduction to Riemnn Integrtion The PROOFS of the stndrd lemms nd theorems concerning the Riemnn Integrl re NEB, nd you will not be sked to reproduce proofs of these in full in the exmintion in
More informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which
More informationMath 61CM - Solutions to homework 9
Mth 61CM - Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ
More informationMATH 423 Linear Algebra II Lecture 28: Inner product spaces.
MATH 423 Liner Algebr II Lecture 28: Inner product spces. Norm The notion of norm generlizes the notion of length of vector in R 3. Definition. Let V be vector spce over F, where F = R or C. A function
More informationFourier series. Preliminary material on inner products. Suppose V is vector space over C and (, )
Fourier series. Preliminry mteril on inner products. Suppose V is vector spce over C nd (, ) is Hermitin inner product on V. This mens, by definition, tht (, ) : V V C nd tht the following four conditions
More informationInner-product spaces
Inner-product spces Definition: Let V be rel or complex liner spce over F (here R or C). An inner product is n opertion between two elements of V which results in sclr. It is denoted by u, v nd stisfies:
More informationSTUDY GUIDE FOR BASIC EXAM
STUDY GUIDE FOR BASIC EXAM BRYON ARAGAM This is prtil list of theorems tht frequently show up on the bsic exm. In mny cses, you my be sked to directly prove one of these theorems or these vrints. There
More informationThe Riemann-Lebesgue Lemma
Physics 215 Winter 218 The Riemnn-Lebesgue Lemm The Riemnn Lebesgue Lemm is one of the most importnt results of Fourier nlysis nd symptotic nlysis. It hs mny physics pplictions, especilly in studies of
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More informationTRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS
TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS S.S. DRAGOMIR AND A. SOFO Abstrct. In this pper by utilising result given by Fink we obtin some new results relting to the trpezoidl inequlity
More informationarxiv:math/ v2 [math.ho] 16 Dec 2003
rxiv:mth/0312293v2 [mth.ho] 16 Dec 2003 Clssicl Lebesgue Integrtion Theorems for the Riemnn Integrl Josh Isrlowitz 244 Ridge Rd. Rutherford, NJ 07070 jbi2@njit.edu Februry 1, 2008 Abstrct In this pper,
More informationAnalytical Methods Exam: Preparatory Exercises
Anlyticl Methods Exm: Preprtory Exercises Question. Wht does it men tht (X, F, µ) is mesure spce? Show tht µ is monotone, tht is: if E F re mesurble sets then µ(e) µ(f). Question. Discuss if ech of the
More informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationHomework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.
Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function
More informationAppendix to Notes 8 (a)
Appendix to Notes 8 () 13 Comprison of the Riemnn nd Lebesgue integrls. Recll Let f : [, b] R be bounded. Let D be prtition of [, b] such tht Let D = { = x 0 < x 1
More informationMath Solutions to homework 1
Mth 75 - Solutions to homework Cédric De Groote October 5, 07 Problem, prt : This problem explores the reltionship between norms nd inner products Let X be rel vector spce ) Suppose tht is norm on X tht
More information7 Improper Integrals, Exp, Log, Arcsin, and the Integral Test for Series
7 Improper Integrls, Exp, Log, Arcsin, nd the Integrl Test for Series We hve now ttined good level of understnding of integrtion of nice functions f over closed intervls [, b]. In prctice one often wnts
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationSome estimates on the Hermite-Hadamard inequality through quasi-convex functions
Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13-693 Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions Dniel Alexndru Ion Abstrct. In this pper
More informationThe one-dimensional Henstock-Kurzweil integral
Chpter 1 The one-dimensionl Henstock-Kurzweil integrl 1.1 Introduction nd Cousin s Lemm The purpose o this monogrph is to study multiple Henstock-Kurzweil integrls. In the present chpter, we shll irst
More informationProblem Set 4: Solutions Math 201A: Fall 2016
Problem Set 4: s Mth 20A: Fll 206 Problem. Let f : X Y be one-to-one, onto mp between metric spces X, Y. () If f is continuous nd X is compct, prove tht f is homeomorphism. Does this result remin true
More informationNew Integral Inequalities for n-time Differentiable Functions with Applications for pdfs
Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353-362 New Integrl Inequlities for n-time Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute
More informationHandout 4. Inverse and Implicit Function Theorems.
8.95 Hndout 4. Inverse nd Implicit Function Theorems. Theorem (Inverse Function Theorem). Suppose U R n is open, f : U R n is C, x U nd df x is invertible. Then there exists neighborhood V of x in U nd
More informationThe presentation of a new type of quantum calculus
DOI.55/tmj-27-22 The presenttion of new type of quntum clculus Abdolli Nemty nd Mehdi Tourni b Deprtment of Mthemtics, University of Mzndrn, Bbolsr, Irn E-mil: nmty@umz.c.ir, mehdi.tourni@gmil.com b Abstrct
More informationConvergence of Fourier Series and Fejer s Theorem. Lee Ricketson
Convergence of Fourier Series nd Fejer s Theorem Lee Ricketson My, 006 Abstrct This pper will ddress the Fourier Series of functions with rbitrry period. We will derive forms of the Dirichlet nd Fejer
More informationMATH1050 Cauchy-Schwarz Inequality and Triangle Inequality
MATH050 Cuchy-Schwrz Inequlity nd Tringle Inequlity 0 Refer to the Hndout Qudrtic polynomils Definition (Asolute extrem for rel-vlued functions of one rel vrile) Let I e n intervl, nd h : D R e rel-vlued
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationProperties of the Riemann Integral
Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University Februry 15, 2018 Outline 1 Some Infimum nd Supremum Properties 2
More informationON A CONVEXITY PROPERTY. 1. Introduction Most general class of convex functions is defined by the inequality
Krgujevc Journl of Mthemtics Volume 40( (016, Pges 166 171. ON A CONVEXITY PROPERTY SLAVKO SIMIĆ Abstrct. In this rticle we proved n interesting property of the clss of continuous convex functions. This
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationAbsolute values of real numbers. Rational Numbers vs Real Numbers. 1. Definition. Absolute value α of a real
Rtionl Numbers vs Rel Numbers 1. Wht is? Answer. is rel number such tht ( ) =. R [ ( ) = ].. Prove tht (i) 1; (ii). Proof. (i) For ny rel numbers x, y, we hve x = y. This is necessry condition, but not
More informationOn the Generalized Weighted Quasi-Arithmetic Integral Mean 1
Int. Journl of Mth. Anlysis, Vol. 7, 2013, no. 41, 2039-2048 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2013.3499 On the Generlized Weighted Qusi-Arithmetic Integrl Men 1 Hui Sun School
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationFUNDAMENTALS OF REAL ANALYSIS by. III.1. Measurable functions. f 1 (
FUNDAMNTALS OF RAL ANALYSIS by Doğn Çömez III. MASURABL FUNCTIONS AND LBSGU INTGRAL III.. Mesurble functions Hving the Lebesgue mesure define, in this chpter, we will identify the collection of functions
More informationNEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS. := f (4) (x) <. The following inequality. 2 b a
NEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS MOHAMMAD ALOMARI A MASLINA DARUS A AND SEVER S DRAGOMIR B Abstrct In terms of the first derivtive some ineulities of Simpson
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationS. S. Dragomir. 2, we have the inequality. b a
Bull Koren Mth Soc 005 No pp 3 30 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Abstrct Compnions of Ostrowski s integrl ineulity for bsolutely
More informationA basic logarithmic inequality, and the logarithmic mean
Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, Bbeş-Bolyi University Str. Koglnicenu
More informationHilbert Spaces. Chapter Inner product spaces
Chpter 4 Hilbert Spces 4.1 Inner product spces In the following we will discuss both complex nd rel vector spces. With L denoting either R or C we recll tht vector spce over L is set E equipped with ddition,
More informationSOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL (1 + µ(f n )) f(x) =. But we don t need the exact bound.) Set
SOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL 28 Nottion: N {, 2, 3,...}. (Tht is, N.. Let (X, M be mesurble spce with σ-finite positive mesure µ. Prove tht there is finite positive mesure ν on (X, M such
More informationEntropy and Ergodic Theory Notes 10: Large Deviations I
Entropy nd Ergodic Theory Notes 10: Lrge Devitions I 1 A chnge of convention This is our first lecture on pplictions of entropy in probbility theory. In probbility theory, the convention is tht ll logrithms
More informationII. Integration and Cauchy s Theorem
MTH6111 Complex Anlysis 2009-10 Lecture Notes c Shun Bullett QMUL 2009 II. Integrtion nd Cuchy s Theorem 1. Pths nd integrtion Wrning Different uthors hve different definitions for terms like pth nd curve.
More informationINNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS AND APPLICATIONS
INNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS AND APPLICATIONS S. S. DRAGOMIR Abstrct. Some inequlities for two inner products h i nd h i which generte the equivlent norms kk nd kk with pplictions
More informationAN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir
RGMIA Reserch Report Collection, Vol., No., 999 http://sci.vu.edu.u/ rgmi AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS I. Fedotov nd S. S. Drgomir Astrct. An
More informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationGeometrically Convex Function and Estimation of Remainder Terms in Taylor Series Expansion of some Functions
Geometriclly Convex Function nd Estimtion of Reminder Terms in Tylor Series Expnsion of some Functions Xioming Zhng Ningguo Zheng December 21 25 Abstrct In this pper two integrl inequlities of geometriclly
More informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More information440-2 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam
440-2 Geometry/Topology: Differentible Mnifolds Northwestern University Solutions of Prctice Problems for Finl Exm 1) Using the cnonicl covering of RP n by {U α } 0 α n, where U α = {[x 0 : : x n ] RP
More informationINEQUALITIES FOR GENERALIZED WEIGHTED MEAN VALUES OF CONVEX FUNCTION
INEQUALITIES FOR GENERALIZED WEIGHTED MEAN VALUES OF CONVEX FUNCTION BAI-NI GUO AND FENG QI Abstrct. In the rticle, using the Tchebycheff s integrl inequlity, the suitble properties of double integrl nd
More information1 i n x i x i 1. Note that kqk kp k. In addition, if P and Q are partition of [a, b], P Q is finer than both P and Q.
Chpter 6 Integrtion In this chpter we define the integrl. Intuitively, it should be the re under curve. Not surprisingly, fter mny exmples, counter exmples, exceptions, generliztions, the concept of the
More informationEntrance Exam, Real Analysis September 1, 2009 Solve exactly 6 out of the 8 problems. Compute the following and justify your computation: lim
1. Let n be positive integers. ntrnce xm, Rel Anlysis September 1, 29 Solve exctly 6 out of the 8 problems. Sketch the grph of the function f(x): f(x) = lim e x2n. Compute the following nd justify your
More informationINDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Analysis Autumn 2012
Lecture 6: Line Integrls INDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Anlysis Autumn 2012 August 8, 2012 Lecture 6: Line Integrls Lecture 6: Line Integrls Lecture 6: Line Integrls Integrls of complex
More information7.2 Riemann Integrable Functions
7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous
More informationGENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS. (b a)3 [f(a) + f(b)] f x (a,b)
GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR Abstrct. In this pper, we estblish some generliztions
More informationMAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL
MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL DR. RITU AGARWAL MALVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR, INDIA-302017 Tble of Contents Contents Tble of Contents 1 1. Introduction 1 2. Prtition
More informationCalculus I-II Review Sheet
Clculus I-II Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing
More informationGENERALIZED ABSTRACTED MEAN VALUES
GENERALIZED ABSTRACTED MEAN VALUES FENG QI Abstrct. In this rticle, the uthor introduces the generlized bstrcted men vlues which etend the concepts of most mens with two vribles, nd reserches their bsic
More informationFor a continuous function f : [a; b]! R we wish to define the Riemann integral
Supplementry Notes for MM509 Topology II 2. The Riemnn Integrl Andrew Swnn For continuous function f : [; b]! R we wish to define the Riemnn integrl R b f (x) dx nd estblish some of its properties. This
More informationS. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality:
FACTA UNIVERSITATIS NIŠ) Ser Mth Inform 9 00) 6 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Dedicted to Prof G Mstroinni for his 65th birthdy
More informationSome inequalities of Hermite-Hadamard type for n times differentiable (ρ, m) geometrically convex functions
Avilble online t www.tjns.com J. Nonliner Sci. Appl. 8 5, 7 Reserch Article Some ineulities of Hermite-Hdmrd type for n times differentible ρ, m geometriclly convex functions Fiz Zfr,, Humir Klsoom, Nwb
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More informationNOTES AND PROBLEMS: INTEGRATION THEORY
NOTES AND PROBLEMS: INTEGRATION THEORY SAMEER CHAVAN Abstrct. These re the lecture notes prepred for prticipnts of AFS-I to be conducted t Kumun University, Almor from 1st to 27th December, 2014. Contents
More informationMATH 174A: PROBLEM SET 5. Suggested Solution
MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion
More informationKeywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.
Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. Lmi-Athens Lmi 3500 Greece Abstrct Using
More information. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =
Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos( - 1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin( - 1 ) = -π 2 6 2 6 Cn you do similr problems?
More informationChapter 6. Riemann Integral
Introduction to Riemnn integrl Chpter 6. Riemnn Integrl Won-Kwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl
More informationRegulated functions and the regulated integral
Regulted functions nd the regulted integrl Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics University of Toronto April 3 2014 1 Regulted functions nd step functions Let = [ b] nd let X be normed
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry dierentil eqution (ODE) du f(t) dt with initil condition u() : Just
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More information