a multidimensional half-discrete hilbert-type inequality and the riemann zeta function

15
A multidimensional half-discrete Hilbert-type inequality and the Riemann zeta function Michael Th. Rassias a,, Bicheng Yang b a Department of Mathematics, ETH-Zürich, CH-8092 Zurich, Switzerland b Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, PR China article info Keywords: Gamma function Riemann zeta function Numerical estimates Half-discrete Hilbert’s inequality Weight function Hermite–Hadamard’s inequality Hilbert’s operator Equivalent form abstract In this paper, by applying methods of weight functions and techniques of real analysis, a more accurate multidimensional half-discrete Hilbert’s inequality with the best possible constant factor related to the Riemann zeta function is proved. Equivalent forms and some reverses are also obtained. Additionally, we consider the operator expressions with the norms and finally present a corollary related to the non-homogeneous kernel. Ó 2013 Elsevier Inc. All rights reserved. 1. Introduction Assuming that p > 1; 1 p þ 1 q ¼ 1; f ðxÞ; gðyÞ P 0; f 2 L p ðR þ Þ; g 2 L q ðR þ Þ, kf k p ¼ Z 1 0 f p ðxÞdx 1 p > 0; kgk q > 0; we have the following Hardy–Hilbert’s integral inequality (cf. [1]): Z 1 0 Z 1 0 f ðxÞgðyÞ x þ y dxdy < p sinðp=pÞ kf k p kgk q ; ð1:1Þ where the constant factor p= sinðp=pÞ is the best possible. If a m ; b n P 0; a ¼fa m g 1 m¼1 2 l p ; b ¼fb n g 1 n¼1 2 l q ; kak p ¼ P 1 m¼1 a p m 1 p > 0; kbk q > 0, then we still have the following dis- crete variant of the above inequality with the same best constant factor p= sinðp=pÞ: X 1 m¼1 X 1 n¼1 a m b n m þ n < p sinðp=pÞ kak p kbk q 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 0; 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 k 1 ; k 2 ; k 2 R; k 1 þ k 2 ¼ k; k k ðx; yÞ is a non-negative homogeneous function of degree k, 0096-3003/$ - see front matter Ó 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). Applied Mathematics and Computation 225 (2013) 263–277 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

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Applied Mathematics and Computation 225 (2013) 263–277

Contents lists available at ScienceDirect

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 1

0

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:

X1m¼1

X1n¼1

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 1

0

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

X1m¼1

X1n¼1

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 1

0f 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

1

0f 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 1

1hðtÞdt 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

X1n¼1

hðnÞ <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

ZR

i0þ

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

Xn

csc 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 <n

xdðr;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

Jq

2 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 1

0csc 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:

X1n¼1

Z 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:

Xn

ZR

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:

X1n¼1

Z 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 of

J < 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:

Xn

ZR

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:

X1n¼1

Z 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).

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