HILBERT-PACHPATTE TYPE INEQUALITIES FROM BONSALL’S FORM OF HILBERT’S INEQUALITY

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Hilbert-Pachpatte Type Inequalities Josip Peˇcari´c, Ivan Peri´c

and Predrag Vukovi´c vol. 9, iss. 1, art. 9, 2008

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HILBERT-PACHPATTE TYPE INEQUALITIES FROM BONSALL’S FORM OF HILBERT’S INEQUALITY

JOSIP PE ˇCARI ´C IVAN PERI ´C

Faculty of Textile Technology Faculty of Food Technology and Biotechnology

University of Zagreb University of Zagreb

Pierottijeva 6, 10000 Zagreb, Croatia Pierottijeva 6, 10000 Zagreb, Croatia

EMail:pecaric@hazu.hr EMail:iperic@pbf.hr

PREDRAG VUKOVI ´C

Faculty of Teacher Education

University of Zagreb, Ante Starˇcevi´ca 55 40000 ˇCakovec, Croatia

EMail:predrag.vukovic@vus-ck.hr

Received: 19 June, 2007

Accepted: 09 October, 2007

Communicated by: B. Yang 2000 AMS Sub. Class.: 26D15.

Key words: Inequalities, Hilbert’s inequality, sequences and functions, homogeneous kernels, conjugate and non-conjugate exponents, the Beta function.

Abstract: The main objective of this paper is to deduce Hilbert-Pachpatte type inequalities using Bonsall’s form of Hilbert’s and Hardy-Hilbert’s inequalities, both in discrete and continuous case.

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and Predrag Vukovi´c

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1. Introduction

An interesting feature of one of the forms of Hilbert-Pachpatte type inequalities, is that it controls the size (in the sense of L^p or l^p spaces ) of the modified Hilbert transform of a function or of a series with the size of its derivate or its backward differences, respectively. We start with the following results of Zhongxue Lü from [9], for both continuous and discrete cases. For a sequencea :N⁰ →R, the sequence

∇a:N→Ris defined by∇a(n) = a(n)−a(n−1). For a functionu: (0,∞)→R, u⁰denotes the usual derivative ofu.

Theorem A. Letp >1,¹_p+¹_q = 1, s >2−min{p, q},andf(x), g(y)be real-valued continuous functions defined on[0,∞),respectively, and letf(0) =g(0) = 0,and

0<

Z ∞

0

Z x

0

x^1−s|f⁰(τ)|^pdτ dx <∞, 0<

Z ∞

0

Z y

0

y^1−s|g⁰(δ)|^qdδdy <∞, then

(1.1) Z ∞

0

Z ∞

0

|f(x)||g(y)|

(qx^p−1+py^q−1)(x+y)^sdxdy

≤ B

q+s−2

q ,^p+s−2_p

pq ·

Z ∞

0

Z x

0

x^1−s|f⁰(τ)|^pdτ dx ¹_p

× Z ∞

0

Z y

0

y^1−s|g⁰(δ)|^qdδdy ¹_q

. Theorem B. Letp >1, ¹_p +¹_q = 1, s >2−min{p, q},and{a(m)}and{b(n)}be

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two sequences of real numbers wherem, n∈N⁰,anda(0) =b(0) = 0,and 0<

∞

X

m=1 m

X

τ=1

m^1−s|∇a(τ)|^p <∞, 0<

∞

X

n=1 n

X

δ=1

n^1−s|∇b(δ)|^q <∞, then

(1.2)

∞

X

m=1

∞

X

n=1

|am||bn|

(qm^p−1+pn^q−1)(m+n)^s

≤

B_q+s−2

q ,^p+s−2_p

pq ·

∞

X

m=1 m

X

τ=1

m^1−s|∇a(τ)|^p

!¹_p

×

∞

X

n=1 n

X

δ=1

n^1−s|∇b(δ)|^q

!¹_q .

Note that the condition s > 2−min{p, q} from Theorem B is not sufficient.

Namely, the author of the proof of TheoremBused the following result (1.3)

∞

X

n=1

1 (m+n)^s

m n

^2−s_q

< B

q+s−2

q ,p+s−2 p

m^1−s,

form ∈ {1,2, . . .} ands > 2−min{p, q}.Forp = q = 2, s = 18 and m = 1, the left-hand side of (1.3) is greater than the right-hand side of (1.3). Therefore, we refer to a paper of Krni´c and Peˇcari´c, [4], where the next inequality is given:

(1.4)

∞

X

n=1

1 (m+n)^s

m^α¹

n^α² < m^1−s+α¹^−α²B(1−α₂, s+α₂−1),

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where0< s≤14,1−s < α₂ <1fors≤2and−1≤α₂ <1fors >2.By using this result, here we shall obtain a generalization of TheoremBbut with the condition 2−min{p, q}< s≤2 + min{p, q}.Also, the following result is given in [9]:

(1.5)

∞

X

n=1

1 m^s+n^s

m n

^2−s_q

< 1 sB

q+s−2

sq ,p+s−2 sp

m^1−s,

form∈ {1,2, . . .}ands >2−min{p, q}.Similarly as before, forp=q = 2, s= 6 andm = 1, the left-hand side of (1.5) is greater than the right-hand side of (1.5).

The case of nontrivial weights is essential in TheoremAand TheoremB, since for s= 1only the trivial functions and sequences satisfy the assumptions.

In 1951, Bonsall established the following conditions for non-conjugate exponents (see [1]). Letpandqbe real parameters, such that

(1.6) p >1, q > 1, 1

p +1 q ≥1,

and letp⁰ andq⁰ respectively be their conjugate exponents, that is, ¹_p + _p¹0 = 1 and

1

q +_q¹0 = 1. Further, define

(1.7) λ= 1

p⁰ + 1 q⁰

and note that0< λ ≤ 1for allpandqas in (1.6). In particular,λ = 1holds if and only ifq =p⁰, that is, only whenpandqare mutually conjugate. Otherwise, we have 0< λ <1, and in such casespandqwill be referred to as non-conjugate exponents.

Also, in this paper we shall obtain some generalizations of (1.1). It will be done in simplier way than in [9]. Our results will be based on the following results of Peˇcari´c et al., [2], for the non-conjugate and conjugate exponents.

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Theorem C. Let p, p⁰, q, q⁰ and λ be as in (1.6) and (1.7). If K, ϕ, ψ, f and g are non-negative measurable functions, then the following inequalities hold and are equivalent

(1.8) Z

Ω²

K^λ(x, y)f(x)g(y)dxdy ≤ Z

Ω

(ϕF f)^p(x)dx ¹_pZ

Ω

(ψGg)^q(y)dy ¹_q

and

(1.9)

Z

Ω

1 ψG(y)

Z

Ω

K^λ(x, y)f(x)dx q⁰

dy

!_q¹0

≤ Z

Ω

(ϕF f)^p(x)dx ¹_p

, where the functionsF, Gare defined by

F(x) = Z

Ω

K(x, y) ψ^q⁰(y) dy

_q¹0

and G(y) = Z

Ω

K(x, y) ϕ^p⁰(x) dx

_p¹0

.

The next inequalities from [5] can be seen as a special case of (1.8) and (1.9) respectively for the conjugate exponents:

(1.10) Z

Ω²

K(x, y)f(x)g(y)dxdy

≤ Z

Ω

ϕ^p(x)F(x)f^p(x)dx ¹_pZ

Ω

ψ^q(y)G(y)g^q(y)dy ¹_q

and (1.11)

Z

Ω

G^1−p(y)ψ^−p(y) Z

Ω

K(x, y)f(x)dx p

dy≤ Z

Ω

ϕ^p(x)F(x)f^p(x)dx,

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where

(1.12) F(x) =

Z

Ω

K(x, y)

ψ^p(y) dy and G(y) = Z

Ω

K(x, y) ϕ^q(x) dx.

In particular, inequalities (1.10) and (1.11) are equivalent.

On the other hand, here we also refer to a paper of Brneti´c et al., [8], where a general Hilbert-type inequality was obtained for n ≥ 2 conjugate exponents, that is, real parameters p₁, . . . , p_n > 1, such that Pn

i=1 1

pi = 1. Namely, we let K : Ωⁿ →Randφ_ij : Ω→R,i, j = 1, . . . , n, be non-negative measurable functions. If Qn

i,j=1φ_ij(x_j) = 1, then the inequality (1.13)

Z

Ωⁿ

K(x₁, . . . , x_n)

n

Y

i=1

f_i(x_i)dx₁. . . dx_n

≤

n

Y

i=1

Z

Ω

F_i(x_i)(φ_iif_i)^pⁱ(x_i)dx_i _pi¹

,

holds for all non-negative measurable functionsf₁, . . . , f_n : Ω→R, where (1.14) Fi(xi) =

Z

Ωⁿ⁻¹

K(x1, . . . , xn)

n

Y

j=1,j6=i

φ^p_ijⁱ(xj)dx1. . . dxi−1dxi+1. . . dxn,

fori= 1, . . . , n.

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2. Integral Case

In this section we shall state our main results. We suppose that all integrals converge and shall omit these types of conditions. Thus, we have the following.

Theorem 2.1. Let ¹_p + ¹_q = 1with p > 1.IfK(x, y), ϕ(x), ψ(y)are non-negative functions and f(x), g(y) are absolutely continuous functions such that f(0) = g(0) = 0,then the following inequalities hold

Z ∞

0

Z ∞

0

K(x, y)|f(x)| |g(y)|

qx^p−1+py^q−1 dxdy (2.1)

≤ Z ∞

0

Z ∞

0

K(x, y)|f(x)| |g(y)|d x¹^p

d

y¹^q

≤ 1 pq

Z ∞

0

Z x

0

ϕ^p(x)F(x)|f⁰(τ)|^pdτ dx ¹_p

× Z ∞

0

Z y

0

ψ^q(y)G(y)|g⁰(δ)|^qdδdy ¹_q

, and

(2.2)

Z ∞

0

G^1−p(y)ψ^−p(y) Z ∞

0

K(x, y)|f(x)|d x¹^pp

dy

≤ 1 p^p

Z ∞

0

Z x

0

ϕ^p(x)F(x)|f⁰(τ)|^pdτ dx, whereF(x)andG(y)are defined as in (1.12).

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Proof. By using Hölder’s inequality, (see also [9]), we have (2.3) |f(x)| |g(y)| ≤x¹^qy¹^p

Z x

0

|f⁰(τ)|^pdτ

¹_pZ y

0

|g⁰(δ)|^qdδ ¹_q

.

From (2.3) and using the elementary inequality

(2.4) xy≤ x^p

p +y^q

q , x ≥0, y ≥0, 1 p+ 1

q = 1, p >1,

we observe that (2.5) pq|f(x)| |g(y)|

qx^p−1+py^q−1 ≤ |f(x)| |g(y)|

x¹^qy¹^p

≤ Z x

0

|f⁰(τ)|^pdτ

_p¹ Z y

0

and therefore pq

Z ∞

0

Z ∞

0

K(x, y)|f(x)| |g(y)|

≤ Z ∞

0

Z ∞

0

K(x, y) x¹^qy¹^p

|f(x)||g(y)|dxdy

≤ Z ∞

0

Z ∞

0

K(x, y) Z x

0

|f⁰(τ)|^pdτ

¹_pZ y

0

dxdy.

Applying the substitutions f₁(x) =

Z x

0

|f⁰(τ)|^pdτ ¹_p

, g₁(y) = Z y

0

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and (1.10), we obtain Z ∞

0

Z ∞

0

K(x, y)f₁(x)g₁(y)dxdy (2.7)

≤ Z ∞

0

ϕ^p(x)F(x)f₁^p(x)dx

¹_pZ ∞

0

ψ^q(y)G(y)g₁^q(y)dy ¹_q

= Z ∞

0

Z x

0

ϕ^p(x)F(x)|f⁰(τ)|^pdτ dx ¹_p

× Z ∞

0

Z y

0

ψ^q(y)G(y)|g⁰(δ)|^qdδdy ¹_q

.

By using (2.6) and (2.7) we obtain (2.1). The second inequality (2.2) can be proved by applying (1.11) and the inequality

|f(x)| ≤x¹^q Z x

0

|f⁰(t)|^pdt ¹_p

.

Now we can apply our main result to non-negative homogeneous functions. Re- call that for a homogeneous function of degree−s, s > 0,the equalityK(tx, ty) = t^−sK(x, y)is satisfied. Further, we define

k(α) :=

Z ∞

0

K(1, u)u^−αdu

and suppose thatk(α) < ∞ for1−s < α < 1.To prove first application of our main results we need the following lemma.

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Lemma 2.2. Ifs >0,1−s < α < 1andK(x, y)is a non-negative homogeneous function of degree−s,then

(2.8)

Z ∞

0

K(x, y) x

y α

dy=x^1−sk(α).

Proof. By using the substitutionu = ^y_x and the fact thatK(x, y)is homogeneous function, the equation (2.8) follows easily.

Corollary 2.3. Let s > 0, ¹_p + ¹_q = 1 with p > 1. If f(x), g(y) are absolutely continuous functions such that f(0) = g(0) = 0, and K(x, y) is a non-negative symmetrical and homogeneous function of degree−s,then the following inequalities hold

Z ∞

0

Z ∞

0

K(x, y)|f(x)| |g(y)|

≤ Z ∞

0

Z ∞

0

K(x, y)|f(x)| |g(y)|d x^p¹

d y¹^q

≤ L pq

Z ∞

0

Z x

0

x^1−s+p(A¹^−A²⁾|f⁰(τ)|^pdτ dx ¹_p

× Z ∞

0

Z y

0

y^1−s+q(A²^−A¹⁾|g⁰(δ)|^qdδdy ¹_q

and (2.10)

Z ∞

0

y(p−1)(s−1)+p(A₁−A₂)

Z ∞

0

K(x, y)|f(x)|d(x¹^p) p

dy

≤ L

p

pZ ∞

0

Z x

0

x^1−s+p(A¹^−A²⁾|f⁰(τ)|^pdτ dx,

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whereA₁ ∈(^1−s_q ,¹_q), A₂ ∈(^1−s_p ,_p¹)andL=k(pA₂)¹^pk(qA₁)¹^q.

Proof. LetF(x), G(y)be the functions defined as in (1.12). Settingϕ(x) =x^A¹ and ψ(y) =y^A²,by Lemma2.2we obtain

Z ∞

0

Z x

0

ϕ^p(x)F(x)|f^0pdτ dx (2.11)

= Z ∞

0

Z x

0

|f⁰(τ)|^p Z ∞

0

K(x, y) x

y pA2

dy

!

x^p(A¹^−A²⁾dτ dx

=k(pA2) Z ∞

0

Z x

0

and similarly (2.12)

Z ∞

0

Z y

0

ψ^q(y)G(y)|g⁰(δ)|^qdδdy

=k(qA₁) Z ∞

0

Z y

0

y^1−s+q(A²^−A¹⁾|g⁰(δ)|^qdδdy.

From (2.1), (2.11) and (2.12), we get (2.9). Similarly, the inequality (2.10) follows from (2.2).

We proceed with some special homogeneous functions. First, by puttingK(x, y) =

1

(x+y)^s in Corollary2.3, we get the following.

Corollary 2.4. Let s > 0, ¹_p + ¹_q = 1 with p > 1. If f(x), g(y) are absolutely continuous functions such that f(0) = g(0) = 0, then the following inequalities

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hold

Z ∞

0

Z ∞

0

|f(x)| |g(y)|

(qx^p−1 +py^q−1)(x+y)^sdxdy

≤ Z ∞

0

Z ∞

0

|f(x)| |g(y)|

(x+y)^s d x¹^p

d y¹^q

≤ L1

pq Z ∞

0

Z x

0

x^1−s+p(A¹^−A²⁾|f⁰(τ)|^pdτ dx _p¹

× Z ∞

0

Z y

0

and Z ∞

0

y(p−1)(s−1)+p(A1−A2)

Z ∞

0

|f(x)|

(x+y)^sd x¹^pp

dy

≤ L₁

p

pZ ∞

0

Z x

0

whereA₁ ∈(^1−s_q ,¹_q), A₂ ∈(^1−s_p ,_p¹)and

L₁ = [B(1−pA₂, pA₂+s−1)]¹^p[B(1−qA₁, qA₁ +s−1)]¹^q.

Remark 1. By putting A₁ = A₂ = ^2−s_pq in Corollary 2.4, with the condition s >

2−min{p, q},we obtain TheoremAfrom the introduction.

Since the function K(x, y) = ^ln

y x

y−x is symmetrical and homogeneous of degree

−1,by using Corollary2.3we obtain:

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Corollary 2.5. Let ¹_p + ¹_q = 1withp > 1. Iff(x), g(y)are absolutely continuous functions such thatf(0) =g(0) = 0,then the following inequalities hold

Z ∞

0

Z ∞

0

ln^y_x|f(x)| |g(y)|

(qx^p−1+py^q−1)(y−x)dxdy

≤ Z ∞

0

Z ∞

0

ln_x^y|f(x)| |g(y)|

y−x d

x¹^p d

y¹^q

≤ L₂ pq

Z ∞

0

Z x

0

x^p(A¹^−A²⁾|f⁰(τ)|^pdτ dx

¹_pZ ∞

0

Z y

0

y^q(A²^−A¹⁾|g⁰(δ)|^qdδdy ¹_q

and Z ∞

0

y^p(A¹^−A²⁾ Z ∞

0

|f(x)|ln^y_x y−x d

x¹^p^p dy ≤

L₂ p

pZ ∞

0

Z x

0

x^p(A¹^−A²⁾|f⁰(τ)|^pdτ dx,

whereA₁ ∈(0,¹_q), A₂ ∈(0,¹_p)and

L₂ =π²(sinpA₂π)⁻²^p(sinqA₁π)⁻²^q.

Similarly, for the symmetrical homogeneous function of degree−s, K(x, y) =

1

max{x,y}^s,we have:

Corollary 2.6. Let s > 0, ¹_p + ¹_q = 1 with p > 1. If f(x), g(y) are absolutely continuous functions such that f(0) = g(0) = 0, then the following inequalities

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hold

Z ∞

0

Z ∞

0

|f(x)| |g(y)|

(qx^p−1 +py^q−1) max{x, y}^sdxdy

≤ Z ∞

0

Z ∞

0

|f(x)| |g(y)|

max{x, y}^sd x¹^p

d y¹^q

≤ L₃ pq

Z ∞

0

Z x

0

x^1−s+p(A¹^−A²⁾|f⁰(τ)|^pdτ dx _p¹

× Z ∞

0

Z y

0

and Z ∞

0

y(p−1)(s−1)+p(A₁−A₂)

Z ∞

0

|f(x)|

max{x, y}^sd x¹^pp

dy

≤ L3

p

pZ ∞

0

Z x

0

x^1−s+p(A¹^−A²⁾|f⁰(τ)|^pdτ dx, whereA1 ∈

1−s q ,¹_q

, A2 ∈

1−s p ,¹_p

andL3 = k(pA2)^p¹k(qA1)¹^q,wherek(α) =

s (1−α)(s+α−1).

At the end of this section we give a generalization of the inequality (2.1) from Theorem2.1. In the proof we used a general Hilbert-type inequality (1.13) of Brneti´c et al., [8].

Theorem 2.7. Letn∈ N, n≥2,Pn i=1

1

pi = 1withp_i >1, i = 1, . . . , n.Letq_i, α_i, i = 1, . . . , n,are defined with _q¹

i = 1− _p¹

i andα_i = Qn

j=1,j6=ip_j.IfK(x₁, . . . , x_n), φij(xj), i, j = 1, . . . , n, are non-negative functions such that Qn

i,j=1φij(xj) = 1,

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andf_i(x_i), i = 1, . . . , n,are absolutely continuous functions such thatf_i(0) = 0, i= 1, . . . , n,then the following inequality holds

Z ∞

0

. . . Z ∞

0

K(x₁, . . . , x_n)Qn

i=1|f_i(x_i)|

Pn

i=1α_ix^p_iⁱ⁻¹ dx1. . . dxn

≤ Z ∞

0

. . . Z ∞

0

K(x₁, . . . , x_n)

n

Y

i=1

|f_i(x_i)|d

x

1 p1

1

. . . d

x

1

npn

≤ 1

p₁. . . p_n

n

Y

i=1

Z ∞

0

Z xi

0

φ^p_iiⁱ(x_i)F_i(x_i)|f_i⁰(τ_i)|^pⁱdτ_idx_i _pi¹

, whereF_i(x_i)are defined as in (1.14) fori= 1, . . . , n.

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3. Discrete Case

We also give results for the discrete case. For that, we apply the following result from [5].

Theorem 3.1. If {a(m)} and {b(n)} are non-negative real sequences, K(x, y) is non-negative homogeneous function of degree−sstrictly decreasing in both param- etersxand y, ¹_p + ¹_q = 1, p > 1, A, B, α, β > 0,then the following inequalities hold and are equivalent

(3.1)

∞

X

m=1

∞

X

n=1

K(Am^α, Bn^β)a_mb_n

< N

∞

X

m=1

mα(1−s)+αp(A₁−A₂)+(p−1)(1−α)

a^p_m

!¹_p

·

∞

X

n=1

nβ(1−s)+βq(A2−A1)+(q−1)(1−β)

b^q_n

!¹_q , and

(3.2)

∞

X

n=1

nβ(s−1)(p−1)+pβ(A1−A2)+β−1

∞

X

m=1

K(Am^α, Bn^β)a_m

!p

< N^p

∞

X

m=1

mα(1−s)+αp(A₁−A₂)+(p−1)(1−α)

a^p_m,

whereA1 ∈(max{^1−s_q ,^α−1_αq },¹_q), A2 ∈ max

n1−s p ,^β−1_βp

o ,¹_p

and

(3.3) N =α⁻¹^qβ⁻¹^pA^2−s^p ^+A¹^−A²⁻¹B^2−s^q ^+A²^−A¹⁻¹k(pA2)¹^pk(2−s−qA1)¹^q.

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Applying Theorem3.1, we obtain the following.

Corollary 3.2. Lets > 0, ¹_p + ¹_q = 1 withp > 1. Let{a(m)} and{b(n)}be two sequences of real numbers where m, n ∈ N0, and a(0) = b(0) = 0. If K(x, y) is a non-negative homogeneous function of degree −s strictly decreasing in both parametersxandy, A, B, α, β >0,then the following inequalities hold

∞

X

m=1

∞

X

n=1

K(Am^α, Bn^β)|a_m| |b_n| qm^p−1+pn^q−1

≤ 1 pq

∞

X

m=1

∞

X

n=1

K(Am^α, Bn^β)|a_m| |b_n| m¹^qn^p¹

(3.4)

< N pq

∞

X

m=1 m

X

τ=1

mα(1−s)+αp(A1−A2)+(p−1)(1−α)|∇a(τ)|^p

!¹_p

·

∞

X

n=1 n

X

δ=1

nβ(1−s)+βq(A₂−A₁)+(q−1)(1−β)|∇b(δ)|^q

!¹_q , and

(3.5)

∞

X

n=1

nβ(s−1)(p−1)+pβ(A₁−A₂)+β−1

∞

X

m=1

K(Am^α, Bn^β)|a_m| m¹^q

!p

< N^p

∞

X

m=1 m

X

τ=1

mα(1−s)+αp(A₁−A₂)+(p−1)(1−α)|∇a(τ)|^p,

whereA₁ ∈ maxn

1−s q ,^α−1_αq o

,¹_q

, A₂ ∈ maxn

1−s p ,^β−1_βp o

,¹_p

and the constant N is defined as in (3.3).

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Proof. By using Hölder’s inequality, (see also [9]), we have

(3.6) |a_m| |b_n| ≤m¹^qn¹^p

m

X

τ=1

|∇a(τ)|^p

!¹_p _n X

δ=1

|∇b(δ)|^q

!¹_q .

From (2.4) and (3.6), we get (3.7) pq|a_m| |b_n|

qm^p−1+pn^q−1 ≤ |a_m| |b_n| m¹^qn¹^p

≤

m

X

τ=1

|∇a(τ)|^p

!¹_p _n X

δ=1

|∇b(δ)|^q

!¹_q ,

and therefore pq

∞

X

m=1

∞

X

n=1

K(Am^α, Bn^β)|am| |bn| qm^p−1+pn^q−1 (3.8)

≤

∞

X

m=1

∞

X

n=1

K(Am^α, Bn^β)|a_m| |b_n| m¹^qn¹^p

≤

∞

X

m=1

∞

X

n=1

K(Am^α, Bn^β)

m

X

τ=1

|∇a(τ)|^p

!¹_p _n X

δ=1

|∇b(δ)|^q

!¹_q .

Applying the substitutionsea_m = (Pm

τ=1|∇a(τ)|^p)¹^p,eb_n = (Pn

δ=1|∇b(δ)|^q)¹^q and (3.1), we have

∞

X

m=1

∞

X

n=1

K(Am^α, Bn^β)ea_meb_n (3.9)

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< N

∞

X

m=1

mα(1−s)+αp(A1−A2)+(p−1)(1−α)

ea^p_m

!¹_p

·

∞

X

n=1

nβ(1−s)+βq(A2−A1)+(q−1)(1−β)

eb^q_n

!¹_q ,

=

∞

X

m=1 m

X

τ=1

mα(1−s)+αp(A₁−A₂)+(p−1)(1−α)|∇a(τ)|^p

!¹_p

·

∞

X

n=1 n

X

δ=1

nβ(1−s)+βq(A2−A1)+(q−1)(1−β)|∇b(δ)|^q

!¹_q ,

whereA1 ∈ (max{^1−s_q ,^α−1_αq },¹_q), A2 ∈ (max{^1−s_p ,^β−1_βp },¹_p)and the constantN is defined as in (3.3). Now, by applying (3.8) and (3.9) we obtain (3.4). The second inequality (3.5) can be proved by using (3.2) and the inequality

|a_m| ≤m¹^q

m

X

τ=1

|∇a(τ)|^p

!¹_p .

Remark 2. If the functionK(x, y)from the previous corollary is symmetrical, then k(2−s−qA₁) =k(qA₁).So, ifK(x, y) = _(x+y)¹ s,then we can putA₁ =A₂ = ^2−s_pq , A= B =α =β = 1in Corollary3.2and obtain TheoremBfrom the introduction but with the condition2−min{p, q}< s <2.

By using (1.4), see [4], we will obtain a larger interval for the parameters.More precisely, we have:

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Corollary 3.3. Let ¹_p +¹_q = 1withp >1and2−min{p, q}< s ≤2 + min{p, q}.

Let {a(m)} and{b(n)} be two sequences of real numbers wherem, n ∈ N0, and a(0) =b(0) = 0. Then the following inequalities hold

∞

X

m=1

∞

X

n=1

|a_m| |b_n|

(qm^p−1+pn^q−1)(m+n)^s (3.10)

≤ 1 pq

∞

X

m=1

∞

X

n=1

|a_m| |b_n| m¹^qn¹^p(m+n)^s

< N₁ pq

∞

X

m=1 m

X

τ=1

m^1−s|∇a(τ)|^p

!¹_p

·

∞

X

n=1 n

X

δ=1

n^1−s|∇b(δ)|^q

!¹_q , and

(3.11)

∞

X

n=1

n^{(s−1)(p−1)}

∞

X

m=1

|a_m| m¹^q(m+n)^s

!p

< N₁^p

∞

X

m=1 m

X

τ=1

m^1−s|∇a(τ)|^p,

whereN₁ =B(^s+q−2_q ,^s+p−2_p ).

Proof. As in the proof of Corollary3.2, by using Hölder’s inequality we obtain

∞

X

m=1

∞

X

n=1

|am| |bn|

(qm^p−1+pn^q−1)(m+n)^s

≤ 1 pq

∞

X

m=1

∞

X

n=1

|a_m| |b_n| m¹^qn¹^p(m+n)^s

≤ 1 pq

∞

X

m=1

∞

X

n=1

1 (m+n)^s

m

X

τ=1

|∇a(τ)|^p

!¹_p _n X

δ=1

|∇b(δ)|^q

!¹_q m

n

^2−s_pq n m

^2−s_pq

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≤ 1 pq

∞

X

m=1 m

X

τ=1

∞

X

n=1

1 (m+n)^s

m n

^2−s_q

!

|∇a(τ)|^p

!¹_p

·

∞

X

n=1 n

X

δ=1

∞

X

m=1

1 (m+n)^s

n m

^2−s_p

!

|∇b(δ)|^q

!¹_q .

Now, the inequality (3.10) follows from (1.4). Let us show that the inequality (3.11) is valid. For this purpose we use the following inequality from [4]

(3.12)

∞

X

n=1

n^{(s−1)(p−1)}

∞

X

m=1

a_m (m+n)^s

!p

< L₁

∞

X

m=1

m^1−sa^p_m,

where2−min{p, q}< s≤2 + min{p, q}andL1 =B(^s+p−2_p ,^s+q−2_q ).Setting

a_m =

m

X

τ=1

|∇a(τ)|^p

!¹_p

in (3.12) and using

|a_m| ≤m¹^q

m

X

τ=1

|∇a(τ)|^p

!¹_p , the inequality (3.11) follows easily.

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4. Non-conjugate Exponents

Let p, p⁰, q, q⁰ and λ be as in (1.6) and (1.7). To obtain an analogous result for the case of non-conjugate exponents, we introduce real parametersr⁰, r such that p≤r⁰ ≤q⁰and_r¹0+¹_r = 1.For example, we can define_r¹0 = _q¹0+^1−λ₂ orr⁰ = (2−λ)p.

It is easy to see that (4.1) x^p¹⁰y^q¹⁰ ≤ 1

rr⁰

rx^r

0 p0

+r⁰y^q^r⁰

, x≥0, y≥0,

and

(4.2) |f(x)| |g(y)| ≤x^p¹⁰y^q¹⁰ Z x

0

|f⁰(τ)|^pdτ

¹_pZ y

0

, hold, wheref(x), g(y)are absolutely continuous functions on(0,∞).

Applying TheoremC, (4.1) and (4.2) in the same way as in the proof of Theorem 2.1, we obtain the following result for non-conjugate exponents.

Theorem 4.1. Let p, p⁰, q, q⁰ and λ be as in (1.6) and (1.7). Let r⁰, r be real parameters such that p ≤ r⁰ ≤ q⁰ and _r¹0 + ¹_r = 1. If K(x, y), ϕ(x), ψ(y) are non-negative functions andf(x), g(y)are absolutely continuous functions such that f(0) =g(0) = 0,then the following inequalities hold

Z ∞

0

Z ∞

0

K^λ(x, y)|f(x)| |g(y)|

rx^r

0 p0

+r⁰y^q^r⁰ (4.3) dxdy

≤ pq rr⁰

Z ∞

0

Z ∞

0

K^λ(x, y)|f(x)| |g(y)|d x¹^p

d y¹^q

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≤ 1 rr⁰

Z ∞

0

Z x

0

(ϕF)^p(x)|f⁰(τ)|^pdτ dx ¹_p

× Z ∞

0

Z y

0

(ψG)^q(y)|g⁰(δ)|^qdδdy ¹_q

, and

(4.4)

Z ∞

0

1 ψG(y)

Z ∞

0

K^λ(x, y)|f(x)|d(x¹^p) q⁰

dy

!_q¹0

≤ 1 p

Z ∞

0

Z x

0

(ϕF)^p(x)|f⁰(τ)|^pdτ dx ¹_p

,

whereF(x)andG(y)are defined as in TheoremC.

Obviously, Theorem4.1is the generalization of Theorem2.1. Namely, ifλ= 1, r⁰ = p and r = q, then the inequalities (4.3) and (4.4) become respectively the inequalities (2.1) and (2.2). IfK(x, y)is a non-negative symmetrical and homogeneous function of degree−s, s >0,then we obtain:

Corollary 4.2. Lets >0, p, p⁰, q, q⁰ andλbe as in (1.6) and (1.7). Iff(x), g(y)are absolutely continuous functions such thatf(0) = g(0) = 0,andK(x, y)is a non- negative symmetrical and homogeneous function of degree −s, then the following inequalities hold

Z ∞

0

Z ∞

0

K^λ(x, y)|f(x)| |g(y)|

qx^{(p−1)(2−λ)}+py^{(q−1)(2−λ)}dxdy (4.5)