Octogon Mathematical Magazine - Volume 17, No. 1, April 2009

484 Pages·2009·2.93 MB·English

by  Mihály Bencze (Editor in Chief)

OCTOGON
Mathematical Magazine
Brassó Kronstadt Braşov
Vol. 17, No. 1, April 2009
Editor-In-Chief Mihály Bencze
Str. Hărmanului 6, 505600 Săcele-Négyfalu,
Jud. Braşov, Romania
Phone: [phone]
E-mail: [email protected]
[email protected]

Editorial Board
Šefket Arslanagić, University of Sarajevo, Sarajevo, Bosnia and Preda Mihăilescu, Matematisches Institut, Universitaet
Herzegovina Goettingen
D.M. Bătineţu-Giurgiu, Editorial Board of Josip Pečaric, University of Zagreb,
Gazeta Matematică, Bucharest, Romania Zagreb, Croatia
José Luis Díaz-Barrero, Universitat Politechnica de Catalunya, Themistocles M. Rassias, National Technical University of
Barcelona, Spain Athens, Athen, Greece
Zhao Changjian, China Jiliang University, Hangzhou, China Ovidiu T. Pop, National College
Mihai Eminescu, Satu Mare, Romania
Constantin Corduneanu, University of Texas József Sándor, Babeş-Bolyai University,
at Arlington, Arlington, USA Cluj-Napoca, Romania
Sever S. Dragomir, School of Computer Science and Florentin Smarandache, University of
Mathematics, Victoria University, Melbourne, Australia New Mexico, New Mexico, USA
Péter Körtesi, University of Miskolc, László Zsidó, University of Rome,
Miskolc, Hungary Tor Vergata, Roma, Italy
Maohua Le, Zhangjiang Normal College, Zhangjiang, China Shanhe Wu, Longyan University,
Longyan, China
The Octogon Mathematical Magazine is a continuation of Gamma Mathematical Magazine (1978-1989)
Manuscripts: Should be sent either to the Editor-in-chief. Instruction for authors are
given inside the back cover.
ISSN 1222-5657
ISBN 978-973-88255-5-0
© Fulgur Publishers
Contents
Gao Mingzhe
Some new Hilbert type inequalities and applications . . . . . . . 4
Song-Zhimin, Dou-Xiangkai and Yin Li
On some new inequalities for the Gamma function . . . . . . . 14
Mih´aly Bencze
The integral method in inequalities theory . . . . . . . . . . . . 19
Vlad Ciobotariu-Boer
Hermite-Hadamard and Fej´er Inequalities for Wright-Convex
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Mih´aly Bencze, Nicu¸sor Minculete and Ovidiu T. Pop
New inequalities for the triangle . . . . . . . . . . . . . . . . . . 70
J.Earnest Lazarus Piriyakumar and R. Helen
Helly‘s theorem on fuzzy valued functions . . . . . . . . . . . . 90
Mih´aly Bencze
About AM-HM inequality . . . . . . . . . . . . . . . . . . . . . 106
Mih´aly Bencze
A Generalization of the logarithmic and the Gauss means . . . 117
J. O. Ad´en´ıran, J. T. Akinmoyewa, A. R. T. S.`ol´ar`ın, T. G. Jaiy´eo.la´
On some algebraic properties of generalized groups . . . . . . . 125
Mih´aly Bencze
New identities and inequalities in triangle . . . . . . . . . . . . 135
Mih´aly Bencze and D.M. Ba˘tine¸tu-Giurgiu
A cathegory of inequalities . . . . . . . . . . . . . . . . . . . . . 149
O.O. Fabelurin, A. G. Adeagbo-Sheikh
On Hardy-type integral inequalities involving many functions . 164
Mih´aly Bencze
A method to generate new inequalities in triangle . . . . . . . . 173
Jos´e Luis D´ıaz-Barrero and Eusebi Jarauta-Bragulat
Some Related Results to CBS Inequality . . . . . . . . . . . . . 182
Mih´aly Bencze and Wei-Dong Jiang
One some new type inequalities in triangle . . . . . . . . . . . . 189
Yu-Lin Wu
Two geometric inequalities involved two triangles . . . . . . . . 193
Mih´aly Bencze and Nicu¸sor Minculete
Some applications of certain inequalities . . . . . . . . . . . . . 199
Mih´aly Bencze and Zhao Changjian
A refinement of Jensen‘s inequality . . . . . . . . . . . . . . . . 209
2 Octogon Mathematical Magazine, Vol. 17, No.1, April 2009
Fuhua Wei and Shanhe Wu
Generalizations and analogues of the Nesbitt’s inequality . . . . 215
Hui-Hua Wu and Shanhe Wu
Various proofs of the Cauchy-Schwarz inequality . . . . . . . . 221
Mih´aly Bencze
About a trigonometrical inequality . . . . . . . . . . . . . . . . 230
Jian Liu
On Bergstro¨m’s inequality involving six numbers . . . . . . . . 237
Mih´aly Bencze and Yu-Dong Wu
New refinements for some classical inequalities . . . . . . . . . 250
J´ozsef S´andor and Ildik´o Bakcsi
On the equation ax2+by2 = z2, where a+b = c2 . . . . . . . . 255
Mih´aly Bencze and Yu-Dong Wu
About Dumitru Acu‘s inequality . . . . . . . . . . . . . . . . . . 257
J´ozsef S´andor
Euler and music. A forgotten arithmetic function by Euler . . 265
Mih´aly Bencze
About a partition inequality . . . . . . . . . . . . . . . . . . . . 272
J´ozsef S´andor
A divisibility property of σ (n) . . . . . . . . . . . . . . . . . . 275
k
Mih´aly Bencze and D.M. Ba˘tine¸tu-Giurgiu
New refinements for AM-HM type inequality . . . . . . . . . . . 277
K.P.Pandeyend
Characteristics of triangular numbers . . . . . . . . . . . . . . 282
J´ozsef S´andor
A double-inequality for σ (n) . . . . . . . . . . . . . . . . . . . 285
k
Nicu¸sor Minculete
Improvement of one of Sa´ndor‘s inequalities . . . . . . . . . . . 288
Sˇefket Arslanagi´c
About one algebraic inequality . . . . . . . . . . . . . . . . . . . 291
J´ozsef S´andor
On certain inequalities for the σ function . . . . . . . . . . . . 294
−
J´ozsef S´andor
A note on the inequality (x +x +...+x )2 n x2+...+x2 297
1 2 n ≤ 1 n
J´ozsef S´andor
(cid:0) (cid:1)
A note on inequalities for the logarithmic function . . . . . . . 299
J´ozsef S´andor
On the inequality (f(x))k < f xk . . . . . . . . . . . . . . . . 302
(cid:0) (cid:1)
Contents 3
J´ozsef S´andor
A note on Bang‘s and Zsigmond‘s theorems . . . . . . . . . . . 304
Mih´aly Bencze
Jo´zsef Wildt International Mathematical Competition . . . . . . 306
Book reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Proposed problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
R´obert Sza´sz and Aurel P´al Kup´an
Solution of the OQ. 2283 . . . . . . . . . . . . . . . . . . . . . 464
Kramer Alp´ar-Vajk
A conjecture on a number theoretical function and the OQ. 1240471
Kramer Alp´ar-Vajk
The solution of OQ 1156 . . . . . . . . . . . . . . . . . . . . . 475
Kramer Alp´ar-Vajk
The solution of OQ 1141 . . . . . . . . . . . . . . . . . . . . . 476
Gabriel T. Pr˘ajitura and Tsvetomira Radeva
A logarithmic equation (OQ 19) . . . . . . . . . . . . . . . . . . 477
OCTOGON MATHEMATICAL MAGAZINE
Vol. 17, No.1, April 2009, pp 4-13
4
ISSN 1222-5657, ISBN 978-973-88255-5-0,
www.hetfalu.ro/octogon
Some new Hilbert type inequalities and
applications
Gao Mingzhe1
ABSTRACT. In this paper it is shown that some new Hilbert type integral
inequalities can be established by introducing a proper logarithm function. And the
constant factor is proved to be the best possible. In particular, for case , the
classical Hilbert inequality and its equivalent form are obtained. As applications,
some new inequalities which they are equivalent each other are built.
1. INTRODUCTION
Let f(x),g(x) L2(0,+ ). Then
∈ ∞
1 1
2 2
∞ ∞f(x)g(x) ∞ ∞
dxdy π f2(x)dx g2(x)dx (1.1)
x+y ≤    
Z0 Z0 Z0  Z0 
This is the famous Hilbert integral inequality, where the coefficient π is the
   
best possible.
In the papers [1-2], the following inequality of the form
1 1
∞ ∞ ln x f(x)g(y) ∞ 2 ∞ 2
y
dxdy π2 f2(x)dx g2(x)dx (1.2)
(cid:16) (cid:17)x y ≤    
Z0 Z0 − Z0  Z0 
was established, and the coefficient π2 is also thebestpossible. 
Owing to the importance of the Hilbert inequality and the Hilbert type
inequality in analysis and applications, some mathematicians have been
studying them. Recently, various improvements and extensions of (1.1) and
(1.2) appear in a great deal of papers (see [3]-[8]etc.).
The aim of the present paper is to build some new Hilbert type integral
inequalities by introducing a proper integral kernel function and by using the
1Received: 20.02.2009
2000 Mathematics Subject Classification. 26D15
Keywordsandphrases. Hilberttypeintegralinequality,logarithmfunction,euler
number, the best constant, equivalent inequalities.
Some new Hilbert type inequalities and applications 5
technique of analysis, and to discuss the constant factor of which is related
to the Euler number, and then to study some equivalent forms of them.
In the sake of convenience, we introduce some notations and define some
functions.
Let 0 < α < 1 and n be a positive integer. Define a function ζ by
∗
∞ ( 1)k
ζ∗(n,α) = (α−+k)n. (1.3)
k=0
X
And further define the function ζ by
2
1
ζ = (2n)! 2ζ 2n+1, ,(n N ) (1.4)
2 ∗ 0
2 ∈
(cid:26) (cid:18) (cid:19)(cid:27)
In order to prove our main results, we need the following lemmas.
Lemma 1.1. Let 0 < α < 1 and n be a nonnegative integer. Then
1
n
1 1
tα 1 ln dt = n!ζ (n+1,α). (1.5)
− ∗
t 1+t
Z (cid:18) (cid:19)
0
where ζ is defined by (1.3).
∗
This result has been given in the paper [9]. Hence its proof is omitted here.
Lemma 1.2. With the assumptions as Lemma 1.1, then
∞ 1 2n 1
uα 1 ln du = (2n)! ζ (2n+1,α)+ζ (2n+1,1 α) (1.6)
− ∗ ∗
u 1+u { − }
Z (cid:18) (cid:19)
0
where ζ is defined by (1.3).
∗
Proof. It is easy to deduce that
1
∞ 1 2n 1 1 2n 1
uα 1 ln du = uα 1 ln du+
− −
u 1+u u 1+u
Z (cid:18) (cid:19) Z (cid:18) (cid:19)
0 0
1
∞ 1 2n 1 1 2n 1
+ uα 1 ln du = uα 1 ln du+
− −
u 1+u u 1+u
Z (cid:18) (cid:19) Z (cid:18) (cid:19)
1 0
6 Gao Mingzhe
1 1
2n
1 1 1
+ v α(lnv)2n dv = uα 1 ln du+
− −
1+v u 1+u
Z Z (cid:18) (cid:19)
0 0
1
2n
1 1
+ v(1 α) 1 ln dv.
− −
v 1+v
Z (cid:18) (cid:19)
0
By using Lemma 1.1, the equality (1.6) is obtained at once.
0
Throughout the paper, we define ln x = 1, when x = y.
y
(cid:16) (cid:17)
2. MAIN RESULTS
We are ready now to formulate our main results.
Theorem 2.1. Let f and g be two real functions, and n be a nonnegative
integer, If
∞ ∞
f2(x)dx < + and g2(x)dx < + , then
∞ ∞
Z Z
0 0
2n
∞ ∞ ln x f(x)g(y)
y
dxdy
(cid:16) (cid:17)x+y ≤
Z Z
0 0
1 1
2 2
∞ ∞
π2n+1E f2(x)dx g2(x)dx , (2.1)
n
≤    
(cid:0) (cid:1)Z0  Z0 
where the constant factor π2n+1E is thebest possible, and that E = 1 and
n 0
E is the Euler number, viz. E = 1, E = 5, E = 61, E = 1385,
n 1 2 3 4
E = 50521, etc.
5
Proof. We may apply the Cauchy inequality to estimate the left-hand side of
(2.1) as follows:
2n
∞ ∞ ln x f(x)g(y)
y
dxdy =
(cid:16) (cid:17)x+y
Z Z
0 0
Some new Hilbert type inequalities and applications 7
1 1
= ∞ ∞ ln xy 2n 2 x 41 f(x) ln xy 2n 2 y 41 gydxdy
(cid:16)x+(cid:17)y  y (cid:16)x+(cid:17)y  x ≤
Z0 Z0 (cid:18) (cid:19) (cid:16) (cid:17)
   
   
1 1
∞ ∞ ln xy 2n x 21 2 ∞ ∞ ln xy 2n x 21 2
f2(x)dxdy g2(x)dxdy =
≤  (cid:16)x+(cid:17)y y   (cid:16)x+(cid:17)y y 
Z0 Z0 (cid:18) (cid:19)  Z0 Z0 (cid:18) (cid:19) 
 1  1 
2 2
∞ ∞
= ω(x)f2(x)dx ω(x)g2(x)dx (2.2)
   
Z Z
0 0
   
lnx 2n 1
where ω(x) = ∞ y x 2 dy,
“ x+”y y
0
By using LemmRa 1.2, it i(cid:16)s e(cid:17)asy to deduce that
2n
∞ ln xy x 21 ∞ 1 1 2n 1
ω(x) = x(cid:16) 1+(cid:17)y y dy = u−2 ln u 1+udu = ζ2. (2.3)
Z x (cid:18) (cid:19) Z (cid:18) (cid:19)
0 0
(cid:0) (cid:1)
where ζ is defined by (1.4). Based on (1.3) and (1.4), we have
2
1 ∞ ( 1)k
ζ2 = (2n)! 2ζ∗ 2n+1, 2 = (2n)!2 1 +−k 2n+1 =
(cid:26) (cid:18) (cid:19)(cid:27) k=0 2
X
∞ ( 1)k (cid:0) (cid:1)
= (2n)!22n+2 − .
(2k+1)2n+1
k=0
X
It is known from the paper [10] that
∞ ( 1)k π2n+1
− = E . (2.4)
1 +k 2n+1 22n+2(2n)! n
k=0 2
X
where En is the Euler num(cid:0)ber,vi(cid:1)z. E1 = 1, E2 = 5, E3 = 61, E4 = 1385,
E = 50521, etc.
5
Since ∞ (2−k1+)1k = π4, we can define E0 = 1. As thus, the relation (2.4) is also
k=0
valid wPhen n = 0. So, we get from (2.3) and (2.4) that
8 Gao Mingzhe
ω(x) = π2n+1E , (2.5)
n
It follows from (2.2) and (2.5) that the inequality (2.1) is valid.
It remains to need only to show that π2n+1E in (2.1) is the best possible.
n
0 if x (0,1)
ε > 0. Define two functions by f(x) = ∈ and
1+ε
∀ ( x− 2 if x [1, )
∈ ∞
0 if y (0,1) e
g(y) = ∈ . It is easy to deduce that
1+ε
( y− 2 if y [1, )
∈ ∞
+ +
e ∞ ∞ 1
f2(x)dx = g2(y)dy = .
ε
Z Z
0 0
f e
If π2n+1E is not the best possible, then there exists C > 0, such that
n
C < π2n+1E and
n
2n
∞ ∞ ln x f(x)g(y)
y
S f,g = dxdy
(cid:16) (cid:17)x+y ≤
(cid:16) (cid:17) Z0 Z0 e e
e e
1 1
2 2
∞ ∞ C
C f2(x)dx g2(y)dy = . (2.6)
≤     ε
Z Z
0 0
 f   e 
On the other hand, we have
∞ ∞ x−1+2ε ln xy 2ny−1+2ε
S f,g = n o(cid:26)(cid:16) (cid:17) (cid:27)dxdy =
x+y
(cid:16) (cid:17) Z0 Z0
e e
∞ ∞ ln xy 2ny−1+2ε 1+ε
=  (cid:16) x(cid:17)+y dy x− 2 dx =
Z0 Z0 n o
= ∞ ∞ ln u1 2nu−1+2εdu x 1 ε dx =
− −
 1+u 
Z0 Z0 (cid:0) (cid:1) (cid:8) (cid:9)
 1 ∞ 1+ε 1 2n 1
= u− 2 ln du. (2.7)
ε u 1+u
Z (cid:18) (cid:19)
0