Proceedings of the fifth International symposium "Mathematical inequalities" : Sibiu, 25-27 September 2008

149 Pages·2009·0.546 MB·English

by  , Ana Maria Acu, Emil C. Popa, Dumitru Acu, Daniel Florin Sofonea

”Lucian Blaga” University of Sibiu
Department of Mathematics
and
Romanian Mathematical Scientific Society
Proceedings of The Fifth International
Symposium
”Mathematical Inequalities”
Sibiu, 25 - 27 September 2008
Editors:
Dumitru Acu
Emil C. Popa
Ana-Maria Acu
Florin Sofonea
Copyright (cid:176)c 2009 Publishing of ”Lucian Blaga” University from Sibiu
ISBN 978-973-739-740-9
ISSN 2066-2386
COVER DESIGN AND EDITOR-IN-COMPUTER
Ana Maria Acu
Department of Mathematics
Str. Dr. Ioan Ratiu, No. 5-7
550012-Sibiu, Romania
The Fifth International Symposium ”Mathematical Inequalities”
25 - 27 September 2008, Sibiu, Romania
ORGANIZING COMMITTEE
Professor Ph.D. Dumitru Acu - Head of the Department of Mathematics
Ph.D. Constantin Oprean - Rector of ”Lucian Blaga” University
Professor Ph.D. Dumitru Batˆar - Dean of the Faculty of Sciences
Professor Ph.D. Ilie Barza - Karlstad University, Sweden
Professor Ph.D. Josip E. Peˇcari´c - University of Zagreb, Croatia
Professor Ph.D. Sever S. Dragomir - Victoria University of Technology, Australia
Assoc.Professor Ph.D. Mihai Damian - Strasbourg University, France
Assoc.Professor Ph.D. Sorina Barza - Karlstad University, Sweden
Professor Ph.D. Emil C. Popa - ”Lucian Blaga” University of Sibiu
Professor Ph.D. Vasile Berinde - North University of Baia Mare, Romania
Assoc.Professor Ph.D. Silviu Cr˘aciuna¸s - ”Lucian Blaga” University of Sibiu
Assoc.Professor Ph.D. Florin Sofonea - ”Lucian Blaga” University of Sibiu
Lecturer Ph.D. Ana-Maria Acu - ”Lucian Blaga” University of Sibiu
Lecturer Lecturer Ph.D. Marian Olaru - ”Lucian Blaga” University of Sibiu
Lecturer Ph.D. Adrian Branga - ”Lucian Blaga” University of Sibiu
Lecturer Ph.D. Eugen Constantinescu - ”Lucian Blaga” University of Sibiu
Asist. Petric˘a Dicu - ”Lucian Blaga” University of Sibiu

5
Participants to The Fifth International Symposium
”Mathematical Inequalities” Sibiu, 25 - 27 September 2008
No.
Name/ e-mail Affiliation
Crt.
Ana Maria Acu ”LucianBlaga”University
1
[email protected] fromSibiu,Romania
Mugur Acu ”LucianBlaga”University
2
[email protected] fromSibiu,Romania
Dumitru Acu ”LucianBlaga”University
3
[email protected] ofSibiu,Romania
Adrian Branga ”LucianBlaga”University
4
[email protected] fromSibiu,Romania
Daniel Breaz ”1Decembrie1918”University
5
[email protected] ofAlba-Iulia,Romania
Nicoleta Breaz ”1Decembrie1918”University
6
[email protected] ofAlba-Iulia,Romania
Amelia Bucur ”LucianBlaga”University
7
[email protected] fromSibiu,Romania
Eugen Constantinescu ”LucianBlaga”University
8
[email protected] fromSibiu,Romania
Daniela Dicu LiceulTeoretic”GheorgheLaz˘ar”
9
[email protected] Avrig,Romania
Gheorghe Dicu Gr. S¸c. Forestier
10
CurteadeArge¸s,Romania
Petric˘a Dicu ”LucianBlaga”University
11
[email protected] ofSibiu,Romania
Irina Dorca ”LucianBlaga”University
12
[email protected] ofSibiu,Romania
Eugen Draghici ”LucianBlaga”University
13
[email protected] fromSibiu,Romania
Ali Ebadian UrmiaUniversity,
14
[email protected] Iran
6
Bogdan Gavrea ”TehnicalUniversity”
15
[email protected] ofCluj-Napoca,Romania
Ioan Gavrea ”TehnicalUniversity”
16
[email protected] ofCluj-Napoca,Romania
Heiner Gonska UniversityofDuisburg-Essen
17
[email protected] Germania
Jose Luis Lopez-Bonilla EscuelaSuperiordeIngenieria
18
joseluis.lopezbonilla@gm MecanicayElectricaIns,Mexic
Vasile Mihe¸san ”TehnicalUniversity”
19
[email protected] ofCluj-Napoca,Romania
Nicu¸sor Minculete University”DimitrieCantemir”
20
[email protected] ofBrasov,Romania
Shahram Najafzadeh UniversityofMaragheh,
21
[email protected] Iran
Marian Olaru ”LucianBlaga”University
22
[email protected] fromSibiu,Romania
S¸tefan Poka GheorgheSincaiHighScholl
23
Cluj-Napoca,Romania
Emil C. Popa ”LucianBlaga”University
24
[email protected] fromSibiu,Romania
Ioan Popa EdmondNicolauCollege
25
[email protected] Cluj-Napoca,Romania
Lumini¸ta Preoteasa Gr. S¸c. Forestier
26
CurteadeArge¸s,Romania
COMSATSInstituteof
Arif Rafiq
27 InformationTechnology
arafi[email protected]
Lahore,Pakistan
Sofonea Florin ”LucianBlaga”University
28
sofoneafl[email protected] fromSibiu,Romania
Gheorghe S¸andru S¸coalaGeneral˘aVi¸steadeJos
29
Bra¸sov,Romania
Doru S¸tefanescu UniversityofBucharest
30
[email protected] Romania
Ioan T¸incu ”LucianBlaga”University
31
[email protected] fromSibiu,Romania
Andrei Vernescu UniversityValahia
32
[email protected] ofTargoviste,Romania
7
Contents
A. M. Acu, M. Acu, A. Rafiq – Some inequalities of Ostrowski type in the case of
weighted integrals....................................................................9
D. Acu – Some interesting elementary inequalities..................................23
A. Branga – An inequality for generalized spline functions.........................33
D. Breaz, N. Breaz – Some starlikeness conditions proved by inequalities.........40
I. Dorca – Note on subclass of β-starlike and β-convex functions with negative
coefficients associated with some hyperbola.........................................47
B. Gavrea – On some inequality for convex functions..............................62
I. Gavrea – On some inequalities for convex functions of higher order..............67
H. Gonska, I. Ra¸sa – A Voronovskaya estimate with second order modulus of
smoothness.........................................................................76
V. Mihe¸san – Popoviciu type inequalities for pseudo arithmetic and
geometric means....................................................................91
V. Mihe¸san – Rado type inequalities for weighted power pseudo means ............98
N. Minculete – Several inequalities about arithmetic functions which use the
e-divisors..........................................................................107
N. Minculete, P.Dicu – Inequalities between some arithmetic functions..........116
I. M. Olaru – An integral inequality for convex functions of three order...........126
E.C. Popa – On a problem of A. Shafie...........................................129
F. Sofonea, A.M. Acu, A. Rafiq – An error analysis for a family of four-point
quadrature formulas...............................................................136
I. T¸incu, G. S¸andru – A proof of an inequality..................................146
8
Proceedings of the Fifth International Symposium
”Mathematical Inequalities” Sibiu, 25 - 27 September 2008
SOME INEQUALITIES OF OSTROWSKI TYPE IN THE
CASE OF WEIGHTED INTEGRALS
Ana Maria Acu, Mugur Acu, Arif Rafiq
Abstract
Some new inequalities of Ostrowski type are established. In this
paperweconsideredtheweightedintegralcase. Someofthisinequal-
ities are obtained using the mean value theorems.
2000 Mathematics Subject Classification: 65D30 , 65D32
Key words and phrases: quadrature rule, Ostrowski inequality
1. INTRODUCTION
In 1938, A. M. Ostrowski proved the following classical inequality [5]:
Theorem 1. Let f : [a,b] → R be continuous on [a,b] and differentiable
on (a,b), whose first derivative f(cid:48) : (a,b) → R is bounded on (a,b), i.e.,
|f(cid:48)(x)| ≤ M < ∞. Then,
 (cid:181) (cid:182) 
a+b 2
(cid:175) (cid:90) (cid:175)  x− 
(cid:175) 1 b (cid:175) 1 2 
(cid:175)f(x)− f(t)dt(cid:175) ≤  + (b−a)M,
(cid:175) (cid:175)
b−a 4 (b−a)2 
a
for all x ∈ [a,b], where M is a constant.
9
A.M. Acu, M. Acu, A. Rafiq - Some inequalities of Ostrowski type...
In [10], N. Ujevic proved the following generalization of Ostrowski(cid:48)s
inequality:
Theorem 2.[10] Let I ⊂ R be a open interval and a,b ∈ I, a < b. If
f : I → R is a differentiable function such that γ ≤ f(cid:48)(t) ≤ Γ, for all
t ∈ [a,b], for some constants Γ,γ ∈ R, then we have
(cid:175) (cid:181) (cid:182) (cid:90) (cid:175) (cid:34) (cid:161) (cid:162) (cid:35)
(cid:175) Γ+γ a+b 1 b (cid:175) 1 1 x−a+b 2
(cid:175)f(x)− x− − f(t)dt(cid:175) ≤ (Γ−γ)(b−a) + 2 .
(cid:175) (cid:175)
2 2 b−a 2 4 (b−a)2
a
2. THE CASE OF WEIGHTED INTEGRALS
In this section we obtain some inequalities of Ostrowski type in the case
of weighted integrals. Let w : [a,b] → R be a nonnegative and integrable
function on [a,b] defined by w(t) = (b−t)(t−a).
Theorem 3. Let f : [a,b] → R be a differentiable mapping on (a,b) and
suppose that γ ≤ f(cid:48)(t) ≤ Γ for all t ∈ (a,b). Then we have
(cid:175)(cid:90) (cid:175)
(cid:175) b (b−a)3 (cid:175) 5
(1) (cid:175) w(t)f(t)dt− (f(a)+f(b))(cid:175) ≤ (b−a)4(Γ−γ),
(cid:175) (cid:175)
12 192
a
(2)
(cid:90)
1 b (b−a)3 1
(b−a)4(γ−S) ≤ w(t)f(t)dt− (f(a)+f(b)) ≤ (b−a)4(Γ−S),
12 12 12
a
f(b)−f(a)
where S = .
b−a
Proof. We define
(cid:181) (cid:182) (cid:181) (cid:182) (cid:181) (cid:182)
b−a 2 a+b 1 a+b 3
P(t) = t− − t− .
2 2 3 2
10