a multidimensional half-discrete hilbert-type inequality and the riemann zeta function
TRANSCRIPT
Applied Mathematics and Computation 225 (2013) 263–277
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Applied Mathematics and Computation
journal homepage: www.elsevier .com/ locate/amc
A multidimensional half-discrete Hilbert-type inequalityand the Riemann zeta function
0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.09.040
⇑ Corresponding author.E-mail addresses: [email protected] (M.Th. Rassias), [email protected] (B. Yang).
Michael Th. Rassias a,⇑, Bicheng Yang b
a Department of Mathematics, ETH-Zürich, CH-8092 Zurich, Switzerlandb Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, PR China
a r t i c l e i n f o a b s t r a c t
Keywords:Gamma functionRiemann zeta functionNumerical estimatesHalf-discrete Hilbert’s inequalityWeight functionHermite–Hadamard’s inequalityHilbert’s operatorEquivalent form
In this paper, by applying methods of weight functions and techniques of real analysis, amore accurate multidimensional half-discrete Hilbert’s inequality with the best possibleconstant factor related to the Riemann zeta function is proved. Equivalent forms and somereverses are also obtained. Additionally, we consider the operator expressions with thenorms and finally present a corollary related to the non-homogeneous kernel.
� 2013 Elsevier Inc. All rights reserved.
1. Introduction
Assuming that p > 1; 1pþ 1
q ¼ 1; f ðxÞ; gðyÞP 0; f 2 LpðRþÞ; g 2 LqðRþÞ,
kfkp ¼Z 1
0f pðxÞdx
� �1p
> 0; kgkq > 0;
we have the following Hardy–Hilbert’s integral inequality (cf. [1]):
Z 10
Z 1
0
f ðxÞgðyÞxþ y
dxdy <p
sinðp=pÞ kfkpkgkq; ð1:1Þ
where the constant factor p= sinðp=pÞ is the best possible.If am; bn P 0; a ¼ famg1m¼1 2 lp
; b ¼ fbng1n¼1 2 lq; kakp ¼
P1m¼1ap
m
� �1p > 0; kbkq > 0, then we still have the following dis-
crete variant of the above inequality with the same best constant factor p= sinðp=pÞ:
X1m¼1X1n¼1
ambn
mþ n<
psinðp=pÞ kakpkbkq ð1:2Þ
Inequalities (1.1) and (1.2) are important in Analysis and its applications (cf. [1–5,31]).In 1998, by introducing an independent parameter k 2 ð0;1�, Yang [6] presented an extension of (1.1) for p ¼ q ¼ 2. In
2009–2011, Yang [3,4] obtained some extensions of (1.1) and (1.2) as follows:
If k1; k2; k 2 R; k1 þ k2 ¼ k; kkðx; yÞ is a non-negative homogeneous function of degree �k,
264 M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277
kðk1Þ ¼Z 1
0kkðt;1Þtk1�1dt 2 Rþ; /ðxÞ ¼ xpð1�k1Þ�1; wðyÞ ¼ yqð1�k2Þ�1; f ðxÞ; gðyÞP 0;
f 2 Lp;/ðRþÞ ¼ f ; kfkp;/ :¼Z 1
0/ðxÞjf ðxÞjpdx
� �1p
<1( )
;
g 2 Lq;wðRþÞ; kfkp;/; kgkq;w > 0;
then we have
Z 10
Z 1
0kkðx; yÞf ðxÞgðyÞdxdy < kðk1Þkfkp;/kgkq;w; ð1:3Þ
where the constant factor kðk1Þ is the best possible. Moreover, if kkðx; yÞ is finite and kkðx; yÞxk1�1 ðkkðx; yÞyk2�1Þ is decreasingwith respect to x > 0 ðy > 0Þ, then for am;bn P 0,
a 2 lp;/ ¼ a; kakp;/ :¼X1n¼1
/ðnÞjanjp !1
p
<1
8<:9=;;
b ¼ fbng1n¼1 2 lq;w; kakp;/; kbkq;w > 0;
we have
X1m¼1X1n¼1
kkðm;nÞambn < kðk1Þkakp;/kbkq;w; ð1:4Þ
where, the constant factor kðk1Þ is still the best possible.Clearly, for
k ¼ 1; k1ðx; yÞ ¼1
xþ y; k1 ¼
1q; k2 ¼
1p;
(1.3) reduces to (1.1), while (1.4) reduces to (1.2). Some other results including the multidimensional Hilbert-type integralinequalities are provided in [7–20].
Regarding the topic of half-discrete Hilbert-type inequalities with non-homogeneous kernels, Hardy, Littlewood andPólya provided a few results in Theorem 351 of [1]. However, it was not proved that the constant factors are the best pos-sible. Yang [21] presented a result with the kernel 1=ð1þ nxÞk by introducing an interval variable and proved that the con-stant factor is the best possible. In 2011 Yang [22] proved the following half-discrete Hardy–Hilbert’s inequality with thebest possible constant factor B k1; k2ð Þ:
Z 10f xð Þ
X1n¼1
an
xþ nð Þkdx < B k1; k2ð Þkfkp;/kakq;w; ð1:5Þ
where,
k1k2 > 0; 0 6 k2 6 1; k1 þ k2 ¼ k; B u;vð Þ ¼Z 1
0
1ð1þ tÞuþv tu�1dtðu;v > 0Þ
is the beta function. Zhong et al. [23–28] investigated several half-discrete Hilbert-type inequalities with particular kernels.Using the method of weight functions and techniques of discrete and integral Hilbert-type inequalities with some addi-
tional conditions on the kernel, the following half-discrete Hilbert-type inequality with a general homogeneous kernel ofdegree �k 2 R and the best constant factor k k1ð Þ is obtainedZ
10f ðxÞ
X1n¼1
kkðx;nÞandx < kðk1Þkfkp;/kakq;w; ð1:6Þ
which is an extension of (1.5) (see Yang and Chen [29]). Moreover, a half-discrete Hilbert-type inequality with a general non-homogeneous kernel and the best constant factor is given by Yang [30].
Remark
(1) Several different kinds of Hilbert-type discrete, half-discrete and integral inequalities with applications have beenproved during the last twenty years. In this paper, special attention is given to new results proved during 2009–2012. Several generalizations, extensions and refinements of Hilbert-type discrete, half-discrete and integral inequal-ities involving many special functions such as beta, gamma, hypergeometric, trigonometric, hyperbolic, Riemann zeta,Bernoulli functions, Bernoulli numbers, Euler constant etc., are discussed (cf. [2,3,5,6,15,18,33,37]).
M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277 265
(2) Yang presented several new results on Hilbert-type operators with general homogeneous kernels of degree real num-bers and two pairs of conjugate exponents as well as the related inequalities (see [3–5,31,32]).
In this paper, by the use of the method of weight functions and techniques of real analysis, a more accurate multidimen-sional half-discrete Hilbert-type inequality with the best possible constant factor related to the Riemann zeta function is pro-vided. Equivalent forms and some reverses are obtained. We also consider the operator expressions with the norm and acorollary related to the non-homogeneous kernel.
2. Some lemmas
Lemma 2.1. Suppose that ð�1ÞihðiÞðtÞ > 0ðt > 0; i ¼ 0;1;2Þ. Then
(i) for b > 0;0 < a 6 1, we have
ð�1Þi di
dxih ðbþ xaÞ
1a
� �> 0 ðx > 0; i ¼ 1;2Þ;
(ii) forR1
12
hðtÞdt <1, we haveZ 1
1hðtÞdt <
X1n¼1
hðnÞ <Z 1
12
hðtÞdt: ð2:1Þ
Proof
(i) It holds
ddx
h ðbþ xaÞ1a
� �¼ h0ððbþ xaÞ
1aÞðbþ xaÞ
1a�1xa�1 < 0;
d2
dx2 h ðbþ xaÞ1a
� �¼ d
dxh0 ðbþ xaÞ
1a
� �ðbþ xaÞ
1a�1xa�1
h i¼ h00 ðbþ xaÞ
1a
� �ðbþ xaÞ
2a�2x2a�2 þ a
1a� 1
� �h0 ðbþ xaÞ
1a
� �ðbþ xaÞ
1a�2x2a�2
þ ða� 1Þh0 ðbþ xaÞ1a
� �ðbþ xaÞ
1a�1xa�2
¼ h00 ðbþ xaÞ1a
� �ðbþ xaÞ
2a�2x2a�2 þ bða� 1Þh0 ðbþ xaÞ
1a
� �ðbþ xaÞ
1a�2xa�2 > 0:
(ii) Since hðtÞ is a decreasing convex function, by Hermite–Hadamard’s inequality (cf. [34]), we haveZ nþ1
nhðtÞdt < hðnÞ <
Z nþ12
n�12
hðtÞdt ðn 2 NÞ
and thus
Z 11hðtÞdt ¼
X1n¼1
Z nþ1
nhðtÞdt <
X1n¼1
hðnÞ <X1n¼1
Z nþ12
n�12
hðtÞdt ¼Z 1
12
hðtÞdt:
Hence (2.1) follows. h
Note. If ð�1ÞihðiÞðtÞ > 0ðt > 0; i ¼ 0;1Þ;R1
0 hðtÞdt <1, since hðnÞ <R n
n�1 hðtÞdt ðn 2 NÞ, we still have
X1n¼1hðnÞ <X1n¼1
Z n
n�1hðtÞdt ¼
Z 1
0hðtÞdt:
If i0; j0 2 N;a; b > 0, we define
kxka :¼Xi0
k¼1
jxkja !1
a
ðx ¼ ðx1; . . . ; xi0Þ 2 Ri0 Þ;
266 M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277
kykb :¼Xj0
k¼1
jykjb
!1b
ðy ¼ ðy1; . . . ; yj0Þ 2 Rj0Þ: ð2:2Þ
Lemma 2.2. If s 2 N; c;M > 0;WðuÞ is a non-negative measurable function in ð0;1�, and DM :¼ x 2 Rsþ; 0 < u ¼
Psi¼1
xiM
� �c6 1
n o,
then we have (cf. [31])
Z� � �Z
DM
WXs
i¼1
xi
M
� �c !
dx1 � � �dxs ¼MsCs 1
c
� �csC s
c
� � Z 1
0WðuÞus
c�1du: ð2:3Þ
In view of (2.3) and the condition, it follows that:
(i) For Rsþ ¼ x 2 Rs
þ; 0 < u ¼Ps
i¼1xiM
� �c6 1ðM !1Þ
n o, we have
Z� � �Z
Rsþ
WXs
i¼1
xi
M
� �c !
dx1 � � �dxs ¼ limM!1
MsCs 1c
� �csC s
c
� � Z 1
0WðuÞus
c�1du; ð2:4Þ
(ii) for fx 2 Rsþ; kxkc P 1g ¼ x 2 Rs
þ; M�c < u ¼Ps
i¼1xiM
� �c6 1ðM !1Þ
n o, setting WðuÞ ¼ 0ðu 2 ð0;M�cÞÞ, we haveZ
� � �ZfkxkcP1g
WXs
i¼1
xi
M
� �c !
dx1 � � � dxs; ðM > 1; M !1Þ ¼ limM!1
MsCsð1cÞ
csC sc
� � Z 1
M�cWðuÞus
c�1du ; ð2:5Þ
(iii) for fx 2 Rsþ; kxkc 6 1g ¼ x 2 Rs
þ; 0 < u ¼Ps
i¼1xiM
� �c6
1Mc
n o, setting WðuÞ ¼ 0ðu 2 ð 1
Mc ;1ÞÞ, we have
Z� � �Zfkxkc61g
WXs
i¼1
xi
M
� �c !
dx1 � � � dxs ¼MsCs 1
c
� �csC s
c
� � Z M�c
0WðuÞus
c�1du; ð2:6Þ
(iv) for
fx 2 Rsþ; xi P 1g ¼ x 2 Rs
þ;s
Mc < u ¼Xs
i¼1
xi
M
� �c6 1ðMc > s; M !1Þ
( );
setting WðuÞ ¼ 0 ðu 2 0; sMc
� �Þ, we have
Z� � �ZfxiP1g
WXs
i¼1
xi
M
� �c !
dx1 � � �dxs ¼ limM!1
MsCs 1c
� �csC s
c
� � Z 1
sM�cWðuÞus
c�1du; ð2:7Þ
Lemma 2.3. If s 2 N; c > 0; e > 0; c ¼ ðc1; . . . ; csÞ 2 0; 12
� s, then we have
Z� � �Zfx2Rs
þ ;xiP1gkxk�s�e
c dx1 � � �dxs ¼Cs 1
c
� �ese=ccs�1C s
c
� � ; ð2:8Þ
Xm
km� ck�s�ec ¼
Cs 1c
� �ese=ccs�1C s
c
� �þ Oð1Þðe! 0þÞ; ð2:9Þ
Proof. By (2.7), it follows that
Z� � �Zfx2Rs
þ ;xiP1gkxk�s�e
c dx1 � � �dxs ¼Z� � �Zfx2Rs
þ ;xiP1gM
Xs
i¼1
xi
M
� �c !1
c0@ 1A�s�e
dx1 � � �dxs
¼ limM!1
MsCs 1c
� �csC s
c
� � Z 1
sM�cðMu1=cÞ�s�e
usc�1du ¼
Cs 1c
� �ese=ccs�1C s
c
� � : ð2:10Þ
Hence we have (2.8).
M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277 267
By (2.1) and (2.8), we obtain
Xm
km� ck�s�ec >
Zfx2Rs
þ ;xiP1þcigkx� ck�s�e
c dx ¼Zfu2Rs
þ ;uiP1gkuk�s�e
c du ¼ limM!1
Z� � �Z
DM
MXs
i¼1
ui
M
� �c !1
c0@ 1A�s�e
du1 � � �dus ¼ limM!1
MsCs 1c
� �csC s
c
� � Z 1
sM�cðMv1=cÞ�s�e
v sc�1dv ¼
Csð1cÞese=ccs�1CðscÞ
: ð2:11Þ
For s ¼ 1; 0 <Pþ1
m¼1km� ck�1�ec <1; for s P 2, by (2.7), we get
0 <X
fm2Ns ;9i0 ;mi0¼1;2gkm� ck�s�e
c 6 aþX
fm2Ns�1 ;miP3g
km� ck�ðs�1Þ�ð1þeÞc
6 aþZfx2Rs�1
þ ;xiP1þcigkx� ck�ðs�1Þ�ð1þeÞ
c dx
¼ aþZfu2Rs�1
þ ;uiP1gkuk�ðs�1Þ�ð1þeÞ
c du
¼ aþCs�1 1
c
� �ð1þ eÞðs� 1Þð1þeÞ=ccs�2C s�1
c
� � < þ1 ða 2 RþÞ:
Thus we obtain
0 <X
fm2Ns ;miP1gkm� ck�s�e
c ¼X
fm2Ns ;9i0 ;mi0¼1;2gkm� ck�s�e
c þX
fm2Ns ;miP3gkm� ck�s�e
c
6eOð1Þ þ Z
fx2Rsþ ;xiP1þcig
kx� ck�s�ec dx ¼ eOð1Þ þ Cs 1
c
� �ese=ccs�1C s
c
� � ðe! 0þÞ: ð2:12Þ
In view of (2.11) and (2.12), (2.9) follows. h
3. A more accurate multidimensional Hilbert-type inequality with parameters
In this section, we set i0; j0 2 N; a; b > 0; p 2 R n f0;1g; 1p þ 1
q ¼ 1.
Definition 3.1. Let x ¼ ðx1; . . . ; xi0Þ 2 Ri0
þ;n ¼ ðn1; . . . ;nj0Þ 2 Nj0 ; s ¼ ðs1; . . . ; sj0
Þ 2 0; 12
� j0 ; d 2 f�1;1g;q > 0;0 < g < r< j0; csc hðuÞ :¼ 2=ðeu � e�uÞ be the hyperbolic cosecant function. Let the two weight coefficients xdðr;nÞ and -dðr; xÞ bedefined as follows:
xdðr;nÞ :¼ kn� skrbZ
Ri0þ
csc hqkn� skgbkxkdga
!kxk�dr�i0
a dx; ð3:1Þ
-dðr; xÞ :¼ kxk�dra
Xn
csc hqkn� skgbkxkdga
!kn� skr�j0
b ; ð3:2Þ
where,P
n ¼P1
nj0¼1 � � �
P1n1¼1:
Lemma 3.2. By the assumptions of Definition 1, if both xdðr;nÞ and -dðr; xÞ are finite, f ðxÞ ¼ f ðx1; . . . ; xi0 ÞP0; an ¼ aðn1 ;...;nj0
Þ P 0, then
(i) for p > 1, we have the following inequality:
J1 :¼X
n
kn� skpr�j0b
½xdðr;nÞ�p�1
ZR
i0þ
csc hqkn� skgbkxkdga
!f ðxÞdx
!p !1p
6
ZR
i0þ
-dðr; xÞkxkpði0þdrÞ�i0a f pðxÞdx
!1p
; ð3:3Þ
J2 :¼Z
Ri0þ
kxk�dqr�i0a
ð-dðr; xÞÞq�1
Xn
csc hqkn� skgbkxkdga
!an
!q
dx
!1q
6
Xn
xdðr;nÞkn� skqðj0�rÞ�j0b aq
n
!1q
; ð3:4Þ
(ii) for p < 0, or 0 < p < 1, we have the reverses of (3.3) and (3.4).
268 M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277
Proof.
(i) For p > 1, by the weighted Hölder’s inequality (cf. [34]), it follows
ZRi0þ
csc hqkn� skgbkxkdga
!f ðxÞdx ¼
ZR
i0þ
csc hqkn� skgbkxkdga
!kxkði0þdrÞ=q
a f ðxÞkn� skðj0�rÞ=p
b
!kn� skðj0�rÞ=p
b
kxkði0þdrÞ=qa
!dx
6
ZR
i0þ
csc hqkn� skgbkxkdga
!kxkði0þdrÞðp�1Þ
a
kn� skj0�rb
f pðxÞdx
!1p
�Z
Ri0þ
csc hqkn� skgbkxkdga
!kn� skðj0�rÞðq�1Þ
b
kxki0þdra
dx
!1q
¼ ðxdðr;nÞÞ1qkn� sk
j0p�rb �
ZR
i0þ
csc hqkn� skgbkxkdga
!kxkði0þdrÞðp�1Þ
a
kn� skj0�rb
f pðxÞdx
!1p
: ð3:5Þ
Then by the Lebesgue term by term integration theorem (cf. [35]), we have ! !1 ! !1
J1 6X
n
ZR
i0þ
csc hqkn� skgbkxkdga
kxkði0þdrÞðp�1Þa
kn� skj0�rb
f pðxÞdxp
¼Z
Ri0þ
Xn
csc hqkn� skgbkxkdga
kxkði0þdrÞðp�1Þa
kn� skj0�rb
f pðxÞdxp
¼Z
Ri0þ
-dðr; xÞkxkpði0þdrÞ�i0a f pðxÞdx
!1p
: ð3:6Þ
Hence, (3.3) follows. Similarly, we obtain ! ! !1
Xncsc hqkn� skgbkxkdga
an 6 ð-dðr; xÞÞ1pkxk
i0qþdra �
Xn
csc hqkn� skgbkxkdga
kn� skðj0�rÞðq�1Þb
kxki0þdra
aqn
q
:
Thus, by the Lebesgue term by term integration theorem, similarly to the way we obtained (3.6), we get (3.4).
(ii) For p < 0, or 0 < p < 1, by the reverse weighted Hölder’s inequality (cf. [34]), we obtain the reverse of (3.5). Then bythe Lebesgue term by term integration theorem, we can obtain the reverse of (3.4). h
Lemma 3.3. With the assumptions of Lemma 3.2, we have
(i) for p > 1, the following inequality equivalent to (3.3) and (3.4) holds true
I :¼X
n
ZR
i0þ
csc hqkn� skgbkxkdga
!anf ðxÞdx 6
ZR
i0þ
-dðr; xÞkxkpði0þdrÞ�i0a f pðxÞdx
!1p X
n
xdðr;nÞkn� skqðj0�rÞ�j0b aq
n
!1q
ð3:7Þ
(ii) for p < 0, or 0 < p < 1, we obtain the reverse of (3.7) which is equivalent to the reverses of (3.3) and (3.4).
Proof.
(i) For p > 1, by Hölder’s inequality (cf. [34]), it follows
I ¼X
n
kn� skj0q�ðj0�rÞb
½xdðr;nÞ�1q
ZR
i0þ
csc hqkn� skgbkxkdga
!f ðxÞdx
!� ðxdðr;nÞÞ
1qkn� skðj0�rÞ�j0
qb an
� �
6 J1
Xn
xdðr;nÞkn� skqðj0�rÞ�j0b aq
n
!1q
:
Then by (3.3), we obtain (3.7).On the other hand, assuming that (3.7) is valid, we set
M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277 269
an :¼kn� skpr�j0
b
½xdðr;nÞ�p�1
ZR
i0þ
csc hqkn� skgbkxkdga
!f ðxÞdx
!p�1
; n 2 Nj0 :
Then it follows that
Jp1 ¼
Xn
xdðr;nÞkn� skqðj0�rÞ�j0b aq
n:
If J1 ¼ 0, then (3.3) is trivially valid;if J1 ¼ 1, then by (3.6), inequality (3.3) reduces to an equality (¼ 1).Suppose that 0 < J1 <1. By (3.7), we obtainX
qðj �rÞ�j q p
0 <nxdðr;nÞkn� sk 0 0b an ¼ J1 ¼ I
6
ZR
i0þ
-dðr; xÞkxkpði0þdrÞ�i0a f pðxÞdx
!1p
�X
n
xdðr;nÞkn� skqðj0�rÞ�j0b aq
n
!1q
<1:
Therefore
J1 ¼X
n
xdðr;nÞkn� skqðj0�rÞ�j0b aq
n
!1p
6
ZR
i0þ
-dðr; xÞkxkpði0þdrÞ�i0a f pðxÞdx
!1p
;
and then (3.3) follows.Hence, (3.3) and (3.7) are equivalent.Similarly, by Hölder’s inequality we obtain
I 6Z
Ri0þ
-dðr; xÞkxkpði0þdrÞ�i0a f pðxÞdx
!1p
J2:
Then by (3.4), we have (3.7). On the other hand, assuming that (3.7) is true, we set
f ðxÞ ¼ kxk�qdr�i0a
½-dðr; xÞ�q�1
Xn
csc hqkn� skgbkxkdga
!g
an
!q�1
ðx 2 Ri0þÞ:
Then, it followsZ
Jq2 ¼R
i0þ
-dðr; xÞkxkpði0þdrÞ�i0a f pðxÞdx:
Similarly, by (3.7), we get
J2 ¼Z
Ri0þ
-dðr; xÞkxkpði0þdrÞ�i0a f pðxÞdx
!1q
6
Xn
xdðr;nÞkn� skqðj0�rÞ�j0b aq
n
!1q
and then (3.4) is equivalent to (3.7).Hence (3.3), (3.4) and (3.7) are equivalent.
(ii) For p < 0, or 0 < p < 1, we similarly obtain the reverse of (3.7), that is equivalent to the reverses of (3.3) and (3.4). h
Lemma 3.4. If
q > 0; er > g > 0; ferg
� �¼X1k¼1
1
ker=g ;
where fð�Þ stands for the Riemann zeta function, then we have
xdðer;nÞ ¼ K2ðerÞ :¼Ci0 1
a
� �ai0�1C i0
a
� � kðerÞ;
270 M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277
kðerÞ :¼ 2
gqer=g Cerg
� �1� 1
2er=g !
ferg
� �2 Rþðn 2 Nj0 Þ: ð3:8Þ
Moreover, there exists a eh 2 ð0;1Þ, such that ð1� ehÞer > g.If 0 < b 6 1;g 6 1;0 < g < er < j0, then we obtain
K1ðerÞð1� herðkxkdaÞÞ < -dðer; xÞ < K1ðerÞðx 2 Ri0þÞ: ð3:9Þ
where,
K1ðerÞ :¼Cj0 1
b
� �bj0�1C j0
b
� � kðerÞ 2 Rþ; ð3:10Þ
herðkxkdaÞ :¼ 1kðerÞ
Z 1
kxkda j�1=b0
csc hqtg
� �t�er�1dt ¼ O
1
kxkdehera
0@ 1A 2 ð0;1Þ: ð3:11Þ
Proof. By (2.4), since d ¼ �1, for
k0ðx; yÞ ¼ csc hqyg
xg
� �; k0ðkxkda; ky� skbÞ ¼ csc h
qkn� skgbkxkdga
!;
we find
xdðer;nÞ ¼ kn� skerb ZR
i0þ
k0 MdXi0
i¼1
xi
M
� �a !d
a
; kn� skb
0@ 1A� 1
Mderþi0Pi0
i¼1xiM
� �a� �derþi0a
dx1 � � � dxi0
¼ limM!1
Mi0 Ci0 1a
� �ai0C i0
a
� � kn� skerb Z 1
0
k0ðMduda; kn� skbÞ
Mderþi0 uderþi0
a
ui0a�1du
¼Ci0 1
a
� �ai0�1C i0
a
� � Z 1
0k0ðv ;1Þv�er�1dv
¼Ci0 1
a
� �ai0�1C i0
a
� � Z 1
0csc h
qvg
� �v�er�1dv:
Thus we obtain
Z 10csc h
qvg
� �v�er�1dv ¼
Z 1
0
2
eq
vg � e�qvg
v�er�1dv
¼u¼ q
vg 2
gqer=gZ 1
0
uðer=gÞ�1dueuð1� e�2uÞ ¼
2
gqer=gZ 1
0
X1k¼0
e�ð2kþ1Þuuðer=gÞ�1du
¼ 2
gqer=gX1k¼0
Z 1
0e�ð2kþ1Þuuðer=gÞ�1du
¼t¼ð2kþ1Þu 2
gqer=gX1k¼0
1
ð2kþ 1Þer=gZ 1
0e�ttðer=gÞ�1dt
¼ 2
gqer=g Cerg
� � X1k¼1
1
ker=g �X1
k¼1
1
ð2kÞer=g0@ 1A ¼ 2
gqer=g Cerg
� �1� 1
2er=g !
ferg
� �:
Hence, (3.8) follows. Moreover, since 0 < b 6 1;g 6 1; er < j0, we get
@
@ycsc h
qyg
xg
� �yer�j0
� �< 0;
@2
@y2 csc hqyg
xg
� �yer�j0
� �> 0
and by (2.1) and (2.4), we obtain
M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277 271
-dðer; xÞ < kxk�dera
Zfy2R
j0þ ;yiP
12g
k0ðkxkda; ky� skbÞdy
ky� skj0�erb
¼ kxk�dera
Zfu2R
j0þ ;uiP
12�sig
k0ðkxkda; kukbÞdu
kukj0�erb
6 kxk�dera
ZR
j0þ
k0ðkxkda; kukbÞdu
kukj0�erb
¼Cj0 1
b
� �bj0�1C j0
b
� � Z 1
0k0ðv;1Þv�er�1dv ¼ K1ðerÞ:
By (2.1), (2.7) and (3.8), we find
-dðer; xÞ > kxk�dera
Zfy2R
j0þ yiP1þsig
k0ðkxkda; ky� skbÞdy
ky� skj0�erb
¼ kxk�dera
Zfu2R
j0þ ;uiP1g
k0ðkxkda; kukbÞdu
kukj0�erb
¼ kxk�dera lim
M!1
Mj0 Cj0 ð1bÞbj0�1Cðj0bÞ
Z 1
j0Mb
k0ðkxkda;Mv1bÞ
ðMv1bÞ
j0�er vj0b�1dv
¼Cj0ð1bÞ
bj0�1Cðj0bÞ
Z kxkda j�1=b0
0k0ðt;1Þt�er�1dt
¼Cj0ð1bÞ
bj0�1Cðj0bÞkðerÞ½1� herðkxkdaÞ� > 0;
0 < herðkxkdaÞ ¼ 1kðerÞ
Z 1
kxkda j�1=b0
csc hðqtgÞt�
er�1dt:
Since we have
tðeh�1Þer csc hðqtgÞ ! 0 ðt ! 0þ; or t !1Þ;
there exists L > 0, such that
tðeh�1Þer csc hðqtgÞ 6 L ðt 2 RþÞ
and then
herðkxkdaÞ 6 LkðerÞ
Z 1
kxkda j�1=b0
t�eher�1dt ¼ LjðeherÞ=b
0
kðerÞeh er 1
kxkdehera
:
Hence (3.9), (3.10) and (3.11) follow. h
Note. The following references [36–43] provide an extensive theory and applications of Analytic Number Theory related tothe Riemann zeta function that offers a source of study for further research on Hilbert – type inequalities.
Setting
UdðxÞ :¼ kxkpði0þdrÞ�i0a ;WðnÞ :¼ kn� skqðj0�rÞ�j0
b ;
eUdðxÞ :¼ ð1� hrðkxkdaÞÞkxkpði0þdrÞ�i0a ; ðhk1 ðkxk
daÞ 2 ð0;1Þ; x 2 Ri0
þÞ;
by Lemmas 3.3 and 3.4, we obtain
Theorem 3.5. Suppose that
a > 0; 0 < b 6 1; s ¼ ðs1; . . . ; sj0 Þ 2 0;12
�j0
; d 2 f�1;1g; q > 0; g 6 1; 0 < g < r < j0:
272 M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277
If p > 1; f ðxÞ ¼ f ðx1; . . . ; xi0ÞP 0; an ¼ aðn1 ;...;nj0Þ P 0;
0 < kfkp;Ud¼
ZR
i0þ
UdðxÞf pðxÞdx
!1p
<1;
0 < kakq;W ¼X
n
WðnÞaqn
!1q
<1;
then we have the following equivalent inequalities with the best possible constant factor KðrÞ:
I ¼X
n
ZR
i0þ
csc hqkn� skgbkxkdga
!anf ðxÞdx < KðrÞkfkp;Ud
kakq;W; ð3:12Þ
J :¼X
n
kn� skpr�j0b
ZR
i0þ
csc hqkn� skgbkxkdga
!f ðxÞdx
!p !1p
< KðrÞkfkp;Ud; ð3:13Þ
H :¼Z
Ri0þ
kxk�dqr�i0a
Xn
csc hqkn� skgbkxkdga
!an
!q
dx
!1q
< KðrÞkakq;W; ð3:14Þ
where
kðrÞ ¼ 2gqr=g C
rg
� �1� 1
2r=g
� �f
rg
� �;
KðrÞ :¼ ðK1ðrÞÞ1pðK2ðrÞÞ
1q ¼
Cj0 ð1bÞbj0�1Cðj0bÞ
!1p
Ci0ð1aÞai0�1Cði0aÞ
!1q
kðrÞ:
In particular, for i0 ¼ j0 ¼ 1; s 2 0; 12
� ;q > 0;0 < g < r < 1,
udðxÞ :¼ xpð1þdrÞ�1; wðnÞ :¼ ðn� sÞqð1�rÞ�1;
we have the following equivalent inequalities with the best possible constant factor kðrÞ:
X1n¼1Z 1
0csc h
qðn� sÞg
xdg
� �anf ðxÞdx < kðrÞkfkp;ud
kakq;w; ð3:15Þ
X1n¼1
ðn� sÞpr�1Z 1
0csc h
qðn� sÞg
xdg
� �f ðxÞdx
� �p !1
p
< kðrÞkfkp;ud; ð3:16Þ
Z 1
0x�dqr�1
X1n¼1
csc hqðn� sÞg
xdg
� �an
!q
dx
!1q
< kðrÞkakq;w: ð3:17Þ
Proof. By Lemmas 3.2, 3.3 and 3.4, we get equivalent inequalities (3.12)–(3.14). By Hölder’s inequality, we have
I 6 JX
n
kn� skqðj0�rÞ�j0b aq
n
!1q
; ð3:18Þ
I 6Z
Ri0þ
kxkpði0þdrÞ�i0a f pðxÞdx
!1p
H: ð3:19Þ
For d0 ¼minfg;r� g; j0 � rg;0 < e < qd0, we set ef ðxÞ; ean as follows:
ef ðxÞ :¼0;0 < kxkda < 1;
kxk�dr�dep�i0
a ; kxkda P 1;
(
ean :¼ kn� skðr�eqÞ�j0
b ; n 2 Nj0 :
M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277 273
Then for 0 < er ¼ r� eq ð< j0Þ, in view of (2.8), (2.9) and (3.11), we find
kef kp;Udkeakq;W ¼
Zfx2R
i0þ ;kxkdaP1g
kxk�i0�dea dx
!1p X
n
kn� sk�j0�eb
!1q
¼ 1e
Ci0ð1aÞai0�1id�=a0 Cði0aÞ
!1p Cj0 ð1bÞ
je=b0 bj0�1Cðj0bÞþ eOð1Þ
!1q
;
eI :¼Z
Ri0þ
Xn
csc hqkn� skgbkxkdga
!eanef ðxÞdx
¼Zfx2R
i0þ ;kxkdaP1g
kxk�i0�dea -dðer; xÞdx
P K1ðerÞZfx2R
i0þ ;kxkdaP1g
kxk�i0�dea 1� O
1
kxkdehera
0@ 1A0@ 1Adx
¼ 1e
K1ðerÞ Ci0ð1aÞai0�1C i0
a
� �� eOerð1Þ0@ 1A:
If there exists a constant K 6 KðrÞ, such that (3.12) is valid when replacing KðrÞ by K, then in particular, by the aboveresults, we have
K1ðerÞ Ci0ð1aÞai0�1id�=a0 Cði0aÞ
� eOerð1Þ !
6 eeI < eKkef kp;Ukeakq;W
¼ KCi0ð1aÞ
ai0�1Cði0aÞ
!1p Cj0 ð1bÞ
je=b0 bj0�1Cðj0bÞþ eOð1Þ
!1q
and thus KðrÞ 6 K ðe! 0þÞ. Hence K ¼ KðrÞ is the best possible constant factor of (3.12).By the equivalency, we can provethat the constant factor KðrÞ in (3.13) and (3.14) is the best possible. Otherwise, we would reach a contradiction by (3.18)and (3.19) that the constant factor KðrÞ in (3.12) is not the best possible. h
Theorem 3.6. By the assumptions of Theorem 3.5, if p < 0 ð0 < q < 1Þ,
f ðxÞ ¼ f ðx1; . . . ; xi0 ÞP 0; an ¼ aðn1 ;...;nj0Þ P 0; 0 < kfkp;Ud
<1; 0 < kakq;W <1;
then we have the equivalent reverses of (3.12)–(3.14) with the best possible constant factor KðrÞ:In particular, for
i0 ¼ j0 ¼ 1; s 2 0;12
�; q > 0; 0 < g < r < 1;
udðxÞ and wðnÞ as indicated in Theorem 3.5, we have the equivalent reverse of3.15,3.16,3.17 with the best possible constant fac-tor kðrÞ.
Proof. We only prove that the constant factor KðrÞ in the reverse of (3.12) is the best possible. The rest is omitted. For
0 < e < qd0; er ¼ r� eq, we set ef ðxÞ; ean as in Theorem 3.5. If there exists a constant K P KðrÞ, such that the reverse inequality
of (3.12) is valid when replacing KðrÞ by K then, we obtain
KCi0 ð1aÞ
ai0�1id�=a0 Cði0aÞ
!1p Cj0ð1bÞ
je=b0 bj0�1Cðj0bÞþ eOð1Þ
!1q
¼ eKkef kp;Ukeakq;W < eeI < eZfx2R
i0þ ;kxkdaP1g
kxk�i0�dea -dðer; xÞdx
< eK2ðerÞZfx2R
i0þ ;kxkdaP1g
kxk�i0�dea dx ¼ K2ðerÞ Ci0 ð1aÞ
ai0�1Cði0aÞ
and thus K 6 KðrÞ ðe! 0þÞ. Hence K ¼ KðrÞ is the best possible constant factor of the reverse inequality of (3.12). h
274 M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277
Theorem 3.7. By the assumptions of Theorem 3.5, if 0 < p < 1 ðq < 0Þ,
f ðxÞ ¼ f ðx1; � � � ; xi0 ÞP 0; an ¼ aðn1 ;���;nj0Þ P 0; 0 < kfk
p;eUd
<1;0 < kakq;W <1;
then we have the following equivalent inequalities with the best possible constant factor KðrÞ:
XnZR
i0þ
csc hqkn� skgbkxkdga
!anf ðxÞdx > KðrÞkfk
p;eUd
kakq;W; ð3:20Þ
Xn
kn� skpr�j0b
ZR
i0þ
csc hqkn� skgbkxkdga
!f ðxÞdx
!p !1p
> KðrÞkfkp;eUd
; ð3:21Þ
ZR
i0þ
kxk�dqr�i0a
ð1� hrðkxkdaÞÞq�1
Xn
csc hðqkn� skgbkxkdga
Þan
!q
dx
!1q
> KðrÞkakq;W: ð3:22Þ
In particular, for i0 ¼ j0 ¼ 1; s 2 ½0; 12�;q > 0;0 < g < r < 1;udðxÞ and wðnÞ as defined in Theorem 3.5,
eudðxÞ ¼ ð1� hrðxdÞÞxpð1þdrÞ�1 ðhrðxdÞ 2 ð0;1Þ; x 2 RþÞ;we have the following equivalent inequalities with the best possible constant factor kðrÞ:
X1n¼1Z 1
0csc h
qðn� sÞg
xdg
� �anf ðxÞdx > kðrÞkfkp;ud
kakq;w; ð3:23Þ
X1n¼1
ðn� sÞpr�1Z 1
0csc h
qðn� sÞg
xdg
� �f ðxÞdx
� �p !1
p
> kðrÞkfkp;eud
; ð3:24Þ
Z 1
0
x�dqr�1
ð1� hrðxdÞÞq�1
X1n¼1
csc hqðn� sÞg
xdg
� �an
!q
dx
!1q
> kðrÞkakq;w: ð3:25Þ
Proof. We only prove that the constant factor KðrÞ in (3.20) is the best possible. The rest is omitted. For0 < e < jqjd0; er ¼ r� e
q ð< j0Þ, we set ef ðxÞ; ean as in Theorem 3.5. Then in view of (2.5), (2.6) and (2.9), we get
kef kp;eUd
keakq;W ¼Zfx2R
i0þ ;kxkdaP1g
kxk�i0�dea ð1� Oðkxk�dhr
a ÞÞdx
!1p X
n
kn� sk�j0�eb
!1q
¼Ci0 ð1aÞ
ai0�1id�=a0 Cði0aÞ� eeOð1Þ !1
p Cj0ð1bÞje=b0 bj0�1Cðj0bÞ
þ eOð1Þ !1
q
:
If there exists a constant K P KðrÞ, such that (3.20) is valid when replacing KðrÞ by K, by (3.9) and the above results, wehave
KCi0ð1aÞ
ai0�1id�=a0 Cði0aÞ� eeOð1Þ !1
p Cj0 ð1bÞje=b0 bj0�1Cðj0bÞ
þ eOð1Þ !1
q
¼ eKkef kp;eUd
keakq;W < eeI < eZfx2R
i0þ ;kxkdaP1g
kxk�i0�dea -dðer; xÞdx
< eK2ðerÞZfx2R
i0þ ;kxkdaP1g
kxk�i0�dea dx ¼ K2ðerÞ Ci0ð1aÞ
ai0�1Cði0aÞ
and then K 6 KðrÞðe! 0þÞ. Hence K ¼ KðrÞ is the best possible constant factor of (3.20). h
For p > 1, we set
UdðxÞ ¼ kxkpði0þdrÞ�i0a ðx 2 Ri0
þÞ;WðnÞ ¼ kn� skqðj0�rÞ�j0b ðn 2 Nj0 Þ
and hence
½WðnÞ�1�p ¼ kn� skpr�j0b ; ½UdðxÞ�1�q ¼ kxk�dqr�j0
a :
We define two real weight normal spaces Lp;UdðRi0þÞ and lq;W as follows:
M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277 275
Lp;UdðRi0þÞ :¼ f ; kfkp;Ud
¼Z
Ri0þ
UdðxÞjf ðxÞjpdx
!1p
<1
0@ 1A;lq;W :¼ a ¼ fang; kakq;w ¼
Xn
WðnÞjanjq !1
q
<1
0@ 1A:
By the assumptions of Theorem 3.5, in view ofJ < KðrÞkfkp;Ud; H < KðrÞkakq;W;
we formulate the following definition:
Definition 3.8. We define a first kind of multidimensional half-discrete Hilbert-type operator T1 : Lp;UdðRi0þÞ ! lp;W1�p as
follows: For f 2 Lp;Ud ðRi0þÞ, there exists a unique representation T1f 2 lp;W1�p , satisfying
ðT1f ÞðnÞ :¼Z
Ri0þ
csc hqðn� sÞg
xdg
� �f ðxÞdx ðn 2 Nj0 Þ: ð3:26Þ
For a 2 lq;W, we define the following formal inner product of T1f and a as follows:
ðT1f ; aÞ :¼X
n
an
ZR
i0þ
csc hqðn� sÞg
xdg
� �anf ðxÞdx: ð3:27Þ
We define a second kind of multidimensional half-discrete Hilbert-type operator
T2 : lq;W ! Lq;U1�qdðRi0þÞ
as follows: For a 2 lq;W, there exists a unique representation
T2a 2 Lq;U1�qdðRi0þÞ;
satisfying
ðT2aÞðxÞ :¼X
n
csc hqðn� sÞg
xdg
� �an ðx 2 Ri0
þÞ: ð3:28Þ
For f 2 Lp;UdðRi0þÞ, we define the following formal inner product of f and T2a as follows:
ðf ; T2aÞ :¼Z
Ri0þ
Xn
csc hqðn� sÞg
xdg
� �anf ðxÞdx: ð3:29Þ
Then by Theorem 3.5, for
0 < kfkp;Ud; kakq;W <1;
we have the following equivalent inequalities:
ðT1f ; aÞ ¼ ðf ; T2aÞ < KðrÞkfkp;Udkakq;W; ð3:30Þ
kT1fkp;W1�p < KðrÞkfkp;Ud; ð3:31Þ
kT2akq;U1�qd
< KðrÞkakq;W: ð3:32Þ
It follows that T1 and T2 are bounded with
kT1k :¼ supf ð–hÞ2Lp;Ud
ðRi0þ Þ
kT1fkp;W1�p
kfkp;Ud
6 KðrÞ;
kT2k :¼ supað–hÞ2lq;W
kT2akq;U1�qd
kakq;W6 KðrÞ:
Since the constant factor KðrÞ in (3.31) and (3.32) is the best possible, we obtain
kT1k ¼ kT2k ¼ KðrÞ ¼Cj0 ð1bÞ
bj0�1Cðj0bÞ
!1p
Ci0 ð1aÞai0�1Cði0aÞ
!1q
kðrÞ: ð3:33Þ
276 M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277
4. A Corollary on d ¼ �1
Corollary 4.1. By the assumptions of Theorems 3.5, 3.6, 3.7, for d ¼ �1,
UðxÞ :¼ kxkpði0�rÞ�i0a and eUðxÞ ¼ ð1� hrðkxk�1
a ÞÞUðxÞ:
(i) If p > 1, then we get the following equivalent inequalities with the best possible constant factor KðrÞ:
XnZR
i0þ
csc h qkxkgakn� skgb� �
anf ðxÞdx < KðrÞkfkp;Ukakq;W; ð4:1Þ
Xn
kn� skpr�j0b
ZR
i0þ
csc h qkxkgakn� skgb� �
f ðxÞdx
!p !1p
< KðrÞkfkp;U; ð4:2Þ
ZR
i0þ
kxkqr�i0a
Xn
csc h qkxkgakn� skgb� �
an
!q
dx
!1q
< KðrÞkakq;W; ð4:3Þ
(ii) if p < 0 ð0 < q < 1Þ, then we have the equivalent reverses of (4.1), (4.2) and (4.3) with the same best possible constantfactor KðrÞ;(iii) if 0 < p < 1 ðq < 0Þ, we obtain the following equivalent reverse inequalities with the best possible constant factorKðrÞ:X
n
ZR
i0þ
csc h qkxkgakn� skgb� �
anf ðxÞdx > KðrÞkfkp;eUkakq;W; ð4:4Þ
Xn
kn� skpr�j0b
ZR
i0þ
csc h qjxkgakn� skgb� �
f ðxÞdx
!p !1p
> KðrÞkfkp;eU ; ð4:5Þ
ZR
i0þ
kxkqr�i0a
ð1� hrðkxk�1a ÞÞ
q�1
Xn
csc h qkxkgakn� skgb� �
an
!q
dx
!1q
> KðrÞkakq;W: ð4:6Þ
In particular, for i0 ¼ j0 ¼ 1; s 2 ½0; 12�;q > 0;0 < g < r < 1, we set
pð1�rÞ�1 �1 qð1�rÞ�1
uðxÞ :¼ x ; euðxÞ ¼ ð1� hrðx ÞÞuðxÞðx 2 RþÞ;wðnÞ ¼ ðn� sÞ ðn 2 NÞ:(i) If p > 1, then we have the following equivalent inequalities with the best possible constant factor kðrÞ:
X1n¼1Z 1
0csc hðqxgðn� sÞgÞanf ðxÞdx < kðrÞkfkp;ukakq;w; ð4:7Þ
X1n¼1
ðn� sÞpr�1Z 1
0csc hðqxgðn� sÞgÞf ðxÞdx
� �p !1
p
< kðrÞkfkp;u; ð4:8Þ
Z 1
0xqr�1
X1n¼1
csc hðqxgðn� sÞgÞan
!q
dx
!1q
< kðrÞkakq;w; ð4:9Þ
(ii) if p < 0 ð0 < q < 1Þ, then we have the equivalent reverses of (4.7), (4.8) and (4.9) with the same best possible constantfactor kðrÞ;(iii) if 0 < p < 1 ðq < 0Þ, then we have the following equivalent reverse inequalities with the best possible constant factorkðrÞ:X1
n¼1
Z 1
0csc hðqxgðn� sÞgÞanf ðxÞdx > kðrÞkfk
p;eukakq;w; ð4:10Þ
X1n¼1
ðn� sÞpr�1Z 1
0csc hðqxgðn� sÞgÞf ðxÞdx
� �p !1
p
> kðrÞkfkp;eu ; ð4:11Þ
M.Th. Rassias, B. Yang / Applied Mathematics and Computation 225 (2013) 263–277 277
Z 1
0
xqr�1
ð1� hrðx�1ÞÞq�1
X1n¼1
csc hðqxgðn� sÞgÞan
!q
dx
!1q
> kðrÞkakq;w: ð4:12Þ
Acknowledgments
We would like to express our thanks to professors Tserendorj Batbold, Mario Krnic, Jin-Wen Lan as well as the referee forreading the manuscript and for their very helpful remarks.
M.Th. Rassias: This work is supported by the Greek State Scholarship Foundation (IKY). B. Yang: This work is supported by2012 Knowledge Construction Special Foundation Item of Guangdong Institution of Higher Learning College and University(No. 2012KJCX0079).
References
[1] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1934.[2] D.S. Mitrinovic, J.E. Pecaric, A.M. Fink, Inequalities Involving Functions and their Integrals and Derivatives, Kluwer Acaremic Publishers, Boston, 1991.[3] B.C. Yang, Hilbert-Type Integral Inequalities, Bentham Science Publishers Ltd., Sharjah, 2009.[4] B.C. Yang, Discrete Hilbert-Type Inequalities, Bentham Science Publishers Ltd., Sharjah, 2011.[5] B.C. Yang, The Norm of Operator and Hilbert-Type Inequalities, Science Press, Beijin, 2009 (China).[6] B.C. Yang, On Hilbert’s integral inequality, J. Math. Anal. Appl. 220 (1998) 778–785.[7] B.C. Yang, I. Brnetic, M. Krnic, J.E. Pecaric, Generalization of Hilbert and Hardy–Hilbert integral inequalities, Math. Inequal. Appl. 8 (2) (2005) 259–272.[8] M. Krnic, J.E. Pecaric, Hilbert’s inequalities and their reverses, Publ. Math. Debrecen 67 (3–4) (2005) 315–331.[9] B.C. Yang, Th.M. Rassias, On the way of weight coefficient and research for Hilbert-type inequalities, Math. Inequal. Appl. 6 (4) (2003) 625–658.
[10] B.C. Yang, Th.M. Rassias, On a Hilbert-type integral inequality in the subinterval and its operator expression, Banach J. Math. Anal. 4 (2) (2010) 100–110.
[11] L. Azar, On some extensions of Hardy–Hilbert’s inequality and applications, J. Inequal. Appl. (2009). no. 546829.[12] B. Arpad, O. Choonghong, Best constant for certain multilinear integral operator, J. Inequal. Appl. (2006). no. 28582.[13] J.C. Kuang, L. Debnath, On Hilbert’s type inequalities on the weighted Orlicz spaces. pacific, J. Appl. Math. 1 (1) (2007) 95–103.[14] W.Y. Zhong, The Hilbert-type integral inequality with a homogeneous kernel of Lambda-degree, J. Inequal. Appl. (2008). no. 917392.[15] Y. Hong, On Hardy–Hilbert integral inequalities with some parameters, J. Inequal. Pure Appl. Math. 6 (4) (2005) 1–10. Art. 92.[16] W.Y. Zhong, B.C. Yang, On multiple Hardy–Hilbert’s integral inequality with kernel, J. Inequal. Appl., vol. 2007, Art. ID 27962, p. 17. http://
www.journalofinequalitiesandapplications.com/search/results?terms=27962.[17] B.C. Yang, M. Krnic, On the norm of a mult-dimensional Hilbert-type operator, Sarajevo J. Math. 7 (20) (2011) 223–243.[18] M. Krnic, J.E. Pecaric, P. Vukovic, On some higher-dimensional Hilbert’s and Hardy–Hilbert’s type integral inequalities with parameters, Math. Inequal.
Appl. 11 (2008) 701–716.[19] M. Krnic, P. Vukovic, On a multidimensional version of the Hilbert-type inequality, Anal. Mathematica 38 (2012) 291–303.[20] Y.J. Li, B. He, On inequalities of Hilbert’s type, Bull. Australian Math. Soc. 76 (1) (2007) 1–13.[21] B.C. Yang, A mixed Hilbert-type inequality with a best constant factor, Int. J. Pure Appl. Math. 20 (3) (2005) 319–328.[22] B.C. Yang, A half-discrete Hilbert-type inequality, J. Guangdong Univ. Educ. 31 (3) (2011) 1–7.[23] W.Y. Zhong, A mixed Hilbert-type inequality and its equivalent forms, J. Guangdong Univ. Educ. 31 (5) (2011) 18–22.[24] W.Y. Zhong, A half discrete Hilbert-type inequality and its equivalent forms, J. Guangdong Univ. Educ. 32 (5) (2012) 8–12.[25] J.H. Zhong, B.C. Yang, On an extension of a more accurate Hilbert-type inequality, J. Zhejiang Univ. (Science Edition) 35 (2) (2008) 121–124.[26] W.Y. Zhong, B.C. Yang, A best extension of Hilbert inequality involving several parameters, J. Jinan Univ. (Natural Science) 28 (1) (2007) 20–23.[27] W.Y. Zhong, B.C. Yang, A reverse Hilbert’s type integral inequality with some parameters and the equivalent forms, Pure Appl. Math. 24 (2) (2008) 401–
407.[28] W.Y. Zhong, B.C. Yang, On the reverse Hardy–Hilbert’s integral inequality, J. Southwest Univ. 29 (4) (2007) 44–48.[29] B.C. Yang, Q. Chen, A half-discrete Hilbert-type inequality with a homogeneous kernel and an extension, J. Inequal. Appl. 124 (2011), http://dx.doi.org/
10.1186/1029-242X-2011-124.[30] B.C. Yang, A half-discrete Hilbert-type inequality with a non-homogeneous kernel and two variables, Mediterr. J. Math. (2012), http://dx.doi.org/
10.1007/s00009- 012- 0213-50. online first.[31] B.C. Yang, Hilbert-type integral operators: norms and inequalities, in: P.M. Pardalos, P.G. Georgiev, H.M. Srivastava (Eds.), Nonlinear Analysis-Stability,
Approximation, and Inequalities, Springer, New York, 2012, pp. 771–859.[32] B.C. Yang, Two Types of Multiple Half-Discrete Hilbert-Type Inequalities, Lambert Acad. Publ., 2012.[33] Y.L. Pan, H.T. Wang, F.T. Wang, On Complex Functions, Science Press, Beijing, 2006.[34] J.C. Kuang, Applied Inequalities, Shangdong Science Technic Press, Jinan, China, 2004.[35] J.C. Kuang, Applied Inequalities, Shangdong Science Technic Press, Jinan, China, 2004.[36] G.V. Milovanovic, M.Th. Rassias (Eds.), Analytic Number Theory, Approximation Theory and Special Functions. Springer, New York, in press.[37] T.M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1984.[38] H.M. Edwards, Riemann’s Zeta Function, Dover Publ, New York, 1974.[39] P. Erd}os, J. Suranyi, Topics in the Theory of Numbers, Springer-Verlag, New York, 2003.[40] G.H. Hardy, E.W. Wright, fifth ed., An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1979.[41] H. Iwaniec, E. Kowalski, Analytic Number Theory, vol. 53, Amer. Math. Soc., Colloquium Publications, Rhode Island, 2004.[42] E. Landau, Elementary Number Theory, second ed., Chelsea, New York, 1966.[43] S.J. Miller, R. Takloo-Bighash, An Invitation to Modern Number Theory, Princeton Univ. Press, Princeton and Oxford, 2006.