Available online at J. Math. Comput. Sci. 4 (2014), No. 3, ISSN:

Size: px
Start display at page:

Download "Available online at J. Math. Comput. Sci. 4 (2014), No. 3, ISSN:"

Transcription

1 Available olie at J. Math. Coput. Sci. (1, No. 3, 9-5 ISSN: ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION FUAD. S. M. AL SARARI 1,, S. LATHA 1 Departet of Studies i Matheatics, Uiversity of Mysore, Maasagagotri, Mysore 57 6, Idia Departet of Matheatics, Yuvaraja s Collega, Uiversity of Mysore, Mysore 57 5, Idia Copyright c 1 Sarari ad Latha. This is a ope access article distributed uder the Creative Coos Attributio Licese, which perits urestricted use, distributio, ad reproductio i ay ediu, provided the origial work is properly cited. Abstract. The object of the preset paper is to derive the itegral represetatio for classes ivolvig the otio of (,-syetrical fuctios with bouded boudary rotatio ad bouded radius rotatio. Soe ore properties like radius of uivalet ad starlike are also ivestigated. Keywords: covex fuctios, starlike fuctios, fuctios of bouded boudary rotatio, bouded radius rotatio, (,-syetric poits. 1 AMS Subject Classificatio: 3C5. 1. Itroductio-preliiaries Let A deote the class of fuctios of for (1 f (z = z + = a z, which are aalytic i the ope uit disk U = z : z C ad z < 1. Let S deote the subclass of A cosistig of all fuctios which are uivalet i U. We also deote by S,K the failiar subclasses of it cosistig of fuctios which are respectively starlike ad covex i Correspodig author Received March 3, 1 9

2 ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION 95 U. It is kow that f (z S if ad oly if z f (z = z exp log( it d(t, for soe (t M. Pichuk [1] geeralized the class S by allowig (t to rage over the class M k. More precisely a fuctio f (z is said to be i the class U k if f (z = z exp z log( it d(t, (t M k i.e, (t is a real valued fuctio of bouded variatio o [,π] satisfyig the coditios. ( d(t =, d(t k. Geoetrically the coditio is that the total variatio of the agle which the radius vector f (re iθ akes whit the positive real axis is bouded above by πk as z describes the circle z = r for z < 1. Thus U k the class of fuctios with radius rotatio bouded by πk. Siilarly V k deotes the class of fuctios f defied o U which ap coforally oto a iage doai of boudary rotatio at ost kπ. Hece f (z V k, if ad oly if f (z = exp log( it d(t, (t M k. It is easy to see that U is the class of starlike fuctios ad V is the class of covex fuctios. Let P k deote the class of fuctios which are aalytic i U ad have the represetatio (3 p(z = ze it d(t, it where (t M k. Clearly we have p = p ad f U k ad V k if ad oly if z f f ad 1 + z f f belog to P k. For p P k, the it has the followig properties (1 p( = 1, ( Rp(z dθ kπ, where k ad z = re iθ, r < 1. Liczberski ad Polubiki [] itroduce the otio of (,-syetrical fuctios ( = 1,,3,..., =,1,.., 1 which is geeralizatio of otios of eve odd ad syetrical fuctios. They also geeralized the kow result that each fuctio defied i syetrical subset ca be u- iquely represeted as the su of a eve fuctio ad odd fuctio.

3 96 FUAD. S. M. AL SARARI, S. LATHA Defiitio 1.1. Let ε = (e πi ad =,1,,.., 1 where is a atural uber. A fuctio f : U C is called (, -syetrical if f (εz = ε f (z, z U. The faily of all (,-syetrical fuctios is deoted be S (,. S (,, S (1, ad S (1, are respectively the classes of eve, odd ad -syetric fuctios. We have the followig decopositio theore. Theore 1.. [] For every appig f : U C, there exists exactly the sequece of (,- syetrical fuctios f,, where f (z = f, (z, = ( f, (z = 1 ε v f (ε v z. v= ( f A ; = 1,,...; =,1,,..., 1. The followig idetities follow directly fro ( (5 f,(z = 1 ε v v f (ε v z, f,(z = 1 v= ε v v f (ε v z, v= (6 f, (ε v z = ε v f, (z, f,(ε v z = ε v v f,(z. Defiitio 1.3. Let U k (, deote the class of fuctios f A satisfies f ( =, f ( = 1 ad, where f, (z is defied by (. z f (z f, (z P k, Defiitio 1.. Let V k (, deote the class of fuctios f A satisfies f ( =, f ( = 1 ad where f, (z is defied by (. (z f (z f,(z P k,

4 ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION 97 Reark 1.5. f V k (, if ad oly if z f U k (,. Spacial cases (i For k = 1, = 1 we get Sigh ad Tygel i [8]. (iifor = = 1 we get paatero i []. (iiifor k =, = 1, = we get Sakaguchi i [13]. I our paper, we also eed the the followig leas. Lea 1.6. [3] Suppose p(z P k. The zp (z R r(k r + kr p(z (1 r (1 kr + r, wher z = r,k ad z < R = k k. For k, zp (z R kr + (8 k + k r kr 3 p(z (1 r (1 kr + r. The above iequality is sharp for fuctio p(z = 1 kz+z 1 z.. Mai results Theore.1. A fuctio f A belogs to U k (,, the (7 f, (z = z exp 1 log( v= where f, (z is defied by ( ad (t is defied (. Proof. Suppose that f U k (,. It follows that πv i(t d(t. (8 z f (z f, (z = p (z, where (9 p (z = 1 Substitutig z by ε v z i (8 respectively 1 + ze it it d(t. (1 zε v f (ε v z f, (ε v z = p (ε v z.

5 98 FUAD. S. M. AL SARARI, S. LATHA The (11 zε v v f (εz v = 1 f, (z 1 + zε v e it 1 zε v e it d(t, or (1 zε v v f (εz v = 1 f, (z 1 + ze πv i(t πv i(t Let (v =,1,,... 1 i (1 ad suig the we get (13 by itegral (13 we have (1 f,(z f, (z 1 z = ze z v= πv i(t πv i(t ( f, (z log = 1 z log[ v= fro (1 we get (7. Hece the proof is coplete. Theore.. A fuctio f A belogs to U k (,, the [ ] (15 f (z = 1 z exp 1 v= log(1 ye i(t πv where f, (z is defied by ( ad (t is defied (. Proof. Suppose that f U k (,. It follows that d(t d(t. d(t 1 z, πv i(t. ]d(t, 1 + ye it 1 ye it d(t dy (16 z f (z f, (z = p (z. The By usig Theore.1, we get (17 f (z = exp 1 v= z f (z = f, (zp (z. log( i(t πv fro (17 we get (15. Hece the proof is coplete. d(t ze it it Corollary.3. For = 1 ad = 1 i Theore.1 we get Paatero []. d(t, By usig the sae ethod i Theore.1, we have the followig corollaries.

6 ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION 99 Corollary.. A fuctio f A belogs to V k (,, the f,(z = exp 1 (18 log( v= where f, (z is defied by ( ad (t is defied (. πv i(t Corollary.5. A fuctio f A belogs to V k (,. The [ ] f (z = 1 z exp 1 i(t πv log(1 ye π (19 d(t z v= where f, (z is defied by ( ad (t is defied (. d(t 1 + ye it 1 ye it Theore.6. A fuctio f A belogs to U k (,. The f, (z i U k. Proof. Suppose that f U k (,. It follows that, d(t dy, ( z f (z f, (z = p (z. Substitutig z by ε v z i ( respectively (1 zε v f (ε v z f, (ε v z = p (ε v z. Now let (v =,1,,... 1 i (1 ad suig the we get ( z f,(z f, (z = 1 1 p (εz. v v= It is vivid that 1 1 v= p (ε v z be bogs to P k. Hece the proof is coplete. Theore.7. Let f U k (, ad let F(z = z f (z. The F(z is starlike for z < r, where r is the least positive root of the equatio 1 3kr + (k + 6r 3kr 3 + r =, where z = r ad k. For k, the F(z is starlike for z < r 3 where r 3 is the least positive root of the equatio 6kr + (1 k + 3k r kr 3 + r =. However the boud r 3 is ot sharp whe k <.

7 5 FUAD. S. M. AL SARARI, S. LATHA Proof. Let f U k (,. The It follows that (3 or ( Hece (5 Therefore, we have ad The F(z = zexp 1 v= zf (z F(z = v= R R zf (z F(z = 1 v= zf (z 1 = R F(z log( ze v= πv i(t πv i(t πv i(t p (ε v z + zp (z p (z. p (ε v z d(t d(t + zp (z p (z, + R zp (z. p (z.p (z. zp (z r(k r + kr p (z (1 r (1 kr + r, where z = r,k, R 1 R v= p (ε v z where z = r < R = k k. Hece R 1 kr + r (1 r, where z = r,k. zf (z 1 kr + r F(z (1 r + r(k r + kr (1 r (1 kr + r (1 kr + r r(k r + kr (1 r (1 kr + r, provided Q(r = 1 3kr + (k + 6r zf (z F(z 3kr 3 + r >. The equatio Q(r = has a uique positive root i (,R. For k, by usig (5, we have zf (z 1 kr + r R F(z (1 r + kr + (8 k + k r kr 3 (1 r (1 kr + r, where z = r < R = k k. Hece R > provided Also D(r = has a root i (,R. zf (z F(z D(r = 6kr + (1 k + 3k r kr 3 + r >.

8 ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION 51 Corollary.8. Let f U k (,. The f is covex for z < r, where r is the least positive root of the equatio 1 3kr + (k + 6r 3kr 3 + r =, where z = r ad k. For k, the f (z is covex for z < r 3, where r 3 is the least positive root of the equatio 6kr + (1 k + 3k r kr 3 + r =. However the boud r 3 is ot sharp whe k <. Theore.9. Let f U k (, ad let F(z = z f, (t f, ( t 1 t dt. The F(z is i V k. Proof. Sice f U k (,, we have ad The or (zf (z F (z (zf (z F (z f, (z = z exp F (z = f,(z f, ( z 1 z = v= = 1 1 v= = 1 1 ze 1 + ze v= i(t πv i(t πv i(t πv i(t πv log( d(t 1 d(t + 1, πv i(t v= v= d(t ze 1 + ze 1 + ze 1 p (εz v + 1 v= p ( εz v. v=. i(t πv i(t πv i(t πv i(t πv d(t d(t Sice p (z P k so 1 v= p (ε v z also i P k, by settig q(z = 1 v= p (ε v z, we have (zf (z F = 1 (z q(z + q( z where q(z P k, = 1 ( ( k + k q 1 (z q (z + 1 ( ( k + k q 1 ( z q ( z (zf (z F (z ( k + q1 (z + q 1 ( z = ( k q (z + q ( z,

9 5 FUAD. S. M. AL SARARI, S. LATHA where q i (z P, i = 1,, also q i(z+q i ( z P, i = 1,. Hece (zf (z ( ( k + k F = w 1 (z w (z, (z where w i (z P, i = 1,. Hece which eas F(z V k. (zf (z F (z P k, Coflict of Iterests The authors declare that there is o coflict of iterests. REFERENCES [1] B. Pichuk, Fuctios of bouded boudary rotatio, Isreal J. Math. 1 (1971, [] V. Paatero, Uber Gebiete vo besehrukter Raddrehug, A. Aced. Sci. Fe. Ser A. 37 (1933, 9. [3] K. S. Padaabha ad R. Parvatha, O fuctios with bouded boudary rotatio, Idia J. pure appl. Math. 6 (1975, [] P. Liczberski, J. Polubiski, O ( j,k-sytrical fuctios, Matheatica Boheica 1 (1995, [5] P. L. Dure, Uivalet Fuctios, Spriger-Verlag (1983. [6] K. I. Noor, O quasi-covex fuctios ad related topics, It. j. Math. Math. Sci. 1 (1987, [7] R. M. Goel ad B. S. Mehrok, Soe ivariace properties of a subclass of close-to-covex fuctios, Idia J. Pure Appl. Math. 1 (1981, [8] R. Sigh ad M. Tygel, O soe uivalet fuctios i the uit disc, Idia. J. Pure. Appl. Math. 1 (1981, [9] Z. G. Wag, C. Y. Gao ad S. M. Yua, O certai subclasses of close-to-covex ad quasi-covex fuctios with respect tok-syetric poits, J. Math. Aal. Appl. 3 (6, [1] S. Owa, M. Nuokawa, H. Saitoh ad H. M. srivastava, Close-to-covexity,starlikess,ad covexity of certai aalytic fuctios, Appl. Math. Lett. 15 (, [11] V. Ravichadra, Starlike ad covex fuctios with respect to cojugate poits, Acta Math. Acad. Paedagog. Nyhazi. (, [1] T. N. Shauga, O α-quasi-covex fuctios, Idia J. Pure Appl. Math. (1989, [13] K. Sakaguchi O certai uivalet appig, J. Math. Soc. Japa. 11 (1959, 7-75.

Generalized Fibonacci-Like Sequence and. Fibonacci Sequence

Generalized Fibonacci-Like Sequence and. Fibonacci Sequence It. J. Cotep. Math. Scieces, Vol., 04, o., - 4 HIKARI Ltd, www.-hiari.co http://dx.doi.org/0.88/ijcs.04.48 Geeralized Fiboacci-Lie Sequece ad Fiboacci Sequece Sajay Hare epartet of Matheatics Govt. Holar

More information

SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR

SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR Joural of the Alied Matheatics Statistics ad Iforatics (JAMSI) 5 (9) No SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR SP GOYAL AND RAKESH KUMAR Abstract Here we

More information

Concavity Solutions of Second-Order Differential Equations

Concavity Solutions of Second-Order Differential Equations Proceedigs of the Paista Academy of Scieces 5 (3): 4 45 (4) Copyright Paista Academy of Scieces ISSN: 377-969 (prit), 36-448 (olie) Paista Academy of Scieces Research Article Cocavity Solutios of Secod-Order

More information

Generalized Class of Sakaguchi Functions in Conic Region

Generalized Class of Sakaguchi Functions in Conic Region Iteratioal Joural of Egieerig ad Techical Research (IJETR) ISSN: 3-0869, Volume-3, Issue-3, March 05 Geeralied Class of Saaguchi Fuctios i Coic Regio Saritha. G. P, Fuad. S. Al Sarari, S. Latha Abstract

More information

On Subordination and Superordination of New Multiplier Transformation

On Subordination and Superordination of New Multiplier Transformation It. J. Ope Probles Copt. Math., Vol., No., March 00 ISSN 998-66; Copyright ICSRS Publicatio, 00 www.i-csrs.org O Subordiatio ad Superordiatio of New Multiplier Trasforatio Aabed Mohaed ad Maslia Darus

More information

On Functions -Starlike with Respect to Symmetric Conjugate Points

On Functions -Starlike with Respect to Symmetric Conjugate Points JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 201, 2534 1996 ARTICLE NO. 0238 O Fuctios -Starlike with Respect to Symmetric Cojugate Poits Mig-Po Che Istitute of Mathematics, Academia Siica, Nakag,

More information

An Application of Generalized Bessel Functions on Certain Subclasses of Analytic Functions

An Application of Generalized Bessel Functions on Certain Subclasses of Analytic Functions Turkish Joural of Aalysis ad Nuber Theory, 5, Vol 3, No, -6 Available olie at htt://ubsscieubco/tjat/3// Sciece ad Educatio ublishig DOI:69/tjat-3-- A Alicatio of Geeralized Bessel Fuctios o Certai Subclasses

More information

ON A SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS DEFINED BY CONVOLUTION. G. Shelake, S. Joshi, S. Halim

ON A SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS DEFINED BY CONVOLUTION. G. Shelake, S. Joshi, S. Halim Acta Uiversitatis Apulesis ISSN: 1582-5329 No. 38/2014 pp. 251-262 ON A SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS DEFINED BY CONVOLUTION G. Shelake, S. Joshi, S. Halim Abstract. I this paper, we itroduce

More information

Lecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces

Lecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such

More information

ON ABSOLUTE MATRIX SUMMABILITY FACTORS OF INFINITE SERIES. 1. Introduction

ON ABSOLUTE MATRIX SUMMABILITY FACTORS OF INFINITE SERIES. 1. Introduction Joural of Classical Aalysis Volue 3, Nuber 2 208), 33 39 doi:0.753/jca-208-3-09 ON ABSOLUTE MATRIX SUMMABILITY FACTORS OF INFINITE SERIES AHMET KARAKAŞ Abstract. I the preset paper, a geeral theore dealig

More information

International Journal of Mathematical Archive-7(6), 2016, Available online through ISSN

International Journal of Mathematical Archive-7(6), 2016, Available online through   ISSN Iteratioal Joural of Mathematical Archive-7(6, 06, 04-0 Available olie through www.ijma.ifo ISSN 9 5046 COMMON FIED POINT THEOREM FOR FOUR WEAKLY COMPATIBLE SELFMAPS OF A COMPLETE G METRIC SPACE J. NIRANJAN

More information

Classes of Uniformly Convex and Uniformly Starlike Functions as Dual Sets

Classes of Uniformly Convex and Uniformly Starlike Functions as Dual Sets JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 16, 4047 1997 ARTICLE NO. AY975640 Classes of Uiformly Covex ad Uiformly Starlike Fuctios as Dual Sets I. R. Nezhmetdiov Faculty of Mechaics ad Mathematics,

More information

Refinements of Jensen s Inequality for Convex Functions on the Co-Ordinates in a Rectangle from the Plane

Refinements of Jensen s Inequality for Convex Functions on the Co-Ordinates in a Rectangle from the Plane Filoat 30:3 (206, 803 84 DOI 0.2298/FIL603803A Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat Refieets of Jese s Iequality for Covex

More information

A new sequence convergent to Euler Mascheroni constant

A new sequence convergent to Euler Mascheroni constant You ad Che Joural of Iequalities ad Applicatios 08) 08:7 https://doi.org/0.86/s3660-08-670-6 R E S E A R C H Ope Access A ew sequece coverget to Euler Mascheroi costat Xu You * ad Di-Rog Che * Correspodece:

More information

Available online through ISSN

Available online through   ISSN Iteratioal Research Joural of Pure Algebra-6(7, 06, 34-347 Aailable olie through wwwrjpaifo ISSN 48 9037 MULTIPLICATIVE HYPER-ZAGREB INDICES AND COINDICES OF GRAPHS: COMPUTING THESE INDICES OF SOME NANOSTRUCTURES

More information

Dominant of Functions Satisfying a Differential Subordination and Applications

Dominant of Functions Satisfying a Differential Subordination and Applications Domiat of Fuctios Satisfyig a Differetial Subordiatio ad Applicatios R Chadrashekar a, Rosiha M Ali b ad K G Subramaia c a Departmet of Techology Maagemet, Faculty of Techology Maagemet ad Busiess, Uiversiti

More information

A NOTE ON SPECTRAL CONTINUITY. In Ho Jeon and In Hyoun Kim

A NOTE ON SPECTRAL CONTINUITY. In Ho Jeon and In Hyoun Kim Korea J. Math. 23 (2015), No. 4, pp. 601 605 http://dx.doi.org/10.11568/kjm.2015.23.4.601 A NOTE ON SPECTRAL CONTINUITY I Ho Jeo ad I Hyou Kim Abstract. I the preset ote, provided T L (H ) is biquasitriagular

More information

1.2 AXIOMATIC APPROACH TO PROBABILITY AND PROPERTIES OF PROBABILITY MEASURE 1.2 AXIOMATIC APPROACH TO PROBABILITY AND

1.2 AXIOMATIC APPROACH TO PROBABILITY AND PROPERTIES OF PROBABILITY MEASURE 1.2 AXIOMATIC APPROACH TO PROBABILITY AND NTEL- robability ad Distributios MODULE 1 ROBABILITY LECTURE 2 Topics 1.2 AXIOMATIC AROACH TO ROBABILITY AND ROERTIES OF ROBABILITY MEASURE 1.2.1 Iclusio-Exclusio Forula I the followig sectio we will discuss

More information

Some Results on Certain Symmetric Circulant Matrices

Some Results on Certain Symmetric Circulant Matrices Joural of Iformatics ad Mathematical Scieces Vol 7, No, pp 81 86, 015 ISSN 0975-5748 olie; 0974-875X prit Pulished y RGN Pulicatios http://wwwrgpulicatioscom Some Results o Certai Symmetric Circulat Matrices

More information

Bertrand s postulate Chapter 2

Bertrand s postulate Chapter 2 Bertrad s postulate Chapter We have see that the sequece of prie ubers, 3, 5, 7,... is ifiite. To see that the size of its gaps is ot bouded, let N := 3 5 p deote the product of all prie ubers that are

More information

Double Derangement Permutations

Double Derangement Permutations Ope Joural of iscrete Matheatics, 206, 6, 99-04 Published Olie April 206 i SciRes http://wwwscirporg/joural/ojd http://dxdoiorg/04236/ojd2066200 ouble erageet Perutatios Pooya aeshad, Kayar Mirzavaziri

More information

On Summability Factors for N, p n k

On Summability Factors for N, p n k Advaces i Dyamical Systems ad Applicatios. ISSN 0973-532 Volume Number 2006, pp. 79 89 c Research Idia Publicatios http://www.ripublicatio.com/adsa.htm O Summability Factors for N, p B.E. Rhoades Departmet

More information

MAJORIZATION PROBLEMS FOR SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING

MAJORIZATION PROBLEMS FOR SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING Iteratioal Joural of Civil Egieerig ad Techology (IJCIET) Volume 9, Issue, November 08, pp. 97 0, Article ID: IJCIET_09 6 Available olie at http://www.ia aeme.com/ijciet/issues.asp?jtypeijciet&vtype 9&IType

More information

Some remarks on the paper Some elementary inequalities of G. Bennett

Some remarks on the paper Some elementary inequalities of G. Bennett Soe rears o the paper Soe eleetary iequalities of G. Beett Dag Ah Tua ad Luu Quag Bay Vieta Natioal Uiversity - Haoi Uiversity of Sciece Abstract We give soe couterexaples ad soe rears of soe of the corollaries

More information

On Weak Concircular Symmetries of (LCS) 2n+1 - Manifolds By D. Narain & S. Yadav D.D.U.Gorakhpur University, India

On Weak Concircular Symmetries of (LCS) 2n+1 - Manifolds By D. Narain & S. Yadav D.D.U.Gorakhpur University, India Global Joural of Sciece Frotier Research Mathematics ad Decisio Scieces Volume Issue 0 Versio.0 0 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. (USA Olie ISSN:

More information

Chapter 1. Complex Numbers. Dr. Pulak Sahoo

Chapter 1. Complex Numbers. Dr. Pulak Sahoo Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler

More information

1. Introduction. g(x) = a 2 + a k cos kx (1.1) g(x) = lim. S n (x).

1. Introduction. g(x) = a 2 + a k cos kx (1.1) g(x) = lim. S n (x). Georgia Mathematical Joural Volume 11 (2004, Number 1, 99 104 INTEGRABILITY AND L 1 -CONVERGENCE OF MODIFIED SINE SUMS KULWINDER KAUR, S. S. BHATIA, AND BABU RAM Abstract. New modified sie sums are itroduced

More information

Analytic Continuation

Analytic Continuation Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for

More information

Research Article Approximate Riesz Algebra-Valued Derivations

Research Article Approximate Riesz Algebra-Valued Derivations Abstract ad Applied Aalysis Volume 2012, Article ID 240258, 5 pages doi:10.1155/2012/240258 Research Article Approximate Riesz Algebra-Valued Derivatios Faruk Polat Departmet of Mathematics, Faculty of

More information

On Order of a Function of Several Complex Variables Analytic in the Unit Polydisc

On Order of a Function of Several Complex Variables Analytic in the Unit Polydisc ISSN 746-7659, Eglad, UK Joural of Iforatio ad Coutig Sciece Vol 6, No 3, 0, 95-06 O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc Rata Kuar Dutta + Deartet of Matheatics, Siliguri

More information

Bounds for the Positive nth-root of Positive Integers

Bounds for the Positive nth-root of Positive Integers Pure Mathematical Scieces, Vol. 6, 07, o., 47-59 HIKARI Ltd, www.m-hikari.com https://doi.org/0.988/pms.07.7 Bouds for the Positive th-root of Positive Itegers Rachid Marsli Mathematics ad Statistics Departmet

More information

REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS

REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS LIVIU I. NICOLAESCU ABSTRACT. We ivestigate the geeralized covergece ad sums of series of the form P at P (x, where P R[x], a R,, ad T : R[x] R[x]

More information

Characterizations Of (p, α)-convex Sequences

Characterizations Of (p, α)-convex Sequences Applied Mathematics E-Notes, 172017, 77-84 c ISSN 1607-2510 Available free at mirror sites of http://www.math.thu.edu.tw/ ame/ Characterizatios Of p, α-covex Sequeces Xhevat Zahir Krasiqi Received 2 July

More information

New Results for the Fibonacci Sequence Using Binet s Formula

New Results for the Fibonacci Sequence Using Binet s Formula Iteratioal Mathematical Forum, Vol. 3, 208, o. 6, 29-266 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/imf.208.832 New Results for the Fiboacci Sequece Usig Biet s Formula Reza Farhadia Departmet

More information

New Inequalities of Hermite-Hadamard-like Type for Functions whose Second Derivatives in Absolute Value are Convex

New Inequalities of Hermite-Hadamard-like Type for Functions whose Second Derivatives in Absolute Value are Convex It. Joural of Math. Aalysis, Vol. 8, 1, o. 16, 777-791 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/ijma.1.1 New Ieualities of Hermite-Hadamard-like Type for Fuctios whose Secod Derivatives i

More information

On Some Properties of Tensor Product of Operators

On Some Properties of Tensor Product of Operators Global Joural of Pure ad Applied Matheatics. ISSN 0973-1768 Volue 12, Nuber 6 (2016), pp. 5139-5147 Research Idia Publicatios http://www.ripublicatio.co/gjpa.ht O Soe Properties of Tesor Product of Operators

More information

Sufficient Conditions for Subordination of Meromorphic Functions

Sufficient Conditions for Subordination of Meromorphic Functions Joural of Mathematics ad Statistics 5 (3):4-45 2009 ISSN 549-3644 2009 Sciece Publicatios Sufficiet Coditios for Subordiatio of Meromorphic Fuctios Rabha W. Ibrahim ad Maslia arus School of Mathematical

More information

lim za n n = z lim a n n.

lim za n n = z lim a n n. Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget

More information

Automated Proofs for Some Stirling Number Identities

Automated Proofs for Some Stirling Number Identities Autoated Proofs for Soe Stirlig Nuber Idetities Mauel Kauers ad Carste Scheider Research Istitute for Sybolic Coputatio Johaes Kepler Uiversity Altebergerstraße 69 A4040 Liz, Austria Subitted: Sep 1, 2007;

More information

On Orlicz N-frames. 1 Introduction. Renu Chugh 1,, Shashank Goel 2

On Orlicz N-frames. 1 Introduction. Renu Chugh 1,, Shashank Goel 2 Joural of Advaced Research i Pure Mathematics Olie ISSN: 1943-2380 Vol. 3, Issue. 1, 2010, pp. 104-110 doi: 10.5373/jarpm.473.061810 O Orlicz N-frames Reu Chugh 1,, Shashak Goel 2 1 Departmet of Mathematics,

More information

New Inequalities For Convex Sequences With Applications

New Inequalities For Convex Sequences With Applications It. J. Ope Problems Comput. Math., Vol. 5, No. 3, September, 0 ISSN 074-87; Copyright c ICSRS Publicatio, 0 www.i-csrs.org New Iequalities For Covex Sequeces With Applicatios Zielaâbidie Latreuch ad Beharrat

More information

Lecture Outline. 2 Separating Hyperplanes. 3 Banach Mazur Distance An Algorithmist s Toolkit October 22, 2009

Lecture Outline. 2 Separating Hyperplanes. 3 Banach Mazur Distance An Algorithmist s Toolkit October 22, 2009 18.409 A Algorithist s Toolkit October, 009 Lecture 1 Lecturer: Joatha Keler Scribes: Alex Levi (009) 1 Outlie Today we ll go over soe of the details fro last class ad ake precise ay details that were

More information

Convergence of Random SP Iterative Scheme

Convergence of Random SP Iterative Scheme Applied Mathematical Scieces, Vol. 7, 2013, o. 46, 2283-2293 HIKARI Ltd, www.m-hikari.com Covergece of Radom SP Iterative Scheme 1 Reu Chugh, 2 Satish Narwal ad 3 Vivek Kumar 1,2,3 Departmet of Mathematics,

More information

Common Fixed Point Theorem in Fuzzy Metric Spaces using weakly compatible maps

Common Fixed Point Theorem in Fuzzy Metric Spaces using weakly compatible maps IJ Iforatio Egieerig ad Electroic Busiess 2014 2 64-69 Published Olie April 2014 i MECS (http://wwwecs-pressorg/) DOI: 105815/ijieeb20140208 Coo Fixed Poit Theore i Fuzzy Metric Spaces usig weakly copatible

More information

Complete Convergence for Asymptotically Almost Negatively Associated Random Variables

Complete Convergence for Asymptotically Almost Negatively Associated Random Variables Applied Mathematical Scieces, Vol. 12, 2018, o. 30, 1441-1452 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.810142 Complete Covergece for Asymptotically Almost Negatively Associated Radom

More information

A Note On L 1 -Convergence of the Sine and Cosine Trigonometric Series with Semi-Convex Coefficients

A Note On L 1 -Convergence of the Sine and Cosine Trigonometric Series with Semi-Convex Coefficients It. J. Ope Problems Comput. Sci. Math., Vol., No., Jue 009 A Note O L 1 -Covergece of the Sie ad Cosie Trigoometric Series with Semi-Covex Coefficiets Xhevat Z. Krasiqi Faculty of Educatio, Uiversity of

More information

Different kinds of Mathematical Induction

Different kinds of Mathematical Induction Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}

More information

SOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS

SOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS ARCHIVU ATHEATICU BRNO Tomus 40 2004, 33 40 SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS E. SAVAŞ AND R. SAVAŞ Abstract. I this paper we itroduce a ew cocept of λ-strog covergece with respect to a Orlicz

More information

COMMON FIXED POINT THEOREMS VIA w-distance

COMMON FIXED POINT THEOREMS VIA w-distance Bulleti of Mathematical Aalysis ad Applicatios ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 3, Pages 182-189 COMMON FIXED POINT THEOREMS VIA w-distance (COMMUNICATED BY DENNY H. LEUNG) SUSHANTA

More information

SZEGO S THEOREM STARTING FROM JENSEN S THEOREM

SZEGO S THEOREM STARTING FROM JENSEN S THEOREM UPB Sci Bull, Series A, Vol 7, No 3, 8 ISSN 3-77 SZEGO S THEOREM STARTING FROM JENSEN S THEOREM Cǎli Alexe MUREŞAN Mai îtâi vo itroduce Teorea lui Jese şi uele coseciţe ale sale petru deteriarea uǎrului

More information

A New Type of q-szász-mirakjan Operators

A New Type of q-szász-mirakjan Operators Filoat 3:8 07, 567 568 https://doi.org/0.98/fil7867c Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat A New Type of -Szász-Miraka Operators

More information

Asymptotic distribution of products of sums of independent random variables

Asymptotic distribution of products of sums of independent random variables Proc. Idia Acad. Sci. Math. Sci. Vol. 3, No., May 03, pp. 83 9. c Idia Academy of Scieces Asymptotic distributio of products of sums of idepedet radom variables YANLING WANG, SUXIA YAO ad HONGXIA DU ollege

More information

Bangi 43600, Selangor Darul Ehsan, Malaysia (Received 12 February 2010, accepted 21 April 2010)

Bangi 43600, Selangor Darul Ehsan, Malaysia (Received 12 February 2010, accepted 21 April 2010) O Cesáro Meas of Order μ for Outer Fuctios ISSN 1749-3889 (prit), 1749-3897 (olie) Iteratioal Joural of Noliear Sciece Vol9(2010) No4,pp455-460 Maslia Darus 1, Rabha W Ibrahim 2 1,2 School of Mathematical

More information

Partial Sums of Starlike and Convex Functions

Partial Sums of Starlike and Convex Functions JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 209, 221227 1997 ARTICLE NO. AY975361 Partial Sums of Starlie ad Covex Fuctios H. Silverma* Departmet of Mathematics, Uiersity of Charlesto, Charlesto,

More information

Fixed Point Theorems for Expansive Mappings in G-metric Spaces

Fixed Point Theorems for Expansive Mappings in G-metric Spaces Turkish Joural of Aalysis ad Number Theory, 7, Vol. 5, No., 57-6 Available olie at http://pubs.sciepub.com/tjat/5//3 Sciece ad Educatio Publishig DOI:.69/tjat-5--3 Fixed Poit Theorems for Expasive Mappigs

More information

Research Article Carleson Measure in Bergman-Orlicz Space of Polydisc

Research Article Carleson Measure in Bergman-Orlicz Space of Polydisc Abstract ad Applied Aalysis Volume 200, Article ID 603968, 7 pages doi:0.55/200/603968 Research Article arleso Measure i Bergma-Orlicz Space of Polydisc A-Jia Xu, 2 ad Zou Yag 3 Departmet of Mathematics,

More information

The Hypergeometric Coupon Collection Problem and its Dual

The Hypergeometric Coupon Collection Problem and its Dual Joural of Idustrial ad Systes Egieerig Vol., o., pp -7 Sprig 7 The Hypergeoetric Coupo Collectio Proble ad its Dual Sheldo M. Ross Epstei Departet of Idustrial ad Systes Egieerig, Uiversity of Souther

More information

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

More information

On Generalized Fibonacci Numbers

On Generalized Fibonacci Numbers Applied Mathematical Scieces, Vol. 9, 215, o. 73, 3611-3622 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.5299 O Geeralized Fiboacci Numbers Jerico B. Bacai ad Julius Fergy T. Rabago Departmet

More information

Some Common Fixed Point Theorems in Cone Rectangular Metric Space under T Kannan and T Reich Contractive Conditions

Some Common Fixed Point Theorems in Cone Rectangular Metric Space under T Kannan and T Reich Contractive Conditions ISSN(Olie): 319-8753 ISSN (Prit): 347-671 Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 397: 7 Certified Orgaizatio) Some Commo Fixed Poit Theorems i Coe Rectagular Metric

More information

Common Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces

Common Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:319-765x Volume 10, Issue 3 Ver II (May-Ju 014), PP 69-77 Commo Coupled Fixed Poit of Mappigs Satisfyig Ratioal Iequalities i Ordered Complex

More information

M-Quasihyponormal Composition Operators. on Weighted Hardy Spaces

M-Quasihyponormal Composition Operators. on Weighted Hardy Spaces It. Joural of Math. Aalysis, Vol., 8, o. 4, 1163-117 M-Quasihypoormal ompositio Operators o Weighted Hardy Spaces S. Paayappa Departmet of Mathematics, Govermet Arts ollege oimbatore 641 18, amil Nadu,

More information

Research Article Carleson Measure in Bergman-Orlicz Space of Polydisc

Research Article Carleson Measure in Bergman-Orlicz Space of Polydisc Hidawi Publishig orporatio Abstract ad Applied Aalysis Volume 00, Article ID 603968, 7 pages doi:0.55/00/603968 Research Article arleso Measure i Bergma-Orlicz Space of Polydisc A-Jia Xu, ad Zou Yag 3

More information

Jacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a

Jacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a Jacobi sybols efiitio Let be a odd positive iteger If 1, the Jacobi sybol : Z C is the costat fuctio 1 1 If > 1, it has a decopositio ( as ) a product of (ot ecessarily distict) pries p 1 p r The Jacobi

More information

Binomial transform of products

Binomial transform of products Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {

More information

TR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT

TR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the

More information

On n-collinear elements and Riesz theorem

On n-collinear elements and Riesz theorem Available olie at www.tjsa.com J. Noliear Sci. Appl. 9 (206), 3066 3073 Research Article O -colliear elemets ad Riesz theorem Wasfi Shataawi a, Mihai Postolache b, a Departmet of Mathematics, Hashemite

More information

Problem. Consider the sequence a j for j N defined by the recurrence a j+1 = 2a j + j for j > 0

Problem. Consider the sequence a j for j N defined by the recurrence a j+1 = 2a j + j for j > 0 GENERATING FUNCTIONS Give a ifiite sequece a 0,a,a,, its ordiary geeratig fuctio is A : a Geeratig fuctios are ofte useful for fidig a closed forula for the eleets of a sequece, fidig a recurrece forula,

More information

Some vector-valued statistical convergent sequence spaces

Some vector-valued statistical convergent sequence spaces Malaya J. Mat. 32)205) 6 67 Some vector-valued statistical coverget sequece spaces Kuldip Raj a, ad Suruchi Padoh b a School of Mathematics, Shri Mata Vaisho Devi Uiversity, Katra-82320, J&K, Idia. b School

More information

SUBCLASSES OF CLOSE.TO-CONVEX FUNCTIONS

SUBCLASSES OF CLOSE.TO-CONVEX FUNCTIONS It. J. ath. & Math. Sci. Vol. 6 No. 3 (1983) 449-458 449 SUBCLASSES OF CLOSE.TO-CONVEX FUNCTIONS E.M. $1LVIA Departmet of Mathematics Uiversity of Califoria, Davis Davis, Califoria 95616 (Received lauary

More information

BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS. 1. Introduction. Throughout the paper we denote by X a linear space and by Y a topological linear

BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS. 1. Introduction. Throughout the paper we denote by X a linear space and by Y a topological linear BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS Abstract. The aim of this paper is to give sufficiet coditios for a quasicovex setvalued mappig to be covex. I particular, we recover several kow characterizatios

More information

A Generalization of Ince s Equation

A Generalization of Ince s Equation Joural of Applied Matheatics ad Physics 7-8 Published Olie Deceber i SciRes. http://www.scirp.org/oural/ap http://dx.doi.org/.36/ap..337 A Geeralizatio of Ice s Equatio Ridha Moussa Uiversity of Wiscosi

More information

Uniform Strict Practical Stability Criteria for Impulsive Functional Differential Equations

Uniform Strict Practical Stability Criteria for Impulsive Functional Differential Equations Global Joural of Sciece Frotier Research Mathematics ad Decisio Scieces Volume 3 Issue Versio 0 Year 03 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic (USA Olie

More information

On a Polygon Equality Problem

On a Polygon Equality Problem JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 223, 6775 998 ARTICLE NO. AY985955 O a Polygo Equality Proble L. Elser* Fakultat fur Matheatik, Uiersitat Bielefeld, Postfach 003, 3350 Bielefeld, Geray

More information

Domination Number of Square of Cartesian Products of Cycles

Domination Number of Square of Cartesian Products of Cycles Ope Joural of Discrete Matheatics, 01,, 88-94 Published Olie October 01 i SciRes http://wwwscirporg/joural/ojd http://dxdoiorg/10436/ojd014008 Doiatio Nuber of Square of artesia Products of ycles Morteza

More information

Complex Analysis Spring 2001 Homework I Solution

Complex Analysis Spring 2001 Homework I Solution Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle

More information

ONE MODULO THREE GEOMETRIC MEAN LABELING OF SOME FAMILIES OF GRAPHS

ONE MODULO THREE GEOMETRIC MEAN LABELING OF SOME FAMILIES OF GRAPHS ONE MODULO THREE GEOMETRIC MEAN LABELING OF SOME FAMILIES OF GRAPHS A.Maheswari 1, P.Padiaraj 2 1,2 Departet of Matheatics,Kaaraj College of Egieerig ad Techology, Virudhuagar (Idia) ABSTRACT A graph G

More information

On the Variations of Some Well Known Fixed Point Theorem in Metric Spaces

On the Variations of Some Well Known Fixed Point Theorem in Metric Spaces Turkish Joural of Aalysis ad Number Theory, 205, Vol 3, No 2, 70-74 Available olie at http://pubssciepubcom/tjat/3/2/7 Sciece ad Educatio Publishig DOI:0269/tjat-3-2-7 O the Variatios of Some Well Kow

More information

Al Lehnen Madison Area Technical College 10/5/2014

Al Lehnen Madison Area Technical College 10/5/2014 The Correlatio of Two Rado Variables Page Preliiary: The Cauchy-Schwarz-Buyakovsky Iequality For ay two sequeces of real ubers { a } ad = { b } =, the followig iequality is always true. Furtherore, equality

More information

COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q n } Sang Pyo Jun

COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q n } Sang Pyo Jun Korea J. Math. 23 2015) No. 3 pp. 371 377 http://dx.doi.org/10.11568/kjm.2015.23.3.371 COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q } Sag Pyo Ju Abstract. I this ote we cosider a geeralized

More information

IJITE Vol.2 Issue-11, (November 2014) ISSN: Impact Factor

IJITE Vol.2 Issue-11, (November 2014) ISSN: Impact Factor IJITE Vol Issue-, (November 4) ISSN: 3-776 ATTRACTIVITY OF A HIGHER ORDER NONLINEAR DIFFERENCE EQUATION Guagfeg Liu School of Zhagjiagag Jiagsu Uiversit of Sciece ad Techolog, Zhagjiagag, Jiagsu 56,PR

More information

ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS

ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS It. J. Cotep. Math. Sci., Vol. 1, 2006, o. 1, 39-43 ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS Qaaruddi ad S. A. Mohiuddie Departet of Matheatics, Aligarh Musli Uiversity Aligarh-202002, Idia sdqaar@rediffail.co,

More information

Math 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version]

Math 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version] Math 4707 Sprig 08 Darij Griberg: idter page Math 4707 Sprig 08 Darij Griberg: idter with solutios [preliiary versio] Cotets 0.. Coutig first-eve tuples......................... 3 0.. Coutig legal paths

More information

Abelian Theorem for Generalized Fourier-Laplace Transform

Abelian Theorem for Generalized Fourier-Laplace Transform Iteratioal Joural o Iovatio ad Applied Studies ISSN 2028-9324 Vol. 8 No. 2 Sep. 204, pp. 549-555 204 Iovative Space o Scietiic Research Jourals http://www.ijias.issr-jourals.org/ Abelia Theore or Geeralized

More information

Probability Theory. Exercise Sheet 4. ETH Zurich HS 2017

Probability Theory. Exercise Sheet 4. ETH Zurich HS 2017 ETH Zurich HS 2017 D-MATH, D-PHYS Prof. A.-S. Szita Coordiator Yili Wag Probability Theory Exercise Sheet 4 Exercise 4.1 Let X ) N be a sequece of i.i.d. rado variables i a probability space Ω, A, P ).

More information

Strong Convergence Theorems According. to a New Iterative Scheme with Errors for. Mapping Nonself I-Asymptotically. Quasi-Nonexpansive Types

Strong Convergence Theorems According. to a New Iterative Scheme with Errors for. Mapping Nonself I-Asymptotically. Quasi-Nonexpansive Types It. Joural of Math. Aalysis, Vol. 4, 00, o. 5, 37-45 Strog Covergece Theorems Accordig to a New Iterative Scheme with Errors for Mappig Noself I-Asymptotically Quasi-Noexpasive Types Narogrit Puturog Mathematics

More information

PAPER : IIT-JAM 2010

PAPER : IIT-JAM 2010 MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure

More information

distinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k)

distinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k) THE TWELVEFOLD WAY FOLLOWING GIAN-CARLO ROTA How ay ways ca we distribute objects to recipiets? Equivaletly, we wat to euerate equivalece classes of fuctios f : X Y where X = ad Y = The fuctios are subject

More information

The Choquet Integral with Respect to Fuzzy-Valued Set Functions

The Choquet Integral with Respect to Fuzzy-Valued Set Functions The Choquet Itegral with Respect to Fuzzy-Valued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to real-valued oadditive set fuctios, such as siged efficiecy measures, has bee used i

More information

A Common Fixed Point Theorem Using Compatible Mappings of Type (A-1)

A Common Fixed Point Theorem Using Compatible Mappings of Type (A-1) Aals of Pure ad Applied Mathematics Vol. 4, No., 07, 55-6 ISSN: 79-087X (P), 79-0888(olie) Published o 7 September 07 www.researchmathsci.org DOI: http://dx.doi.org/0.457/apam.v4a8 Aals of A Commo Fixed

More information

International Journal of Mathematical Archive-4(9), 2013, 1-5 Available online through ISSN

International Journal of Mathematical Archive-4(9), 2013, 1-5 Available online through   ISSN Iteratioal Joural o Matheatical Archive-4(9), 03, -5 Available olie through www.ija.io ISSN 9 5046 THE CUBIC RATE OF CONVERGENCE OF GENERALIZED EXTRAPOLATED NEWTON RAPHSON METHOD FOR SOLVING NONLINEAR

More information

Orthogonal Functions

Orthogonal Functions Royal Holloway Uiversity of odo Departet of Physics Orthogoal Fuctios Motivatio Aalogy with vectors You are probably failiar with the cocept of orthogoality fro vectors; two vectors are orthogoal whe they

More information

An Algorithmist s Toolkit October 20, Lecture 11

An Algorithmist s Toolkit October 20, Lecture 11 18.409 A Algorithist s Toolkit October 20, 2009 Lecture 11 Lecturer: Joatha Keler Scribe: Chaithaya Badi 1 Outlie Today we ll itroduce ad discuss Polar of a covex body. Correspodece betwee or fuctios ad

More information

A GRÜSS TYPE INEQUALITY FOR SEQUENCES OF VECTORS IN NORMED LINEAR SPACES AND APPLICATIONS

A GRÜSS TYPE INEQUALITY FOR SEQUENCES OF VECTORS IN NORMED LINEAR SPACES AND APPLICATIONS A GRÜSS TYPE INEQUALITY FOR SEQUENCES OF VECTORS IN NORMED LINEAR SPACES AND APPLICATIONS S. S. DRAGOMIR Abstract. A discrete iequality of Grüss type i ormed liear spaces ad applicatios for the discrete

More information

ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION OF BALÁZS OPERATORS

ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION OF BALÁZS OPERATORS STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVII, Number 4, December 2002 ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION OF BALÁZS OPERATORS OGÜN DOĞRU Dedicated to Professor D.D. Stacu o his 75

More information

Self-normalized deviation inequalities with application to t-statistic

Self-normalized deviation inequalities with application to t-statistic Self-ormalized deviatio iequalities with applicatio to t-statistic Xiequa Fa Ceter for Applied Mathematics, Tiaji Uiversity, 30007 Tiaji, Chia Abstract Let ξ i i 1 be a sequece of idepedet ad symmetric

More information

Chapter 4. Fourier Series

Chapter 4. Fourier Series Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,

More information

If a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?

If a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero? 2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a

More information

Central limit theorem and almost sure central limit theorem for the product of some partial sums

Central limit theorem and almost sure central limit theorem for the product of some partial sums Proc. Idia Acad. Sci. Math. Sci. Vol. 8, No. 2, May 2008, pp. 289 294. Prited i Idia Cetral it theorem ad almost sure cetral it theorem for the product of some partial sums YU MIAO College of Mathematics

More information

Generalization of Contraction Principle on G-Metric Spaces

Generalization of Contraction Principle on G-Metric Spaces Global Joural of Pure ad Applied Mathematics. ISSN 0973-1768 Volume 14, Number 9 2018), pp. 1159-1165 Research Idia Publicatios http://www.ripublicatio.com Geeralizatio of Cotractio Priciple o G-Metric

More information