GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS
|
|
- Catherine Allen
- 6 years ago
- Views:
Transcription
1 - TAMKANG JOURNAL OF MATHEMATICS Volume 5, Number, 7-5, June doi:5556/jkjm555 Avilble online hp://journlsmhkueduw/ GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS MARCELA V MIHAI AND DANIEL ALEXANDRU ION Absrc Some Hermie-Hdmrd ype ineuliies re provided We del wih funcions whose derivives in bsolue vlue re convex or concve By defining wo cumulive gps which enble us o generlize known resuls in he frmework of Riemnn- Liouville frcionl clculus, we open new perspecive on he clssic semen of he ineuliy Inroducion The Hermie-Hdmrd ineuliy ses h if funcion f : [,b] R is convex hen b f f xdx Boh ineuliies hold in he reversed direcion if f is concve f f b, H H This ineuliy received gre del of enion in he ls decde insnce, mny generlizions pplicions being obined See [], [], [], [6], [8], [] he references herein Of specil ineres o us is he following improvemens of he Hermie-Hdmrd ineuliy, h cn be found in he monogrph [7], p 5: b f [ f 3 b f 3b ] f xdx, LH H f xdx [ ] f f b b f f f b, R H H The purpose of he presen pper is o esblish new Hermie-Hdmrd ype ineuliies wihin Riemnn-Liouville frcionl clculus Unlike he clssicl cse, he funcions under enion re no ssumed convex or concve, bu his fc is sked for he bsolue vlue Received November 8, 3, cceped December 3, 3 Mhemics Subjec Clssificion 6A5 Key words phrses Convex funcion, Hermie-Hdmrd ineuliy, Riemnn-Liouville frcionl inegrls 7
2 8 MARCELA V MIHAI AND DANIEL ALEXANDRU ION of heir derivives Under hese circumsnces we will prove he exisence of wo srings of ineuliies refining he ineuliies LH H R H H In wh follows we will consider only rel-vlued funcions defined on inervls [, b] wih < b, n is n odd number Le f L [,b] be n inegrble funcion J α f, of orderα>, ched o f re defined respecively by b The Riemnn-Liouville inegrls J α f J α f x= Γα J α b f x= Γα x x x α f, for x>, x α f, for x < b Here,Γα= e α is he Gmm funcion We mke he convenion J f x= J b f x= f x The heory of Riemnn-Liouville frcionl inegrls cn be found in he book [3] Min resuls As bove, we ssume h [,b] is compc subinervl of [, f : [,b] R is n inegrble funcion We define he cumulive o he lef α,n gp of f by he formul n / L α,n,b= f k= Γα n kbk J α n k bk n f α n / k= [ J α n kbk n f n kbk In he priculr cse whereα= n= 3 we hve L,3,b = [ f 3 b f 3b n k b k ] ] f, so he cumulive o he lef gp L,3,b esimes he precision of he righ h side ineuliy in LH H f xdx [ f 3 b f 3b ]
3 GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS 9 The cumulive o he lef gps L α,n,b hve he sme mening, relive o higher order refinemens of LH H The following echnicl lemm provides suible formul for esiming L α,n,b in bsolue vlue: Lemm We hve L α,n,b= n / [ α f n kbk n k b k k= α f n k bk n kbk Proof Pu I k = I k = α f n kbk n k b k, ] α f n k bk n kbk By using he inegrion by prs he subsiuions n kbk n k b k u =, n k bk n kbk v =, we infer h n kbk I k I k = f Γα [ n k b k J α f n kbk n ] n kbk J α f n k bk n The proof is compleed α We re now in posiion o se prove he following resul: Theorem Assume h f : [,b] R is differenible funcion such h f is convex on [,b] Then Lα,n,b n / k= [ α 5α n kbk αα f
4 MARCELA V MIHAI AND DANIEL ALEXANDRU ION n k b k αα f ] α n k bk α f Proof Using Lemm he convexiy of f we obin Lα,n,b n /[ n kbk f α k= n k b k f α n k bk f α n kbk ] f α The proof ends fer srighforwrd compuion in he righ h side erm The Be funcion is defined by he formul Bx, y= x y for x, y > Our nex resul is s follows: Theorem Assume h f : [,b] R is differenible funcion such h f is convex on [,b] for some exponen > Then Lα,n,b where p = n / { p k= αp n kbk [ f n k b k f ] ] p [ α B p, α [ n k bk f n kbk f Proof According o Lemm Hölder s ineuliy, we hve Lα,n,b n / k= [ α p p ] }
5 GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS f n kbk n k b k α p p f n k bk n kbk ] Since f is convex on [,b], we hve: f n kbk n k b k [ n kbk f n k b k f ] f n k bk n kbk [ n k bk f n kbk f ] A simple compuion shows h he proof is compleed αp = αp, α p = α B p, α Theorem 3 Assume h f : [,b] R is differenible funcion such h f is convex on [, b] for some exponen Then he following ineuliy holds: Lα,n,b α p α { n / [ n kbk f n k b k k= α f α [ n k bk f α3 n kbk α f where p = Proof Using Lemm he power men ineuliy, we hve Lα,n,b α f n / k= [ α p n kbk n k b k ] ] },
6 MARCELA V MIHAI AND DANIEL ALEXANDRU ION α p α f n k bk n kbk ] Since f is convex on [,b], we hve: α f n kbk n k b k n kbk α f n k b k αα f α f n k bk n kbk α n k bk α f α 3α n kbk αα f This complees he proof of he heorem Theorem Assume h f : [,b] R is differenible funcion such h f is concve on [,b] for some exponen > Then where p = Lα,n,b α B n / k= [ p, α /p n k bk f αp /p ] n k bk 3 f, Proof From Lemm Hölder s inegrl ineuliy for > p =, we hve Lα,n,b n / k= [ α p p f n kbk n k b k α p p f n k bk n kbk ]
7 GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS 3 Since f is concve on [,b], we infer from Jensen s ineuliy for concve funcions h f n kbk n k b k n kbk n k b k f = n k bk f In he sme mnner, f n k bk n kbk n k bk 3 f Using we complee he proof αp = Our nex resul is s follows: αp, α p = α B p,, α Theorem 5 Assume h f : [,b] R is differenible funcion such h f is concve on [,b] for some exponen > Then Lα,n,b α n /[ α f k= α α α f α Proof From Lemm we hve Lα,n,b n / k= [ n kbk n k b k α n k bk α3 n kbk α α f n kbk n k b k α f n k bk n kbk Since f is concve, by Jensen s ineuliy we obin Lα,n,b n / [ α I k= α ] I, ] ]
8 MARCELA V MIHAI AND DANIEL ALEXANDRU ION where n kbk n k b k I = f α α = α f n kbk n k b k α α I = f n k bk n kbk α α = α f n k bk α3 n kbk α α he proof is compleed Remrk Forα= in he Theorems,, 3, respecively 5, we recover he resuls sed in [9, Theorems 6-] Also, forα= in Lemm, we ge [9, Lemm ] Remrk For n = 3 in he Theorems,, 3, respecively, we recover he resuls sed in [5, Theorems -] Also forα= in Lemm, we ge [5, Lemm ] We end our pper by considering he cumulive o he righ α,n gp defined by he formul n /[ ] n k b k n k bk R α,n,b= f f k= α n / [ n kbk Γα J α n kb k f k= n ] n k bk J α f n kbk, n In he priculr cse whereα= n= 3 we hve R,3,b = [ ] f f b b f f, so he cumulive o he righ gp R,3,b esimes he precision of he lef h side ineuliy in R H H, f xdx [ ] f f b b f Using he bove echniues, one cn prove compnions of ll he resuls we proved for he cumulive o he lef α,n gp The sring poin is he following formul for compuing R α,n,b
9 GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS 5 Lemm We hve R α,n,b= n / [ k= α f α f n k b k n kbk n kbk n k bk Using Lemm, one cn prove vrious esimes of R α,n,b such s Rα,n,b n /[ n k b k k= α f α α n kbk αα f α 3α n k bk αα f ] ] References [] M Bessenyei, The Hermie Hdmrd Ineuliy on Simplices, Americn Mhemicl Monhly, 5 8, [] S S Drgomir C E M Perce, Seleced Topic on Hermie-Hdmrd Ineuliies Applicions, Melbourne Adelide, December, [3] R Gorenflo F Minrdi, Frcionl Clculus: Inegrl Differenil Euions of Frcionl Order, Springer Verlg, Wien, 997 [] H Kvurmci M Avci, M E Özdemir, New ineuliies of Hermie-Hdmrd ype for convex funcions wih pplicions, rxiv: 6593v [5] M Mihi, Some Hermie-Hdmrd ype ineuliies obined vi Riemnn-Liouville frcionl clculus submied [6] F-C Miroi C I Spiridon, Hermie-Hdmrd ype ineuliies of convex funcions wih respec o pir of usi-rihmeic mens, Mh Rep, [7] C P Niculescu L-E Persson, Convex Funcions heir Applicions A Conemporry Approch, CMS Books in Mhemics vol 3, Springer-Verlg, New York, 6 [8] C P Niculescu, The Hermie-Hdmrd ineuliy for log-convex funcions, Nonliner Anlysis 75, [9] M Emin Özdemir, A Ekinci A Akdemir, Some new inegrl ineuliies for funcions whose derivives of bsolue vlues re convex concve, RGMIA Reserch Repor Collecion, 5, Aricle 8, pp [] S W sowicz A Wikowski, On some ineuliy of Hermie-Hdmrd ype, Opuscul Mh, 3, 59 6 Deprmen of Mhemics, Universiy of Criov, Sree A I Cuz 3, Criov, RO-585, Romni E-mil: mrcelmihi58@yhoocom Deprmen of Mhemics, Universiy of Criov, Sree A I Cuz 3, Criov, RO-585, Romni E-mil: dn_lexion@yhoocom
ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX
Journl of Applied Mhemics, Sisics nd Informics JAMSI), 9 ), No. ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX MEHMET ZEKI SARIKAYA, ERHAN. SET
More informationHermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals
Sud. Univ. Beş-Bolyi Mh. 6(5, No. 3, 355 366 Hermie-Hdmrd-Fejér ype inequliies for convex funcions vi frcionl inegrls İmd İşcn Asrc. In his pper, firsly we hve eslished Hermie Hdmrd-Fejér inequliy for
More informationON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX.
ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX. MEHMET ZEKI SARIKAYA?, ERHAN. SET, AND M. EMIN OZDEMIR Asrc. In his noe, we oin new some ineuliies
More informationON THE OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS
Hceepe Journl of Mhemics nd Sisics Volume 45) 0), 65 655 ON THE OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS M Emin Özdemir, Ahme Ock Akdemir nd Erhn Se Received 6:06:0 : Acceped
More information1. Introduction. 1 b b
Journl of Mhemicl Inequliies Volume, Number 3 (007), 45 436 SOME IMPROVEMENTS OF GRÜSS TYPE INEQUALITY N. ELEZOVIĆ, LJ. MARANGUNIĆ AND J. PEČARIĆ (communiced b A. Čižmešij) Absrc. In his pper some inequliies
More informationOn The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function
Turkish Journl o Anlysis nd Numer Theory, 4, Vol., No. 3, 85-89 Aville online h://us.scieu.com/jn//3/6 Science nd Educion Pulishing DOI:.69/jn--3-6 On The Hermie- Hdmrd-Fejér Tye Inegrl Ineuliy or Convex
More informationNew Inequalities in Fractional Integrals
ISSN 1749-3889 (prin), 1749-3897 (online) Inernionl Journl of Nonliner Science Vol.9(21) No.4,pp.493-497 New Inequliies in Frcionl Inegrls Zoubir Dhmni Zoubir DAHMANI Lborory of Pure nd Applied Mhemics,
More informationContraction Mapping Principle Approach to Differential Equations
epl Journl of Science echnology 0 (009) 49-53 Conrcion pping Principle pproch o Differenil Equions Bishnu P. Dhungn Deprmen of hemics, hendr Rn Cmpus ribhuvn Universiy, Khmu epl bsrc Using n eension of
More informationPositive and negative solutions of a boundary value problem for a
Invenion Journl of Reerch Technology in Engineering & Mngemen (IJRTEM) ISSN: 2455-3689 www.ijrem.com Volume 2 Iue 9 ǁ Sepemer 28 ǁ PP 73-83 Poiive nd negive oluion of oundry vlue prolem for frcionl, -difference
More informationThe Hadamard s inequality for quasi-convex functions via fractional integrals
Annls of the University of Criov, Mthemtics nd Computer Science Series Volume (), 3, Pges 67 73 ISSN: 5-563 The Hdmrd s ineulity for usi-convex functions vi frctionl integrls M E Özdemir nd Çetin Yildiz
More informationResearch Article New General Integral Inequalities for Lipschitzian Functions via Hadamard Fractional Integrals
Hindwi Pulishing orporion Inernionl Journl of Anlysis, Aricle ID 35394, 8 pges hp://d.doi.org/0.55/04/35394 Reserch Aricle New Generl Inegrl Inequliies for Lipschizin Funcions vi Hdmrd Frcionl Inegrls
More informationRefinements to Hadamard s Inequality for Log-Convex Functions
Alied Mhemics 899-93 doi:436/m7 Pulished Online Jul (h://wwwscirporg/journl/m) Refinemens o Hdmrd s Ineuli for Log-Convex Funcions Asrc Wdllh T Sulimn Dermen of Comuer Engineering College of Engineering
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE BOUNDARY-VALUE PROBLEM
Elecronic Journl of Differenil Equions, Vol. 208 (208), No. 50, pp. 6. ISSN: 072-669. URL: hp://ejde.mh.xse.edu or hp://ejde.mh.un.edu EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE
More informationMathematics 805 Final Examination Answers
. 5 poins Se he Weiersrss M-es. Mhemics 85 Finl Eminion Answers Answer: Suppose h A R, nd f n : A R. Suppose furher h f n M n for ll A, nd h Mn converges. Then f n converges uniformly on A.. 5 poins Se
More informationOn Hadamard and Fejér-Hadamard inequalities for Caputo k-fractional derivatives
In J Nonliner Anl Appl 9 8 No, 69-8 ISSN: 8-68 elecronic hp://dxdoiorg/75/ijn8745 On Hdmrd nd Fejér-Hdmrd inequliies for Cpuo -frcionl derivives Ghulm Frid, Anum Jved Deprmen of Mhemics, COMSATS Universiy
More informationNew Integral Inequalities of the Type of Hermite-Hadamard Through Quasi Convexity
Punjb University Journl of Mthemtics (ISSN 116-56) Vol. 45 (13) pp. 33-38 New Integrl Inequlities of the Type of Hermite-Hdmrd Through Qusi Convexity S. Hussin Deprtment of Mthemtics, College of Science,
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 informationFURTHER GENERALIZATIONS. QI Feng. The value of the integral of f(x) over [a; b] can be estimated in a variety ofways. b a. 2(M m)
Univ. Beogrd. Pul. Elekroehn. Fk. Ser. M. 8 (997), 79{83 FUTHE GENEALIZATIONS OF INEQUALITIES FO AN INTEGAL QI Feng Using he Tylor's formul we prove wo inegrl inequliies, h generlize K. S. K. Iyengr's
More informationA LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR IAN KNOWLES
A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR j IAN KNOWLES 1. Inroducion Consider he forml differenil operor T defined by el, (1) where he funcion q{) is rel-vlued nd loclly
More informationConvergence of Singular Integral Operators in Weighted Lebesgue Spaces
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 2, 2017, 335-347 ISSN 1307-5543 www.ejpm.com Published by New York Business Globl Convergence of Singulr Inegrl Operors in Weighed Lebesgue
More informationHow to prove the Riemann Hypothesis
Scholrs Journl of Phsics, Mhemics nd Sisics Sch. J. Phs. Mh. S. 5; (B:5-6 Scholrs Acdemic nd Scienific Publishers (SAS Publishers (An Inernionl Publisher for Acdemic nd Scienific Resources *Corresonding
More informationHow to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh.
How o Prove he Riemnn Hohesis Auhor: Fez Fok Al Adeh. Presiden of he Srin Cosmologicl Socie P.O.Bo,387,Dmscus,Sri Tels:963--77679,735 Emil:hf@scs-ne.org Commens: 3 ges Subj-Clss: Funcionl nlsis, comle
More informationSome Inequalities variations on a common theme Lecture I, UL 2007
Some Inequliies vriions on common heme Lecure I, UL 2007 Finbrr Hollnd, Deprmen of Mhemics, Universiy College Cork, fhollnd@uccie; July 2, 2007 Three Problems Problem Assume i, b i, c i, i =, 2, 3 re rel
More informationEÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Sayı: 3-2 Yıl:
63 EÜFBED - Fen Bilimleri Ensiüsü Dergisi Cil-Syı: 3- Yıl: 63-7 SOME INEQUALITIES OF HERMITE-HADAMARD TYPE FOR FUNCTIONS WHOSE DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX TÜREVİNİN MUTLAK DEĞERİ QUASI-KONVEKS
More informationf t f a f x dx By Lin McMullin f x dx= f b f a. 2
Accumulion: Thoughs On () By Lin McMullin f f f d = + The gols of he AP* Clculus progrm include he semen, Sudens should undersnd he definie inegrl s he ne ccumulion of chnge. 1 The Topicl Ouline includes
More informationAnalytic solution of linear fractional differential equation with Jumarie derivative in term of Mittag-Leffler function
Anlyic soluion of liner frcionl differenil equion wih Jumrie derivive in erm of Mig-Leffler funcion Um Ghosh (), Srijn Sengup (2), Susmi Srkr (2b), Shnnu Ds (3) (): Deprmen of Mhemics, Nbdwip Vidysgr College,
More informationGreen s Functions and Comparison Theorems for Differential Equations on Measure Chains
Green s Funcions nd Comprison Theorems for Differenil Equions on Mesure Chins Lynn Erbe nd Alln Peerson Deprmen of Mhemics nd Sisics, Universiy of Nebrsk-Lincoln Lincoln,NE 68588-0323 lerbe@@mh.unl.edu
More information4.8 Improper Integrals
4.8 Improper Inegrls Well you ve mde i hrough ll he inegrion echniques. Congrs! Unforunely for us, we sill need o cover one more inegrl. They re clled Improper Inegrls. A his poin, we ve only del wih inegrls
More informationProcedia Computer Science
Procedi Compuer Science 00 (0) 000 000 Procedi Compuer Science www.elsevier.com/loce/procedi The Third Informion Sysems Inernionl Conference The Exisence of Polynomil Soluion of he Nonliner Dynmicl Sysems
More informationHermite-Hadamard and Simpson Type Inequalities for Differentiable Quasi-Geometrically Convex Functions
Trkish Jornl o Anlysis nd Nmer Theory, 4, Vol, No, 4-46 Aville online h://ssciecom/jn/// Science nd Edcion Plishing DOI:69/jn--- Hermie-Hdmrd nd Simson Tye Ineliies or Dierenile Qsi-Geomericlly Convex
More informatione t dt e t dt = lim e t dt T (1 e T ) = 1
Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie
More informationApplication on Inner Product Space with. Fixed Point Theorem in Probabilistic
Journl of Applied Mhemics & Bioinformics, vol.2, no.2, 2012, 1-10 ISSN: 1792-6602 prin, 1792-6939 online Scienpress Ld, 2012 Applicion on Inner Produc Spce wih Fixed Poin Theorem in Probbilisic Rjesh Shrivsv
More informationNew general integral inequalities for quasiconvex functions
NTMSCI 6, No 1, 1-7 18 1 New Trends in Mthemticl Sciences http://dxdoiorg/185/ntmsci1739 New generl integrl ineulities for usiconvex functions Cetin Yildiz Atturk University, K K Eduction Fculty, Deprtment
More informationf (a) + f (b) f (λx + (1 λ)y) max {f (x),f (y)}, x, y [a, b]. (1.1)
TAMKANG JOURNAL OF MATHEMATICS Volume 41, Number 4, 353-359, Winter 1 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI, M. DARUS
More informationCALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION
Avilble online hp://scik.org Eng. Mh. Le. 15, 15:4 ISSN: 49-9337 CALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION PANDEY, C. P. 1, RAKESH MOHAN AND BHAIRAW NATH TRIPATHI 3 1 Deprmen o Mhemics, Ajy
More informationHERMITE-HADAMARD TYPE INEQUALITIES OF CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF QUASI-ARITHMETIC MEANS
HERMITE-HADAMARD TYPE INEQUALITIES OF CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF QUASI-ARITHMETIC MEANS FLAVIA CORINA MITROI nd CĂTĂLIN IRINEL SPIRIDON In this pper we estblish some integrl inequlities
More informationLAPLACE TRANSFORM OVERCOMING PRINCIPLE DRAWBACKS IN APPLICATION OF THE VARIATIONAL ITERATION METHOD TO FRACTIONAL HEAT EQUATIONS
Wu, G.-.: Lplce Trnsform Overcoming Principle Drwbcks in Applicion... THERMAL SIENE: Yer 22, Vol. 6, No. 4, pp. 257-26 257 Open forum LAPLAE TRANSFORM OVEROMING PRINIPLE DRAWBAKS IN APPLIATION OF THE VARIATIONAL
More informationREAL ANALYSIS I HOMEWORK 3. Chapter 1
REAL ANALYSIS I HOMEWORK 3 CİHAN BAHRAN The quesions re from Sein nd Shkrchi s e. Chper 1 18. Prove he following sserion: Every mesurble funcion is he limi.e. of sequence of coninuous funcions. We firs
More informationSolutions for Nonlinear Partial Differential Equations By Tan-Cot Method
IOSR Journl of Mhemics (IOSR-JM) e-issn: 78-578. Volume 5, Issue 3 (Jn. - Feb. 13), PP 6-11 Soluions for Nonliner Pril Differenil Equions By Tn-Co Mehod Mhmood Jwd Abdul Rsool Abu Al-Sheer Al -Rfidin Universiy
More informationENGR 1990 Engineering Mathematics The Integral of a Function as a Function
ENGR 1990 Engineering Mhemics The Inegrl of Funcion s Funcion Previously, we lerned how o esime he inegrl of funcion f( ) over some inervl y dding he res of finie se of rpezoids h represen he re under
More informationHermite-Hadamard Type Inequalities for the Functions whose Second Derivatives in Absolute Value are Convex and Concave
Applied Mthemticl Sciences Vol. 9 05 no. 5-36 HIKARI Ltd www.m-hikri.com http://d.doi.org/0.988/ms.05.9 Hermite-Hdmrd Type Ineulities for the Functions whose Second Derivtives in Absolute Vlue re Conve
More informationFractional operators with exponential kernels and a Lyapunov type inequality
Abdeljwd Advnces in Difference Equions (2017) 2017:313 DOI 10.1186/s13662-017-1285-0 RESEARCH Open Access Frcionl operors wih exponenil kernels nd Lypunov ype inequliy Thbe Abdeljwd* * Correspondence: bdeljwd@psu.edu.s
More informationWeighted Inequalities for Riemann-Stieltjes Integrals
Aville hp://pvm.e/m Appl. Appl. Mh. ISSN: 93-9466 ol. Ie Decemer 06 pp. 856-874 Applicion n Applie Mhemic: An Inernionl Jornl AAM Weighe Ineqliie or Riemnn-Sielje Inegrl Hüeyin Bk n Mehme Zeki Sriky Deprmen
More informationFractional Calculus. Connor Wiegand. 6 th June 2017
Frcionl Clculus Connor Wiegnd 6 h June 217 Absrc This pper ims o give he reder comforble inroducion o Frcionl Clculus. Frcionl Derivives nd Inegrls re defined in muliple wys nd hen conneced o ech oher
More informationHUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA
Communicions on Sochsic Anlysis Vol 6, No 4 2012 603-614 Serils Publicions wwwserilspublicionscom THE ITÔ FORMULA FOR A NEW STOCHASTIC INTEGRAL HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA Absrc
More informationResearch Article Generalized Fractional Integral Inequalities for Continuous Random Variables
Journl of Proiliy nd Sisics Volume 2015, Aricle ID 958980, 7 pges hp://dx.doi.org/10.1155/2015/958980 Reserch Aricle Generlized Frcionl Inegrl Inequliies for Coninuous Rndom Vriles Adullh Akkur, Zeynep
More informationINTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).
INTEGRALS JOHN QUIGG Eercise. Le f : [, b] R be bounded, nd le P nd Q be priions of [, b]. Prove h if P Q hen U(P ) U(Q) nd L(P ) L(Q). Soluion: Le P = {,..., n }. Since Q is obined from P by dding finiely
More informationNEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX
Journl of Mthemticl Ineulities Volume 1, Number 3 18, 655 664 doi:1.7153/jmi-18-1-5 NEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX SHAHID
More informationIntegral Transform. Definitions. Function Space. Linear Mapping. Integral Transform
Inegrl Trnsform Definiions Funcion Spce funcion spce A funcion spce is liner spce of funcions defined on he sme domins & rnges. Liner Mpping liner mpping Le VF, WF e liner spces over he field F. A mpping
More informationThink of the Relationship Between Time and Space Again
Repor nd Opinion, 1(3),009 hp://wwwsciencepubne sciencepub@gmilcom Think of he Relionship Beween Time nd Spce Agin Yng F-cheng Compny of Ruid Cenre in Xinjing 15 Hongxing Sree, Klmyi, Xingjing 834000,
More informationSolutions to Problems from Chapter 2
Soluions o Problems rom Chper Problem. The signls u() :5sgn(), u () :5sgn(), nd u h () :5sgn() re ploed respecively in Figures.,b,c. Noe h u h () :5sgn() :5; 8 including, bu u () :5sgn() is undeined..5
More informationMTH 146 Class 11 Notes
8.- Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he
More informationarxiv: v1 [math.ca] 28 Jan 2013
ON NEW APPROACH HADAMARD-TYPE INEQUALITIES FOR s-geometrically CONVEX FUNCTIONS rxiv:3.9v [mth.ca 8 Jn 3 MEVLÜT TUNÇ AND İBRAHİM KARABAYIR Astrct. In this pper we chieve some new Hdmrd type ineulities
More informationRIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROXIMATION OF CSISZAR S f DIVERGENCE
SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17 (2009, 3 12 RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROIMATION OF CSISZAR S f DIVERGENCE GEORGE A. ANASTASSIOU Abstrct. Here re estblished vrious tight probbilistic
More informationApproximation and numerical methods for Volterra and Fredholm integral equations for functions with values in L-spaces
Approximion nd numericl mehods for Volerr nd Fredholm inegrl equions for funcions wih vlues in L-spces Vir Bbenko Deprmen of Mhemics, The Universiy of Uh, Sl Lke Ciy, UT, 842, USA Absrc We consider Volerr
More information( ) ( ) ( ) ( ) ( ) ( y )
8. Lengh of Plne Curve The mos fmous heorem in ll of mhemics is he Pyhgoren Theorem. I s formulion s he disnce formul is used o find he lenghs of line segmens in he coordine plne. In his secion you ll
More informationYan Sun * 1 Introduction
Sun Boundry Vlue Problems 22, 22:86 hp://www.boundryvlueproblems.com/conen/22//86 R E S E A R C H Open Access Posiive soluions of Surm-Liouville boundry vlue problems for singulr nonliner second-order
More informationAverage & instantaneous velocity and acceleration Motion with constant acceleration
Physics 7: Lecure Reminders Discussion nd Lb secions sr meeing ne week Fill ou Pink dd/drop form if you need o swich o differen secion h is FULL. Do i TODAY. Homework Ch. : 5, 7,, 3,, nd 6 Ch.: 6,, 3 Submission
More informationIntegral inequalities
Integrl inequlities Constntin P. Niculescu Bsic remrk: If f : [; ]! R is (Riemnn) integrle nd nonnegtive, then f(t)dt : Equlity occurs if nd only if f = lmost everywhere (.e.) When f is continuous, f =.e.
More informationFRACTIONAL EULER-LAGRANGE EQUATION OF CALDIROLA-KANAI OSCILLATOR
Romnin Repors in Physics, Vol. 64, Supplemen, P. 7 77, Dediced o Professor Ion-Ioviz Popescu s 8 h Anniversry FRACTIONAL EULER-LAGRANGE EQUATION OF CALDIROLA-KANAI OSCILLATOR D. BALEANU,,3, J. H. ASAD
More informationNumerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control
Numericl Approximions o Frcionl Problems of he Clculus of Vriions nd Opiml Conrol Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres To cie his version: Shkoor Pooseh, Ricrdo Almeid, Delfim F. M. Torres.
More informationResearch Article The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses
Hindwi Advnce in Mhemicl Phyic Volume 207, Aricle ID 309473, pge hp://doi.org/0.55/207/309473 Reerch Aricle The Generl Soluion of Differenil Equion wih Cpuo-Hdmrd Frcionl Derivive nd Noninnneou Impule
More informationJournal of Mathematical Analysis and Applications. Two normality criteria and the converse of the Bloch principle
J. Mh. Anl. Appl. 353 009) 43 48 Conens liss vilble ScienceDirec Journl of Mhemicl Anlysis nd Applicions www.elsevier.com/loce/jm Two normliy crieri nd he converse of he Bloch principle K.S. Chrk, J. Rieppo
More informationSeptember 20 Homework Solutions
College of Engineering nd Compuer Science Mechnicl Engineering Deprmen Mechnicl Engineering A Seminr in Engineering Anlysis Fll 7 Number 66 Insrucor: Lrry Creo Sepember Homework Soluions Find he specrum
More information0 for t < 0 1 for t > 0
8.0 Sep nd del funcions Auhor: Jeremy Orloff The uni Sep Funcion We define he uni sep funcion by u() = 0 for < 0 for > 0 I is clled he uni sep funcion becuse i kes uni sep = 0. I is someimes clled he Heviside
More informationAsymptotic relationship between trajectories of nominal and uncertain nonlinear systems on time scales
Asympoic relionship beween rjecories of nominl nd uncerin nonliner sysems on ime scles Fim Zohr Tousser 1,2, Michel Defoor 1, Boudekhil Chfi 2 nd Mohmed Djemï 1 Absrc This pper sudies he relionship beween
More informationGeneralized Hermite-Hadamard-Fejer type inequalities for GA-convex functions via Fractional integral
DOI 763/s4956-6-4- Moroccn J Pure nd Appl AnlMJPAA) Volume ), 6, Pges 34 46 ISSN: 35-87 RESEARCH ARTICLE Generlized Hermite-Hdmrd-Fejer type inequlities for GA-conve functions vi Frctionl integrl I mdt
More informationOn New Inequalities of Hermite-Hadamard-Fejér Type for Harmonically s-convex Functions via Fractional Integrals
Krelm en ve Müh. Derg. 6(:879 6 Krelm en ve Mühendili Dergii Jornl home ge: h://fd.en.ed.r eerch Aricle n New Ineliie of HermieHdmrdejér ye for Hrmoniclly Convex ncion vi rcionl Inegrl Keirli İnegrller
More informationSome Improvements of Hölder s Inequality on Time Scales
DOI: 0.55/uom-207-0037 An. Şt. Univ. Ovidius Constnţ Vol. 253,207, 83 96 Some Improvements of Hölder s Inequlity on Time Scles Cristin Dinu, Mihi Stncu nd Dniel Dănciulescu Astrct The theory nd pplictions
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 informationFM Applications of Integration 1.Centroid of Area
FM Applicions of Inegrion.Cenroid of Are The cenroid of ody is is geomeric cenre. For n ojec mde of uniform meril, he cenroid coincides wih he poin which he ody cn e suppored in perfecly lnced se ie, is
More informationHermite-Hadamard type inequalities for harmonically convex functions
Hcettepe Journl o Mthemtics nd Sttistics Volume 43 6 4 935 94 Hermite-Hdmrd type ineulities or hrmoniclly convex unctions İmdt İşcn Abstrct The uthor introduces the concept o hrmoniclly convex unctions
More informationLAGRANGIAN AND HAMILTONIAN MECHANICS WITH FRACTIONAL DERIVATIVES
LAGRANGIAN AND HAMILTONIAN MEHANIS WITH FRATIONAL DERIVATIVES EMIL POPESU 2,1 1 Asronomicl Insiue of Romnin Acdemy Sr uiul de Argin 5, 40557 Buchres, Romni 2 Technicl Universiy of ivil Engineering, Bd
More informationCharacteristic Function for the Truncated Triangular Distribution., Myron Katzoff and Rahul A. Parsa
Secion on Survey Reserch Mehos JSM 009 Chrcerisic Funcion for he Trunce Tringulr Disriuion Jy J. Kim 1 1, Myron Kzoff n Rhul A. Prs 1 Nionl Cener for Helh Sisics, 11Toleo Ro, Hysville, MD. 078 College
More information1.0 Electrical Systems
. Elecricl Sysems The ypes of dynmicl sysems we will e sudying cn e modeled in erms of lgeric equions, differenil equions, or inegrl equions. We will egin y looking fmilir mhemicl models of idel resisors,
More informationIntegral inequalities for n times differentiable mappings
JACM 3, No, 36-45 8 36 Journl of Abstrct nd Computtionl Mthemtics http://wwwntmscicom/jcm Integrl ineulities for n times differentible mppings Cetin Yildiz, Sever S Drgomir Attur University, K K Eduction
More informationResearch Article On New Inequalities via Riemann-Liouville Fractional Integration
Abstrct nd Applied Anlysis Volume 202, Article ID 428983, 0 pges doi:0.55/202/428983 Reserch Article On New Inequlities vi Riemnn-Liouville Frctionl Integrtion Mehmet Zeki Sriky nd Hsn Ogunmez 2 Deprtment
More informationPhysics 2A HW #3 Solutions
Chper 3 Focus on Conceps: 3, 4, 6, 9 Problems: 9, 9, 3, 41, 66, 7, 75, 77 Phsics A HW #3 Soluions Focus On Conceps 3-3 (c) The ccelerion due o grvi is he sme for boh blls, despie he fc h he hve differen
More informationNEW FRACTIONAL DERIVATIVES WITH NON-LOCAL AND NON-SINGULAR KERNEL Theory and Application to Heat Transfer Model
Angn, A., e l.: New Frcionl Derivives wih Non-Locl nd THERMAL SCIENCE, Yer 216, Vol. 2, No. 2, pp. 763-769 763 NEW FRACTIONAL DERIVATIVES WITH NON-LOCAL AND NON-SINGULAR KERNEL Theory nd Applicion o He
More informationChapter 2. Motion along a straight line. 9/9/2015 Physics 218
Chper Moion long srigh line 9/9/05 Physics 8 Gols for Chper How o describe srigh line moion in erms of displcemen nd erge elociy. The mening of insnneous elociy nd speed. Aerge elociy/insnneous elociy
More information2k 1. . And when n is odd number, ) The conclusion is when n is even number, an. ( 1) ( 2 1) ( k 0,1,2 L )
Scholrs Journl of Engineering d Technology SJET) Sch. J. Eng. Tech., ; A):8-6 Scholrs Acdemic d Scienific Publisher An Inernionl Publisher for Acdemic d Scienific Resources) www.sspublisher.com ISSN -X
More informationA Simple Proof of the Jensen-Type Inequality of Fink and Jodeit
Mediterr. J. Mth. 13 (2016, 119 126 DOI 10.1007/s00009-014-0480-4 0378-620X/16/010119-8 published online October 16, 2014 c Springer Bsel 2014 A Simple Proof of the Jensen-Type Inequlity of Fink nd Jodeit
More informationOn Hermite-Hadamard type integral inequalities for functions whose second derivative are nonconvex
Mly J Mt 34 93 3 On Hermite-Hdmrd tye integrl ineulities for functions whose second derivtive re nonconvex Mehmet Zeki SARIKAYA, Hkn Bozkurt nd Mehmet Eyü KİRİŞ b Dertment of Mthemtics, Fculty of Science
More informationSome new integral inequalities for n-times differentiable convex and concave functions
Avilble online t wwwisr-ublictionscom/jns J Nonliner Sci Al, 10 017, 6141 6148 Reserch Article Journl Homege: wwwtjnscom - wwwisr-ublictionscom/jns Some new integrl ineulities for n-times differentible
More informationOn the Pseudo-Spectral Method of Solving Linear Ordinary Differential Equations
Journl of Mhemics nd Sisics 5 ():136-14, 9 ISS 1549-3644 9 Science Publicions On he Pseudo-Specrl Mehod of Solving Liner Ordinry Differenil Equions B.S. Ogundre Deprmen of Pure nd Applied Mhemics, Universiy
More informationOn Tempered and Substantial Fractional Calculus
On Tempered nd Subsnil Frcionl Clculus Jiniong Co,2, Chngpin Li nd YngQun Chen 2, Absrc In his pper, we discuss he differences beween he empered frcionl clculus nd subsnil frcionl operors in nomlous diffusion
More informationFRACTIONAL ORNSTEIN-UHLENBECK PROCESSES
FRACTIONAL ORNSTEIN-ULENBECK PROCESSES Prick Cheridio Deprmen of Mhemics, ET Zürich C-89 Zürich, Swizerlnd dio@mh.ehz.ch ideyuki Kwguchi Deprmen of Mhemics, Keio Universiy iyoshi, Yokohm 3-85, Jpn hide@999.jukuin.keio.c.jp
More informationTemperature Rise of the Earth
Avilble online www.sciencedirec.com ScienceDirec Procedi - Socil nd Behviorl Scien ce s 88 ( 2013 ) 220 224 Socil nd Behviorl Sciences Symposium, 4 h Inernionl Science, Socil Science, Engineering nd Energy
More informationERROR ESTIMATES FOR APPROXIMATING THE FOURIER TRANSFORM OF FUNCTIONS OF BOUNDED VARIATION
ERROR ESTIMATES FOR APPROXIMATING THE FOURIER TRANSFORM OF FUNCTIONS OF BOUNDED VARIATION N.S. BARNETT, S.S. DRAGOMIR, AND G. HANNA Absrc. I his pper we poi ou pproximio for he Fourier rsform for fucios
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationMatrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality
Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]
More informationA study on Hermite-Hadamard type inequalities for s-convex functions via conformable fractional
Sud. Univ. Babeş-Bolyai Mah. 6(7), No. 3, 39 33 DOI:.493/subbmah.7.3.4 A sudy on Hermie-Hadamard ye inequaliies for s-convex funcions via conformable fracional inegrals Erhan Se and Abdurrahman Gözınar
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 informationA Simple Method to Solve Quartic Equations. Key words: Polynomials, Quartics, Equations of the Fourth Degree INTRODUCTION
Ausrlin Journl of Bsic nd Applied Sciences, 6(6): -6, 0 ISSN 99-878 A Simple Mehod o Solve Quric Equions Amir Fhi, Poo Mobdersn, Rhim Fhi Deprmen of Elecricl Engineering, Urmi brnch, Islmic Ad Universi,
More informationWeighted Hardy-Type Inequalities on Time Scales with Applications
Medierr J Mh DOI 0007/s00009-04-054-y c Sringer Bsel 204 Weighed Hrdy-Tye Ineuliies on Time Scles wih Alicions S H Sker, R R Mhmoud nd A Peerson Absrc In his er, we will rove some new dynmic Hrdy-ye ineuliies
More informationSOME PROPERTIES OF GENERALIZED STRONGLY HARMONIC CONVEX FUNCTIONS MUHAMMAD ASLAM NOOR, KHALIDA INAYAT NOOR, SABAH IFTIKHAR AND FARHAT SAFDAR
Inernaional Journal o Analysis and Applicaions Volume 16, Number 3 2018, 427-436 URL: hps://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-427 SOME PROPERTIES OF GENERALIZED STRONGLY HARMONIC
More informationASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS OF FOURTH-ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH REGULARLY VARYING COEFFICIENTS
Elecronic Journl of Differenil Equions, Vol. 06 06), No. 9, pp. 3. ISSN: 07-669. URL: hp://ejde.mh.xse.edu or hp://ejde.mh.un.edu ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS OF FOURTH-ORDER NONLINEAR
More informationTHE BERNOULLI NUMBERS. t k. = lim. = lim = 1, d t B 1 = lim. 1+e t te t = lim t 0 (e t 1) 2. = lim = 1 2.
THE BERNOULLI NUMBERS The Bernoulli numbers are defined here by he exponenial generaing funcion ( e The firs one is easy o compue: (2 and (3 B 0 lim 0 e lim, 0 e ( d B lim 0 d e +e e lim 0 (e 2 lim 0 2(e
More informationBulletin of the. Iranian Mathematical Society
ISSN: 07-060X Print ISSN: 735-855 Online Bulletin of the Irnin Mthemticl Society Vol 3 07, No, pp 09 5 Title: Some extended Simpson-type ineulities nd pplictions Authors: K-C Hsu, S-R Hwng nd K-L Tseng
More informationMAT 266 Calculus for Engineers II Notes on Chapter 6 Professor: John Quigg Semester: spring 2017
MAT 66 Clculus for Engineers II Noes on Chper 6 Professor: John Quigg Semeser: spring 7 Secion 6.: Inegrion by prs The Produc Rule is d d f()g() = f()g () + f ()g() Tking indefinie inegrls gives [f()g
More information