GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS

Size: px
Start display at page:

Download "GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS"

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

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 information

Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals

Hermite-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 information

ON 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. 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 information

ON THE OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS

ON 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 information

1. Introduction. 1 b b

1. 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 information

On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function

On 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 information

New Inequalities in Fractional Integrals

New 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 information

Contraction Mapping Principle Approach to Differential Equations

Contraction 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 information

Positive and negative solutions of a boundary value problem for a

Positive 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 information

The Hadamard s inequality for quasi-convex functions via fractional integrals

The 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 information

Research Article New General Integral Inequalities for Lipschitzian Functions via Hadamard Fractional Integrals

Research 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 information

Refinements to Hadamard s Inequality for Log-Convex Functions

Refinements 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 information

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE BOUNDARY-VALUE PROBLEM

EXISTENCE 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 information

Mathematics 805 Final Examination Answers

Mathematics 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 information

On Hadamard and Fejér-Hadamard inequalities for Caputo k-fractional derivatives

On 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 information

New Integral Inequalities of the Type of Hermite-Hadamard Through Quasi Convexity

New 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 information

Some estimates on the Hermite-Hadamard inequality through quasi-convex functions

Some 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 information

FURTHER 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)

FURTHER 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 information

A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR IAN KNOWLES

A 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 information

Convergence of Singular Integral Operators in Weighted Lebesgue Spaces

Convergence 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 information

How to prove the Riemann Hypothesis

How 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 information

How to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh.

How 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 information

Some Inequalities variations on a common theme Lecture I, UL 2007

Some 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 information

EÜFBED - Fen Bilimleri Enstitüsü Dergisi Cilt-Sayı: 3-2 Yıl:

EÜFBED - 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 information

f t f a f x dx By Lin McMullin f x dx= f b f a. 2

f 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 information

Analytic solution of linear fractional differential equation with Jumarie derivative in term of Mittag-Leffler function

Analytic 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 information

Green s Functions and Comparison Theorems for Differential Equations on Measure Chains

Green 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 information

4.8 Improper Integrals

4.8 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 information

Procedia Computer Science

Procedia 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 information

Hermite-Hadamard and Simpson Type Inequalities for Differentiable Quasi-Geometrically Convex Functions

Hermite-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 information

e t dt e t dt = lim e t dt T (1 e T ) = 1

e 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 information

Application on Inner Product Space with. Fixed Point Theorem in Probabilistic

Application 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 information

New general integral inequalities for quasiconvex functions

New 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 information

f (a) + f (b) f (λx + (1 λ)y) max {f (x),f (y)}, x, y [a, b]. (1.1)

f (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 information

CALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION

CALDERON 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 information

HERMITE-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 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 information

LAPLACE TRANSFORM OVERCOMING PRINCIPLE DRAWBACKS IN APPLICATION OF THE VARIATIONAL ITERATION METHOD TO FRACTIONAL HEAT EQUATIONS

LAPLACE 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 information

REAL ANALYSIS I HOMEWORK 3. Chapter 1

REAL 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 information

Solutions for Nonlinear Partial Differential Equations By Tan-Cot Method

Solutions 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 information

ENGR 1990 Engineering Mathematics The Integral of a Function as a Function

ENGR 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 information

Hermite-Hadamard Type Inequalities for the Functions whose Second Derivatives in Absolute Value are Convex and Concave

Hermite-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 information

Fractional operators with exponential kernels and a Lyapunov type inequality

Fractional 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 information

Weighted Inequalities for Riemann-Stieltjes Integrals

Weighted 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 information

Fractional Calculus. Connor Wiegand. 6 th June 2017

Fractional 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 information

HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA

HUI-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 information

Research Article Generalized Fractional Integral Inequalities for Continuous Random Variables

Research 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 information

INTEGRALS. 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. 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 information

NEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX

NEW 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 information

Integral Transform. Definitions. Function Space. Linear Mapping. Integral Transform

Integral 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 information

Think of the Relationship Between Time and Space Again

Think 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 information

Solutions to Problems from Chapter 2

Solutions 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 information

MTH 146 Class 11 Notes

MTH 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 information

arxiv: v1 [math.ca] 28 Jan 2013

arxiv: 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 information

RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROXIMATION OF CSISZAR S f DIVERGENCE

RIEMANN-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 information

Approximation and numerical methods for Volterra and Fredholm integral equations for functions with values in L-spaces

Approximation 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 )

( ) ( ) ( ) ( ) ( ) ( 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 information

Yan Sun * 1 Introduction

Yan 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 information

Average & instantaneous velocity and acceleration Motion with constant acceleration

Average & 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 information

Integral inequalities

Integral 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 information

FRACTIONAL EULER-LAGRANGE EQUATION OF CALDIROLA-KANAI OSCILLATOR

FRACTIONAL 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 information

Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control

Numerical 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 information

Research Article The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

Research 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 information

Journal of Mathematical Analysis and Applications. Two normality criteria and the converse of the Bloch principle

Journal 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 information

September 20 Homework Solutions

September 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 information

0 for t < 0 1 for t > 0

0 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 information

Asymptotic relationship between trajectories of nominal and uncertain nonlinear systems on time scales

Asymptotic 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 information

Generalized Hermite-Hadamard-Fejer type inequalities for GA-convex functions via Fractional integral

Generalized 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 information

On New Inequalities of Hermite-Hadamard-Fejér Type for Harmonically s-convex Functions via Fractional Integrals

On 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 information

Some Improvements of Hölder s Inequality on Time Scales

Some 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 information

NEW 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. := 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 information

FM Applications of Integration 1.Centroid of Area

FM 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 information

Hermite-Hadamard type inequalities for harmonically convex functions

Hermite-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 information

LAGRANGIAN AND HAMILTONIAN MECHANICS WITH FRACTIONAL DERIVATIVES

LAGRANGIAN 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 information

Characteristic Function for the Truncated Triangular Distribution., Myron Katzoff and Rahul A. Parsa

Characteristic 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 information

1.0 Electrical Systems

1.0 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 information

Integral inequalities for n times differentiable mappings

Integral 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 information

Research Article On New Inequalities via Riemann-Liouville Fractional Integration

Research 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 information

Physics 2A HW #3 Solutions

Physics 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 information

NEW FRACTIONAL DERIVATIVES WITH NON-LOCAL AND NON-SINGULAR KERNEL Theory and Application to Heat Transfer Model

NEW 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 information

Chapter 2. Motion along a straight line. 9/9/2015 Physics 218

Chapter 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 information

2k 1. . And when n is odd number, ) The conclusion is when n is even number, an. ( 1) ( 2 1) ( k 0,1,2 L )

2k 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 information

A Simple Proof of the Jensen-Type Inequality of Fink and Jodeit

A 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 information

On Hermite-Hadamard type integral inequalities for functions whose second derivative are nonconvex

On 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 information

Some new integral inequalities for n-times differentiable convex and concave functions

Some 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 information

On the Pseudo-Spectral Method of Solving Linear Ordinary Differential Equations

On 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 information

On Tempered and Substantial Fractional Calculus

On 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 information

FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES

FRACTIONAL 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 information

Temperature Rise of the Earth

Temperature 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 information

ERROR ESTIMATES FOR APPROXIMATING THE FOURIER TRANSFORM OF FUNCTIONS OF BOUNDED VARIATION

ERROR 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 information

Chapter 2. First Order Scalar Equations

Chapter 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 information

Matrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality

Matrix 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 information

A study on Hermite-Hadamard type inequalities for s-convex functions via conformable fractional

A 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 information

Some inequalities of Hermite-Hadamard type for n times differentiable (ρ, m) geometrically convex functions

Some 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 information

A Simple Method to Solve Quartic Equations. Key words: Polynomials, Quartics, Equations of the Fourth Degree INTRODUCTION

A 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 information

Weighted Hardy-Type Inequalities on Time Scales with Applications

Weighted 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 information

SOME PROPERTIES OF GENERALIZED STRONGLY HARMONIC CONVEX FUNCTIONS MUHAMMAD ASLAM NOOR, KHALIDA INAYAT NOOR, SABAH IFTIKHAR AND FARHAT SAFDAR

SOME 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 information

ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS OF FOURTH-ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH REGULARLY VARYING COEFFICIENTS

ASYMPTOTIC 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 information

THE 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. 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 information

Bulletin of the. Iranian Mathematical Society

Bulletin 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 information

MAT 266 Calculus for Engineers II Notes on Chapter 6 Professor: John Quigg Semester: spring 2017

MAT 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