Hilbert Inequalities Related to Generalized Hypergeometric Functions

Mathematica Balkanica —————————

New Series Vol. 22, 2008, Fasc. 3-4

Hilbert Inequalities Related to Generalized Hypergeometric Functions m Fn ˇ Mario Krnić 1 , Zivorad Tomovski

2

and Josip Peˇcarić

3

Presented by V. Kiryakova

Using the Poisson-type integral representations of generalized hypergeometric function (see [8]) we derive here some new classes of multidimensional inequalities of Hilbert and Hardy-Hilbert type with some special kernels. These results generalize corresponding inequalities of [9] for Gaussian hypergeometric functions. m Fn

AMS Subj. Classification: 26D15, 33C05, 33C20 Key Words: Hilbert type inequality, Hardy- parameters,

beta function, gamma

function, hypergeometric function, Gaussian hypergeometric function, Poisson-type integral representation, fractional calculus

1. Introduction Hilbert and Hardy-Hilbert type inequalities are very considerable weight inequalities which play an important role in analysis and its applications. First, let us recall on the famous Hilbert’s inequality: Let f ∈ Lp 0, ∞ and g ∈ Lq 0, ∞ be two non-negative functions and 1/p + 1/q = 1, p > 1. Then the following inequality holds (1)

∞ ∞ f (x)g(y) 0

0

π dxdy ≤ x+y sin πp

∞ 0

p

f (x)dx

1 ∞ p 0

q

g (y)dy

1 q

.

Although classical, inequality (1) was a field of interest of numerous mathematicians and was generalized in many different ways. Some possibilities of generalizing (1) are, for example, various choices of kernels and weight functions, 1

Corresponding author

ˇ Tomovski, J. Peˇcarić M. Krnić, Z.

308

extension to multi-dimensional case, extension to non-conjugate exponents etc. The inequalities related to (1) are usually called Hilbert type inequalities and their’s equivalent forms are called Hardy-Hilbert type inequalities. Hardy– Hilbert inequality assigned to (1) has the following form: (2)

∞ ∞ f (x) 0

0

p

x+y

dx

π dy ≤ sin p

p ∞ 0

f p (x)dx

For more details about such inequalities, see [5]. Considering non-conjugate parameters p and q, Hardy, Littlewood and P´ olya (see [5]) obtained extension of (1), although the original proof did not bring any information about the value of the constant on the right-hand side of the inequality. That imperfection was improved by Levin, [10], who obtained an explicit upper bound for the constant, which in the conjugate case reduced to the previously known sharp constant π/ sin (π/p) in the conjugate case. In this paper we refer to the recent paper [4], where we have obtained some general multidimensional inequalities of Hilbert and Hardy–Hilbert type with non-conjugate parameters. So, let us recall some basic definitions about non–conjugate parameters. Suppose pi , i = 1, 2, . . . , n, are real parameters which satisfy (3)

n 1 i=1

pi

≥1

and pi > 1,

i = 1, 2, . . . n.

Further, the parameters pi , i = 1, 2, . . . , n are defined by the equations (4)

1 1 + = 1, pi pi

i = 1, 2, . . . n.

Since pi > 1, i = 1, 2, . . . , n, it is obvious that pi > 1, i = 1, 2, . . . , n. We define (5)

λ :=

n 1 1 . n − 1 i=1 pi

It is easy to deduce that 0 < λ ≤ 1. Also, we introduce parameters qi , i = 1, 2, . . . n, defined by the relation (6)

1 1 = λ− , qi pi

i = 1, 2, . . . , n.

One easily conclude that qi > 1, i = 1, 2, . . . , n. Above conditions were also given by Bonsall (see [3]). It is easy to see that λ = ni=1 1/qi and 1/qi + 1 − λ = 1/pi ,

Hilbert Inequalities Related to Generalized Hypergeometric . . .

309

i = 1, 2, . . . , n. Of course, if λ = 1, then ni=1 1/pi = 1, so the conditions (3)–(6) reduce to the case of conjugate parameters. The main result from the paper [4] is the following general result: Theorem A. Let n ≥ 2 be an integer and pi , pi , qi , i = 1, 2, . . . , n, be the real numbers satisfying (3)–(6). If the functions φij , i, j = 1, 2, . . . , n, satisfy condition ni,j=1 φij (xj ) = 1, then the following inequalities hold and are equivalent Ω

≤

(7)

···

Ω

K λ (x1 , ..., xn )

n Ω

i=1

n

fi (xi )dμ1 (x1 )...dμn (xn )

i=1

1

(φii Fi fi )pi (xi )dμi (xi )

pi

and Ω

1 (φnn Fn )(xn )

·dμn (xn )

1 pn

≤

···

Ω

n−1

K λ (x1 , ..., xn )

Ω

n−1

pn

fi (xi )dμ1 (x1 )...dμn−1 (xn−1 )

i=1

1

Ω

i=1

(φii Fi fi )pi (xi )dμi (xi )

pi

,

(8) where

⎡ ⎢ Fi (xi ) = ⎣

Ω

···

Ω

K(x1 , ..., xn )

n qi j=1 j=i

⎤1

⎥ φij (xj )dμj (xj )⎦

qi

,

and Ω is interval in 0, ∞. In [4], we have applied Theorem A to the homogeneous kernel of degree −s, namely, K(x1 , x2 , . . . , xn ) = (x1 + x2 + . . . + xn )−s (s > 0). In this case we have obtained multidimensional inequalities with the constants expressed in terms of gamma function. The main objective of this paper is application od Theorem A to some special kernels. More precisely, we shall use so called Poisson–type integral representations of hypergeometric functions in obtaining appropriate kernels, such that derived inequalities involve generalized hypergeometric functions. So, before presenting our idea and results, we introduce the notion of generalized hypergeometric function m Fn .


310

2. Integral representations of generalized hypergeometric functions m Fn By a generalized hypergeometric function

m Fn

we mean the sum of the

series m Fn (a1 , . . . , am ; b1 , . . . , bn ; x) =

∞ (a1 )k (a2 )k . . . (am )k k=0

(b1 )k (b2 )k . . . (bn )k

·

xk , k!

where (a)k denotes the familiar Pochhammer symbol, defined by Γ (a + ν) (a)ν = = Γ (a)

1 : (ν = 0; a ∈ C\{0}) a (a + 1) (a + 2) · · · (a + k − 1) : (ν = k ∈ N; a ∈ C)

in the domain of its convergence: Ω = {|x| < ∞} for m ≤ n and Ω = {|x| < 1} for m = n + 1, or its analytical continuation in {|x| > 1, | arg(1 − x)| < π} in the latter case. One may also consider x as a real variable x ∈ [0, ∞. In paper [8], the unified approach to the generalized hypergeometric functions is proposed, by means of generalized fractional calculus. More precisely, hypergeometric functions m Fn are separated into three classes depending on whether m < n, m = n or m = n + 1. Further, hypergeometric functions of each class are represented as generalized fractional integrals or derivatives of three basic elementary functions: cosn−m+1 (x) (m < n)

xα exp x (m = n)

xα (1 − x)β (m = n + 1).

Here cost (x) is the so-called (generalized) cosine function of order t ≥ 2, defined via confluent hypergeometric function

cost (x) =0 F1

k t+1

t

x ; − t 1

t

=

∞ (−1)k xkt k=0

(kt)!

(t ≥ 2) ,

where cos x = cos2 (x) . The above mentioned representations lead to several new integral and differential formulae for m Fn functions and allow their study in a unified way. One of above representations includes generalized cosine function which generalize the elementary cosine function cos x = cos2 (x). For more details see [8]. Further, the generalized fractional calculus is developed in [7]. In this paper we shall be interest in Poisson- type integral representations of above mentioned classes of hypergeometric functions m Fn . So, let us state such representations from the paper [8].


311

2.1. First case: m < n. If the conditions (9) bk >

k , k = 1, 2, . . . , n − m n−m+1

bn−m+k > ak > 0, k = 1, 2, . . . , m

are satisfied, then the following Poisson-type integral representation is valid: m Fn (a1 , . . . , am ; b1 , . . . , bn ; −x) = 1 1 n−m (1 − tk )bk −(k/(n−m+1))−1

= C

...

0

0

n

×

k=n−m+1

k=1

Γ(bk − (k/ (n − m + 1)))

tk

(k/(n−m+1))−1

×

(1 − tk )bk −ak−n+m −1 ak−n+m −1 tk × Γ(bk − ak−n+m)

× cosn−m+1 (n − m + 1)(xt1 . . . tn )1/(n−m+1) dt1 . . . dtn .

(10)

The constant C is defined by

C=

n

(n − m + 1) /(2π)n−m

Γ(bj )/

j=1

m

Γ(aj ).

j=1

2.2. Second case: m = n. If the conditions (11)

bk > ak > 0, k = 1, 2, . . . , n,

are satisfied, then the following Poisson-type integral representation holds:

(12)

n Fn (a1 , . . . , an ; b1 , . . . , bn ; x) =

1 1 n (1 − tk )bk −ak −1 tk ak −1

= E

0

...

0 k=1

Γ(bk − ak )

The constant E is defined by E =

exp(xt1 . . . tn )dt1 . . . dtn .

n

j=1 [Γ(bj )/Γ(aj )] .

2.3. Third case: m = n + 1. If the conditions (13)

bk > ak+1 > 0, k = 1, 2, . . . , n

are satisfied, then the following Poisson-type integral representation is valid: n+1 Fn (a1 , . . . , an+1 ; b1 , . . . , bn ; ±x)

1

(14)= M

0

...

1 n 0 k=1

=

(1 − tk )bk −ak+1 −1 tk ak+1 −1 (1 ∓ xt1 . . . tn )−a1 dt1 . . . dtn .


312 The constant M is defined by M =

n

j=1 Γ(bj )/ [Γ(aj+1 )Γ(bj

− aj+1 )] .

In the next section, we shall use these integral representations in obtaining some new inequalities of Hilbert and Hardy-Hilbert type. More precisely, our next step is to find appropriate kernels such that the definitions of the functions Fi , i = 1, 2, . . . , n, from Theorem A, reduce to above stated Poisson–type integral representations. 3. The kernel involving exponential function Let us find a more appropriate form of Poisson–type integral representation (12). Namely, by using substitution 1 − ti = 1/(1 + xi ), i = 1, 2, . . . , n and the relationship B(x, y) = Γ(x)Γ(y)/Γ(x + y), x, y > 0, between beta and gamma function, we obtain the following integral equality: ∞ 0

(15)

···

∞ n 0

n xi ai −1 xi exp x b (1 + xi ) i 1 + xi i=1 i=1

= n Fn (a; b; x)

n

dx1 dx2 . . . dxn

B (ai , bi − ai ) ,

i=1

where a = (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ). However, the coordinates ai and bi satisfy (11). Now, we shall we previous integral representation in application of Theorem A to the kernel K : 0, ∞n → R, defined by (16)

K(x1 , x2 , . . . , xn ) =

n

xi i=1 1+xi n bi i=1 (1 + xi )

exp

A

.

We define φij (xj ) := xj ij , where Aij are real parameters, i, j = 1, 2, . . . , n. Further, we specialize Theorem A to the case of Lebesgue measures and also put Ω = 0, ∞. Then, the condition n

(17)

φij (xj ) = 1

i,j=1

from Theorem A, leads to n n Aij i=1 j=1

xj

=

n n A i=1 ij j=1

xj

= 1.


313

It is natural to set ni=1 Aij = 0, j = 1, 2, . . . , n, so that the condition (17) is satisfied. So, we obtain the following result: Theorem 1. Suppose pi , pi , qi , i = 1, 2, . . . , n, satisfy the conditions (3)– (6) and ni=1 Aij = 0, j = 1, 2, . . . , n. Then the inequalities ∞ 0

≤ β

···

∞ 0

n ∞

i=1

0

n

fi (xi ) n i=1 (1 + xi )λbi i=1

exp

λ

n

xi 1 + xi i=1

dx1 dx2 . . . dxn

xi pi Aii (1 + xi )(1−λ)pi bi −bi ×

×n−1 Fn−1 1−(1−λ)pi

xi 1 + qi Ai ; bi ; 1 + xi

1

fi pi (xi )dxi

pi

(18) and

×

∞

0

0

∞

xn −pn Ann (1 + xn )bn (λpn −1) n−1 Fn−1 1−λpn ···

∞ 0

1 + qn An ; bn ;

xn × 1 + xn

n

pn 1 n−1 pn xi f (x ) i i λ n i=1 exp dx dx . . . dx dx 1 2 n−1 n λbi 1 + xi i=1 (1 + xi ) i=1 n−1 ∞

≤β

i=1

×

n−1 Fn−1

0

1−(1−λ)pi


xi 1 + qi Ai ; bi ; 1 + xi

1 pi

fi (xi )dxi

pi

,

(19) hold for all real parameters Aij , i = j such that qi Aij ∈ −1, bj − 1, and for all non-negative measurable functions fi , i = 1, 2, . . . , n, on 0, ∞. Moreover, these inequalities are equivalent. The constant β is defined by (20)

β=

n

1

B (1 + qi Aij , bj − 1 − qi Aij ) qi ,

i,j=1 i=j

1+qi Ai = (1 + qi Ai1 , 1 + qi Ai2 , . . . , 1 + qi Ai,i−1 , 1 + qi Ai,i+1 , . . . , 1 + qi Ain ) and bi = (b1 , b2 , . . . , bi−1 , bi+1 , . . . , bn ). P r o o f. We use Theorem A. Namely, by putting the kernel defined by A (16) and the functions φij (xj ) := xj ij , i, j = 1, 2, . . . , n, where ni=1 Aij = 0,


314 j = 1, 2, . . . , n, we have b − qi i

Fi (xi ) = (1 + xi )

∞

0

···

∞ n xj (1+qi Aij )−1 0

(1 + xj )bj

j=1 j=i

⎛ ⎜

exp ⎝

xi 1 + xi

1

qi

dx1 . . . dxi−1 dxi+1 . . . dxn

n j=1 j=i

⎞

xj ⎟ ⎠ 1 + xj

.

Now, by using integral representation (15) we have b

− qi

Fi (xi ) = (1 + xi )

i

1 q n−1 i

xi 1 + qi Ai ; bi ; . 1 + xi

β n−1 F

Finally, by putting the expression for Fi (xi ) in (7) and (8), we easily obtain required inequalities. It is easy to see that parameters Aij , i = j satisfy conditions qi Aij ∈ −1, bj − 1, since the arguments of beta function are positive. Obviously, by taking some special values of the parameters Aij , i, j = 1, 2, . . . , n, we can simplify the constant β from Theorem 1. It is interesting to take the arithmetic means of the borders of intervals defining parameters Aij , i = j. So, if Aij = (bj − 2)/2qi , then 1 + qi Aij = bj /2, so the constant β, defined by (20), becomes β = ni=1 B (bi /2, bi /2)1/pi . In that case, it is easy to see that the parameters Aii , i = 1, 2, . . . , n, are defined by Aii = −(bj − 2)/2pi . Since 0 < xi /(1 + xi ) < 1, xi ∈ 0, ∞, i = 1, 2, . . . , n, it is easy to deduce that n−1 Fn−1 (1 + qi Ai ; bi ; xi /(1 + xi )) < n−1 Fn−1 (1 + qi Ai ; bi ; 1), so we can estimate the weight functions in both inequalities (18) and (19) and obtain the constant involving hypergeometric function n−1 Fn−1 . That is the content of the following Corollary 1. Under the same assumptions as in Theorem 1, the following two inequalities hold and are equivalent: ∞ 0

(21)

≤ βH

···

∞ 0

fi (xi ) n i=1 (1 + xi )λbi i=1

n ∞ i=1

0

n

xi

pi Aii

exp

λ

n

xi 1 + xi i=1

(1−λ)pi bi −bi

(1 + xi )

pi

dx1 dx2 . . . dxn

fi (xi )dxi

1

pi

,


×

∞

0

···

∞

0

xn −pn Ann (1 + xn )bn (λpn −1) ×

n n−1 λ i=1 fi (xi ) n exp λb i=1 (1

0

∞

≤ βH

n−1 i=1

+ xi )

i

xi 1 + xi i=1

pn

dx1 dx2 . . . dxn−1

dxn

1 pn

1

∞

0

315

xi

pi Aii

(1−λ)pi bi −bi

(1 + xi )

pi

fi (xi )dxi

pi

.

(22) The constant βH is defined by the formula (23)

βH = β

n

n−1 Fn−1

1 qi

(1 + qi Ai ; bi ; 1) ,

i=1

where β is given by (20). R e m a r k 1. The main idea of the proof of Theorem A is a reduction of case of n non-conjugate exponents to the case of n + 1 conjugate exponents, and also application of H¨ older’s inequality on such exponents (see [4]). So, the equality in (7) and (8) holds if and only if it holds in H¨ older’s inequality. That condition implies that there exist constant k ∈ R such that K(x1 , x2 , . . . , xn ) = k ni=1 Fi qi (xi ), if fi are not zero functions (see [4], Remark 1.) Obviously, the kernel (16) doesn’t satisfy previous condition, since its variables are not separated. So, the equalities in (18), (19), (21) and (22) hold if and only if there exist index i, i = 1, 2, . . . n such that fi is zero function. 4. The kernel involving cosine function In this section we shall find more appropriate form of Poisson–type integral representation (10). We consider the special case m = n − 1, which give us representation involving elementary cosine function. Namely, by using substitution 1 − ti = 1/(1 + xi ), i = 1, 2, . . . , n and the relationship between beta and gamma function, we obtain the following equality ∞ 0

···

∞ n xi ai−1 −1 0

⎡

n

xi cos ⎣2 x b i (1 + xi ) 1 + xi i=1 i=1 1

(24) = (π + 1/2)− 2

n−1 Fn

1/2 ⎤

(a; b; −x) B (b1 − 1/2, 1/2)

⎦ dx1 dx2 . . . dxn

n i=2

B (ai−1 , bi − ai−1 ) ,


316

where a = (a1 , a2 , . . . , an−1 ), b = (b1 , b2 , . . . , bn ) and a0 = 1/2. The coordinates of vectors a and b satisfy (9). Since elementary cosine is an even function, integral representation (24) holds also when one replace argument −x with argument x in hypergeometric function n−1 Fn . As in the previous section, we shall use previous integral representation in application of Theorem A to the kernel K : 0, ∞n → R defined by

1/2 n xi i=1 1+xi n . bi i=1 (1 + xi )

cos (25)

K(x1 , x2 , . . . , xn ) =

Since 0 < xi /(1 + xi ) < 1, xi ∈ 0, ∞, i = 1, 2, . . . , n, it is easy to deduce that cos ( ni=1 xi /(1 + xi ))1/2 > 0, so the kernel defined by (25) is positive. A

Similarly, as in Section 3, we define φij (xj ) := xj ij and also require i=1 Aij = 0, j = 1, 2, . . . , n, so that the condition (17) is satisfied.

n

Finally, by putting these functions and the kernel (25) in Theorem A we obtain the following result:

Theorem 2. Suppose pi , pi , qi , i = 1, 2, . . . , n, satisfy the conditions (3)– (6) and ni=1 Aij = 0, j = 1, 2, . . . , n. Then the inequalities

∞ 0

···

0

n ∞

≤ γ

i=1

× (26)

∞

0

n

fi (xi ) n i=1 (1 + xi )λbi i=1

cos

λ

n

xi 1 + xi i=1

1 2

dx1 dx2 . . . dxn


n−2 Fn−1

1−(1−λ)pi

xi 1 + qi Ai,i+1 ; bi ; fi pi (xi )dxi 4(1 + xi )

1

pi


317

and

∞

xn

0

⎡

×⎣

∞ 0

−pn Ann

···

∞ 0

bn (λpn −1)

(1 + xn )

n n−1 λ i=1 fi (xi ) n cos λb i=1 (1

≤γ

+ xi )

n−1 i=1

×

n−2 Fn−1

1−λpn

n−2 Fn−1

1−(1−λ)pi

xi 1 + xi i=1

i

∞

0

xn 1 + qn An1 ; bn ; × 4(1 + xn ) ⎤pn

1 2

dx1 dx2 . . . dxn−1 ⎦

dxn

1 pn


xi 1 + qi Ai,i+1 ; bi ; fi pi (xi )dxi 4(1 + xi )

1

pi

,

(27) hold for all real parameters Aij , i, j = 1, 2, . . . , n and bi , i = 1, 2, . . . , n, such / {0, 1, 1 − n}, qi Ai,i+1 = −1/2 and bi > 1/2, that qi Aij ∈ −1, bj − 1, j − i ∈ i = 1, 2, . . . , n. Moreover, these inequalities are equivalent. The constant γ is given by (28)

γ = (π + 1/2)−λ/2 β

n

1

B (bi+1 − 1/2, 1/2) ,qi

i=1

where β is given by (20), 1 + qi Ai,i+1 = (1 + qi Ai1 , . . . , 1 + qi Ai,i−1 , 1 + qi Ai,i+2 , . . . , 1 + qi Ain ) and bi = (b1 , b2 , . . . , bi−1 , bi+1 , . . . , bn ). P r o o f. We specialize Theorem A with the kernel defined by (25) and the A functions φij (xj ) := xj ij , i, j = 1, 2, . . . , n, where ni=1 Aij = 0, j = 1, 2, . . . , n. Hence, we have b

− qi

Fi (xi ) = (1 + xi )

∞

···

i

0

∞ (1+qi Ai,i+1 )−1 xi+1 0

(1 + xi+1 )bi+1

n j=1 j=i,i+1

xj (1+qi Aij )−1 × (1 + xj )bj

⎡ ⎛ ⎞1 ⎤ 2

1 n qi ⎢ ⎜ ⎥ x x i j ⎟ ⎥ 2 dx . . . dx dx . . . dx . × cos ⎢ ⎝ ⎠ 1 i−1 i+1 n ⎣ ⎦ 4(1 + xi ) j=1 1 + xj j=i

Now, by using integral representation (24) we have b

− qi

Fi (xi ) = (1 + xi )

i

1

×n−2 Fn−1 qi

− 2q1

1

B (bi+1 − 1/2, 1/2) qi × xi 1 + qi Ai,i+1; bi ; , 4(1 + xi )

β (π + 1/2)

i


318

where β is constant given by (20). Finally, by putting the expression for Fi (xi ) in (7) and (8), we easily obtain appropriate Hilbert and Hardy–Hilbert type inequalities. It is easy to obtain the conditions for parameters Aij , i, j = 1, 2, . . . , n, and bi , i = 1, 2, . . . , n, since the arguments of beta function are positive. In the previous theorem we assume the congruence modulo n on the parameters Aij . More precisely, we have An,n+1 = An1 . Since 0 < xi /4(1+ xi ) < 1, xi ∈ 0, ∞, i = 1, 2, . . . , n, following estimate holds: n−2 Fn−1 (1 + qi Ai,i+1 ; bi ; xi /4(1 + xi )) < n−2 Fn−1 (1 + qi Ai,i+1 ; bi ; 1). So, similarly as in Corollary 1, we obtain appropriate Hilbert type and Hardy– Hilbert type inequality with the constants expressed in terms of hypergeometric function n−2 Fn−1 . Corollary 2. Under the same assumptions as in Theorem 2, the following two inequalities hold and are equivalent: ∞ 0

≤ γH

(29)

···

∞ 0

n ∞ i=1

0

n

fi (xi ) n i=1 λbi i=1 (1 + xi )

cos

∞ 0

⎡

≤ γH

i=1 (1 n−1

xi 1 + xi i=1

1 2

dx1 dx2 . . . dxn

1

+ xi )

i=1

,

xi 1 + xi i=1

i

∞

0

pi

xn −pn Ann (1 + xn )bn (λpn −1) ×

n ∞ ∞ n−1 λ i=1 fi (xi ) ⎣ n ··· cos × λb 0

n

xi pi Aii (1 + xi )(1−λ)pi bi −bi fi pi (xi )dxi

0

λ

xi

pi Aii

⎤pn

1 2

dx1 dx2 . . . dxn−1 ⎦

(1−λ)pi bi −bi

(1 + xi )

dxn

1 pn

1 pi

fi (xi )dxi

pi

.

(30) The constant γH is defined by the formula (31)

γH = γ

n i=1

n−2 Fn−1

1 qi

(1 + qi Ai,i+1; bi ; 1) ,

where γ is given by (28). R e m a r k 2. Equalities in Theorem 2 and Corollary 2 hold if and only if there exist index i, i = 1, 2, . . . n such that fi is zero function (see Remark 1).


319

5. Fractional kernel Finally, let us find more appropriate form of the integral representation (14). Using the same substitution as before, 1 − ti = 1/(1 + xi ), i = 1, 2, . . . , n, and the relationship between beta and gamma function, we obtain the following integral equality:

∞ 0

(32)

···

∞ n xi ai+1 −1 0

i=1

(1 + xi )bi

= n+1 Fn (a; b; x)

n

n

xi 1−x 1 + xi i=1

−a1

dx1 dx2 . . . dxn

B (ai+1 , bi − ai+1 ) ,

i=1

where a = (a1 , a2 , . . . , an+1 ) and b = (b1 , b2 , . . . , bn ) are vectors which coordinates satisfy condition bi > ai+1 > 0, i = 1, 2, . . . , n. Here, we shall use previous integral representation in application of Theorem A to the kernel K : 0, ∞n → R, defined by the formula

(33)

K(x1 , x2 , . . . , xn ) =

−s xi i=1 1+xi n , bi i=1 (1 + xi )

1−

n

(s > 0)

Similarly as in previous two sections, we obtain the result for the kernel defined by (33): Theorem 3. Suppose pi , pi , qi , i = 1, 2, . . . , n, satisfy the conditions (3)– (6), s > 0 and ni=1 Aij = 0, j = 1, 2, . . . , n. Then the inequalities ∞ 0

< β

···

∞

n ∞

i=1

(34)

and

0

×n Fn−1

0

n

fi (xi ) n i=1 λbi i=1 (1 + xi )

n

xi 1− 1 + xi i=1

−λs

dx1 dx2 . . . dxn


1−(1−λ)pi

xi s, 1 + qi Ai ; bi ; 1 + xi

1

dxi

pi


320 ⎡

×⎣

∞

0

∞ 0

xn ···

−pn Ann

∞ 0

bn (λpn −1)

(1 + xn )

n Fn−1

n−1 n i=1 fi (xi ) n 1 − λb i=1 (1

+ xi )