We also show that the constant factors on the right-hand sides of the established inequalities are the best possible

A HALF-DISCRETE VERSION OF THE MULHOLLAND INEQUALITY

BICHENG YANG and MARIO KRNI ´C

Communicated by the former editorial board

In this paper, we derive a half-discrete version of the Mulholland inequality and its equivalent forms. We also show that the constant factors on the right-hand sides of the established inequalities are the best possible. As an application, we consider some operator expressions that correspond to the half-discrete Mulhol- land inequality.

AMS 2010 Subject Classification: 26D15.

Key words: Mulholland inequality, Hilbert-type inequality, weight function, equiv- alent form, the best possible constant factor.

1. INTRODUCTION

In this paper, we refer to a Mulholland inequality ([1], see also paper [14]) (1)

∞

X

m=2

∞

X

n=2

ambn

lnmn < π

" _∞ X

m=2

ma²_m

∞

X

n=2

nb²_n

#¹₂ ,

which holds for all non-negative sequences (a_m)_m∈N\{1} and (b_n)_n∈N\{1} such that 0 < P∞

m=2ma²_m < ∞ and 0 < P∞

n=2nb²_n < ∞. The above inequality includes the best possible constant factor π in the sense that it cannot be replaced with a smaller constant so that (1) still holds.

The above inequality belongs to a more general class of inequalities, these are the Hilbert-type inequalities. Generally speaking (see paper [4]), the Hilbert-type inequality asserts that

Z

Ω

Z

Ω

K(x, y)f(x)g(y)dµ₁(x)dµ₂(y)

≤ Z

Ω

ϕ^p(x)F(x)f^p(x)dµ1(x) ¹

pZ

Ω

ψ^q(y)G(y)g^q(y)dµ2(y) ¹

q

, (2)

where p > 1 and q are conjugate exponents, i.e. 1/p + 1/q = 1, Ω is a measure space with positive σ-finite measuresµ1 and µ2,K : Ω×Ω→R and

MATH. REPORTS16(66),2(2014), 163–174

ϕ, ψ, f, g : Ω → R are non-negative measurable functions, and F(x) = R

ΩK(x, y)ψ^−p(y)dµ₂(y), G(y) =R

ΩK(x, y)ϕ^−q(x)dµ₁(x).

In the last few decades, the Hilbert-type inequalities were extensively studied in numerous particular settings in both integral and discrete case, and are still of interest. These particular settings include various choices of kernel K, the weight functions ϕ, ψ, and the sets of integrations, as well as the extension to the multidimensional case. A considerable attention is also focused on the case of the homogeneous kernels and the power weight functions. For example, considering the homogeneous kernel K(x, y) = (x+y)^−λ, λ > 0, and the power weight functions, in the corresponding Hilbert-type inequality appears the constant factor expressed in terms of the well-known Beta function

B(a, b) = Z ∞

0

t^a−1

(1 +t)^a+bdt, a, b >0.

Of course, numerous researches are focused on determining the inequali- ties with the best possible constant factors and on refinements of the existing Hilbert-type inequalities. For some recent results in the above mentioned di- rections, the reader is referred, for example, to papers [2, 3, 5, 7, 9–11]. In addition, for comprehensive accounts on the Hilbert-type inequalities includ- ing history, different proofs, refinements and diverse applications, we refer to [1, 8, 13, 15].

Hardy, Littlewood, and P`olya [1], already considered some particular cases of inequality (2) equipped with one Lebesgue and the other counting measure. Such inequalities will be referred to as the half-discrete Hilbert-type inequalities. Recently, Yang [16] (see also paper [12]), established the following half-discrete Hilbert-type inequality

Z ∞

0

f(x)

∞

X

n=1

a_n (x+n)^λdx

< B(λ1, λ2) Z ∞

0

x^{p(1−λ1)−1}f^p(x)dx ¹_p"_∞

X

n=1

n^{q(1−λ2)−1}a^q_n

#¹_q , (3)

where p > 1 and q are conjugate exponents, λ1 > 0, 0 < λ2 ≤ 1, λ = λ1+ λ2, and 0 < R∞

0 x^{p(1−λ1)−1}f^p(x)dx < ∞, 0 < P∞

n=1n^{q(1−λ2)−1}a^qn < ∞. In addition, the constant factorB(λ₁, λ₂) is the best possible in (3).

The main objective of this paper is to derive the half-discrete version of the Mulholland inequality (1). The paper is organized in the following way:

after this Introduction, in Section 2 we derive some auxiliary results needed for establishing our main results. Further, in Section 3 we derive the half-discrete version of the Mulholland inequality as well as its equivalent forms. Moreover,

we also show that the constant factors included in such inequalities are the best possible. Finally, in Section 4 we consider some operators on certain weighted spaces that correspond to established inequalities.

2. AUXILIARY RESULTS

In order to obtain our main results, we need to establish some auxiliary results. We begin with the following lemma.

Lemma 2.1. Suppose λ1 >0, 0< λ2 ≤1, λ=λ1+λ2, α ≤1/2, and let the weight functions ω(n) and $(x) be defined as follows:

ω(n) = ln^λ2(n−α) Z ∞

1+α

ln^λ1−1(x−α)dx

(x−α) ln^λ(x−α)(n−α), n∈N\{1}, (4)

$(x) = ln^λ1(x−α)

∞

X

n=2

ln^λ2−1(n−α)

(n−α) ln^λ(x−α)(n−α), x∈ h1 +α,∞i.

(5)

Then,

(6) $(x)< ω(n) =B(λ₁, λ₂).

Proof. Applying the substitutiont= _ln(n−α)^ln(x−α) to relation (4), we obtain ω(n) =

Z ∞

0

1

(1 +t)^λt^λ1−1dt=B(λ1, λ2).

Now, in order to obtain inequality (6), we consider the function h_x(y) = ln^λ2−1(y−α)

(y−α) ln^λ(x−α)(y−α),

for a fixed valuex >1 +α. It is easy to show that the above function is convex on the interval h3/2,∞i. Namely, the above function can be represented as h_x(y) =a(y)b(y)c(y), where a(y) = (y−α)⁻¹, b(y) = ln^−λ(x−α)(y−α) and c(y) = ln^λ2−1(y−α). It follows from a straightforward calculation that the first derivatives of the functions a(y), b(y), c(y) are negative, while the second derivatives are positive, so that

h⁰⁰_x(y) = X

cyclic

a⁰⁰(y)b(y)c(y) + 2X

cyclic

a⁰(y)b⁰(y)c(y)>0

on the intervalh3/2,∞i. Utilizing the fact thath⁰_x(y) is strictly increasing on intervalhn−1/2, n+ 1/2ifor eachn∈N\{1},we haveh_x(n) +h⁰_x(n)(y−n)<

h_x(y), y6=n, that is, hx(n) =

Z n+¹₂

n−¹₂

hx(n) +h⁰_x(n)(y−n) dy <

Z n+¹₂

n−¹₂

hx(y)dy, n∈N\{1}.

The previous inequality can also be obtained by applying the Hermite- Hadamard inequality (see, e.g., [8]) to the function hx(y) on the interval hn−1/2, n+ 1/2i, n ∈ N\{1}. Summing the above inequalities we have P∞

n=2h_x(n)<R∞

3 2

h_x(y)dy, i.e.

$(x)<ln^λ1(x−α) Z ∞

3 2

ln^λ2−1(y−α)

(y−α) ln^λ(x−α)(y−α)dy

t=ln(y−α)/ln(x−α)

=

Z ∞

ln(3/2−α) ln(x−α)

t^λ2−1dt

(1 +t)^λ ≤B(λ₂, λ₁) =B(λ₁, λ₂), which completes the proof.

Lemma 2.2. Suppose that the assumptions as in Lemma 2.1 are fulfilled, and letpandqbe conjugate exponents, i.e. 1/p+1/q = 1,p >1. If(a_n)_n∈N\{1}

is a non-negative sequence of real numbers, and f :h1 +α,∞i →R is a non- negative measurable function, then the following two inequalities hold:

( _∞ X

n=2

ln^pλ2−1(n−α) n−α

Z ∞

1+α

f(x)dx ln^λ(x−α)(n−α)

p)¹

p

≤[B(λ₁, λ₂)]^1q Z ∞

1+α

$(x)(x−α)^p−1ln^{p(1−λ1)−1}(x−α)f^p(x)dx ¹_p

, (7)

(Z ∞

1+α

ln^qλ1−1(x−α) (x−α)[$(x)]^q−1

" _∞ X

n=2

an

ln^λ(x−α)(n−α)

#q

dx )¹_q

≤[B(λ₁, λ₂)]^1q ( _∞

X

n=2

(n−α)^q−1ln^{q(1−λ2)−1}(n−α)a^q_n )¹_q

. (8)

Proof. We first prove inequality (7). By using the well-known H¨older inequality (see,e.g., [8]) and the relation (6), we obtain

Z ∞

1+α

f(x)dx ln^λ(x−α)(n−α)

p

= (Z ∞

1+α

1

ln^λ(x−α)(n−α)

"

ln^(1−λ1)/q(x−α)

ln^(1−λ2)/p(n−α) ·(x−α)^1/q (n−α)^1/pf(x)

#

×

"

ln^(1−λ2)/p(n−α)

ln^(1−λ1)/q(x−α) ·(n−α)^1/p (x−α)^1/q

# dx

)p

≤ Z ∞

1+α

(x−α)^p−1

ln^λ(x−α)(n−α) · ln^{(1−λ1)(p−1)}(x−α)

(n−α) ln^1−λ2(n−α)f^p(x)dx

× (Z ∞

1+α

(n−α)^q−1

ln^λ(x−α)(n−α) · ln^{(1−λ2)(q−1)}(n−α) (x−α) ln^1−λ1(x−α)dx

)p−1

= (

ω(n)ln^{q(1−λ2)−1}(n−α) (n−α)^1−q

)p−1

× Z ∞

1+α

(x−α)^p−1ln^{(1−λ1)(p−1)}(x−α)

(n−α) ln^λ(x−α)(n−α) ln^1−λ2(n−α)f^p(x)dx

= [B(λ1, λ2)]^p−1 ln^pλ2−1(n−α)

Z ∞

1+α

(x−α)^p−1ln^{(1−λ1)(p−1)}(x−α)f^p(x)dx ln^λ(x−α)(n−α) ln^1−λ2(n−α) .

Moreover, denoting the left-hand side of inequality (7) byJ, the Lebesgue term by term integration theorem (see,e.g., [6]) yields

J ≤[B(λ1, λ2)]^1q ( _∞

X

n=2

Z ∞

1+α

(x−α)^p−1ln^{(1−λ1)(p−1)}(x−α)f^p(x)dx (n−α) ln^λ(x−α)(n−α) ln^1−λ2(n−α)

)¹_p

= [B(λ₁, λ₂)]^1q (Z ∞

1+α

∞

X

n=2

(x−α)^p−1ln^{(1−λ1)(p−1)}(x−α)f^p(x)dx (n−α) ln^λ(x−α)(n−α) ln^1−λ2(n−α)

)¹_p

= [B(λ₁, λ₂)]^1q Z ∞

1+α

$(x)(x−α)^p−1ln^{p(1−λ1)−1}(x−α)f^p(x)dx ¹_p

, so (7) follows.

It remains to prove inequality (8). Another use of the H¨older inequality yields

"_∞ X

n=2

a_n

ln^λ(x−α)(n−α)

#q

= ( _∞

X

n=2

1

ln^λ(x−α)(n−α)

"

ln^(1−λ1)/q(x−α)

ln^(1−λ2)/p(n−α)· (x−α)^1/q (n−α)^1/p

#

×

"

ln^(1−λ2)/p(n−α)

ln^(1−λ1)/q(x−α) ·(n−α)^1/p (x−α)^1/qa_n

#)q

≤ ( _∞

X

n=2

1

ln^λ(x−α)(n−α) ·(x−α)^p−1ln^{(1−λ1)(p−1)}(x−α) (n−α) ln^1−λ2(n−α)

)q−1

×

∞

X

n=2

(n−α)^q−1

ln^λ(x−α)(n−α) · ln^{(1−λ2)(q−1)}(n−α) (x−α) ln^1−λ1(x−α)a^q_n

= [$(x)]^q−1 ln^qλ1−1(x−α)

∞

X

n=2

(n−α)^q−1ln^λ1−1(x−α)

ln^λ(x−α)(n−α) ln^{(q−1)(1−λ2)}(n−α)a^q_n. Finally, utilizing the Lebesgue term by term integration theorem, we have that the left-hand side of inequality (8) is not greater than

(Z ∞

1+α

∞

X

n=2

(n−α)^q−1ln^λ1−1(x−α)

(x−α) ln^λ(x−α)(n−α)ln^{(q−1)(1−λ2)}(n−α)a^q_ndx )¹

q

= ( _∞

X

n=2

"

Z ∞

1+α

ln^λ2(n−α) ln^λ1−1(x−α)dx (x−α) ln^λ(x−α)(n−α)

#ln^{q(1−λ2)−1}(n−α) (n−α)^1−q a^q_n

)¹_q

= ( _∞

X

n=2

ω(n)(n−α)^q−1ln^{q(1−λ2)−1}(n−α)a^q_n )¹_q

,

which completes the proof.

3. MAIN RESULTS

In this section, we derive a half-discrete version of the Mulholland inequal- ity for non-negative measurable functions and sequences. These functions and sequences belong to the corresponding weighted spaces.

For 0 < r < ∞ and a weight function w on hα, βi ⊆ R, we denote by Lr,w(α, β) the weighted Lebesgue space, consisting of all functionsf on Ω with a finite norm

kfk_r,w = Z β

α

|f(x)|^rw(x)dx ¹_r

.

Similarly, for 0< r <∞ and a weight sequence w = (wn)n∈N we denote byl_r,wthe weighted sequencel_p-space of all sequencesa= (a_n)n∈Nwith a finite norm

kak_r,w=

"_∞ X

n=1

|a_n|^rwn

#¹_r .

Regarding the results from the previous section, we shall here be focused on the weight function Φ defined by

Φ(x) = (x−α)^p−1ln^{p(1−λ1)−1}(x−α), x∈ h1 +α,∞i,

and the weight sequence Ψ defined by

Ψn= (n−α)^q−1ln^{q(1−λ2)−1}(n−α), n∈N\{1}.

Moreover, the quantities Φ^1−q(x) = ^lnqλ1_x−α^−1(x−α),and Ψ^1−pn = ^lnpλ2_n−α^−1(n−α) will also appear in the results that follow.

Now, we are ready to establish the half-discrete version of the Mulholland inequality in the above described setting. In addition, we also derive its two equivalent forms.

Theorem 3.1. Suppose p > 1 and q are conjugate exponents and let λ₁ > 0, 0 < λ₂ ≤ 1, λ = λ₁ +λ₂, α ≤ 1/2. If f ∈ L_p,Φ(1 +α,∞) is a non-negative function such that ||f||_p,Φ > 0, and a = (an)n∈N\{1} ∈ lq,Ψ is a non-negative sequence such that ||a||_q,Ψ > 0, then the following inequalities hold and are equivalent:

∞

X

n=2

Z ∞

1+α

a_nf(x)dx ln^λ(x−α)(n−α)

= Z ∞

1+α

∞

X

n=2

f(x)andx

ln^λ(x−α)(n−α) < B(λ1, λ2)||f||_p,Φ||a||_q,Ψ, (9)

(10)

( _∞ X

n=2

Ψ^1−p_n Z ∞

1+α

f(x)dx ln^λ(x−α)(n−α)

p)¹_p

< B(λ₁, λ₂)||f||_p,Φ,

(11)

(Z ∞

1+α

Φ^1−q(x)

"_∞ X

n=2

a_n

ln^λ(x−α)(n−α)

#q

dx )¹_q

< B(λ₁, λ₂)||a||_q,Ψ. Proof. Let I, J, and L respectively, denote the left-hand sides of in- equalities (9), (10), and (11). Clearly, by virtue of the Lebesgue term by term integration theorem, the two expressions for I in (9) are equal.

By Lemma 2.1,$(x)< B(λ₁, λ₂), hence, inequality (10) follows immedi- ately from (7). Now, the application of the H¨older inequality to the left-hand side of (9) yields inequality

I =

∞

X

n=2

Ψ⁻

1

nq

Z ∞

1+α

f(x)dx ln^λ(x−α)(n−α)

[Ψ

1

nqa_n]≤J||a||_q,Ψ, which implies inequality (9), due to (10).

On the other hand, assuming that (9) is valid, and considering the se- quencea= (an)n∈N\{1}, defined by

a_n= Ψ^1−p_n Z ∞

1+α

f(x)dx ln^λ(x−α)(n−α)

p−1

,

we have J^p−1 = ||a||_q,Ψ.Clearly, taking into account the relation (7), we find that J < ∞. Now, if J = 0 then (10) holds trivially, otherwise, utilizing the inequality (9), we have ||a||^q_q,Ψ = J^p = I < B(λ1, λ2)||f||_p,Φ||a||_q,Ψ, that is, ||a||^q−1_q,Ψ = J < B(λ₁, λ₂)||f||_p,Φ. Therefore, inequalities (9) and (10) are equivalent.

It remains to establish inequality (11) and show its equivalence with (9) and (10). Sinceq >1, Lemma 2.1 implies that$^1−q(x)> B^1−q(λ1, λ2), hence, inequality (11) follows from (8).

Now, another application of the H¨older inequality to the left-hand side of (9) yields

I = Z ∞

1+α

[Φ^1p(x)f(x)]

"

Φ^−1p(x)

∞

X

n=2

an

ln^λ(x−α)(n−α)

#

dx≤ ||f||_p,ΦL, which provides inequality (9) as a consequence of (11).

On the other hand, assuming that the inequality (9) is valid, and consid- ering the function

f(x) = Φ^1−q(x)

"_∞ X

n=2

an

ln^λ(x−α)(n−α)

#q−1

, x∈ h1 +α,∞i,

we have L^q−1 = ||f||_p,Φ. Obviously, relation (8) implies that L < ∞. If L = 0, then inequality (11) holds trivially, otherwise, taking into account (9), we have ||f||^p_p,Φ = L^q = I < B(λ1, λ2)||f||_p,Φ||a||_q,Ψ, i.e. ||f||^p−1_p,Φ = L <

B(λ1, λ2)||a||_q,Ψ, which provides equivalence of the inequalities (9) and (11).

The proof is now completed.

Similarly as in the classical case, the half-discrete inequalities (9), (10), and (11) involve the best possible constant factor.

Theorem 3.2. The constant factor B(λ₁, λ₂) is the best possible in in- equalities (9), (10), and (11).

Proof. Due to the equivalence, it is enough to show that the constant factor B(λ1, λ2) is the best possible in inequality (9).

Suppose that the constant factorB(λ₁, λ₂) is not the best possible in (9).

This means that there exists a positive constant 0< k < B(λ1, λ2) such that the inequality

(12)

∞

X

n=2

Z ∞

1+α

anf(x)dx

ln^λ(x−α)(n−α) < k||f||_p,Φ||a||_q,Ψ

holds for all non-negative f ∈L_p,Φ(1 +α,∞) and a= (a_n)_n∈N\{1} ∈l_q,Ψ such that ||f||_p,Φ >0 and ||a||_q,Ψ>0.

Further, consider the sequenceeadefined asean= _n−α¹ ln^{λ2−εq−1}(n−α), n∈ N\{1},and the function

f(x) =e

( 0, x∈ h1 +α, e+αi

1

x−αln^{λ1−pε−1}(x−α), x∈[e+α,∞) ,

where 0< ε < pλ₁.Our next step is to substitute the above sequence and func- tion in inequality (12). In described setting, the right-hand side of inequality (12) yields the estimate

k||fe||_p,Φ||ea||_q,Ψ =k Z ∞

e+α

dx

(x−α) ln^ε+1(x−α) ¹_p

×

( 1

(2−α) ln^ε+1(2−α)+

∞

X

n=3

1

(n−α) ln^ε+1(n−α) )¹_q

< k(1 ε)^1p

1

(2−α) ln^ε+1(2−α)+ Z ∞

2

dy

(y−α) ln^ε+1(y−α) ¹_q

= k ε

ε

(2−α) ln^ε+1(2−α)+ 1 ln^ε(2−α)

¹_q . (13)

On the other hand, considering the left-hand side of (12), denoted here by I, in the same setting, we havee

Ie=

∞

X

n=2

ln^{λ2−εq−1}(n−α) n−α

Z ∞

e+α

ln^{λ1−εp−1}(x−α)dx (x−α) ln^λ(x−α)(n−α)

=^{t=ln(x−α)/ln(n−α)}

∞

X

n=2

1

(n−α) ln^ε+1(n−α) Z ∞

1 ln(n−α)

t^{λ1−εp−1}dt (t+ 1)^λ

=B

λ1−ε

p, λ2+ ε p

∞

X

n=2

1

(n−α) ln^ε+1(n−α) −A(ε)

> B

λ₁−ε

p, λ₂+ ε p

Z ∞

2

dy

(y−α) ln^ε+1(y−α) −A(ε)

= 1

εln^ε(2−α)B

λ1− ε

p, λ2+ε p

−A(ε), (14)

where

A(ε) =

∞

X

n=2

1

(n−α) ln^ε+1(n−α)

Z 1/ln(n−α) 0

t^{λ1−εp−1} (t+ 1)^λdt.

Obviously, 0 < A(ε)≤

∞

X

n=2

1

(n−α) ln^ε+1(n−α)

Z 1/ln(n−α) 0

t^{λ1−pε−1}dt

= 1

λ1−^ε_p

∞

X

n=2

1

(n−α) ln^λ1+εq+1(n−α) <∞, (15)

that is, A(ε) =O(1), ε→0⁺. Hence, taking into account the estimates (13), (14), and (15), we have

1 ln^ε(2−α)B

λ₁−ε

p, λ₂+ε p

−εO(1)

< k

ε

(2−α) ln^ε+1(2−α) + 1 ln^ε(2−α)

¹_q . (16)

Finally, letting ε→0⁺, the above relation (16) asserts that B(λ₁, λ₂)≤ k, which contradicts with our assumption 0 < k < B(λ1, λ2). Therefore, B(λ1, λ2) is the best possible constant factor in (9).

4.APPLICATIONS AND CONCLUDING REMARKS

Inequalities (10) and (11) enable us to establish some interesting operators on the corresponding weighted spaces. Due to Theorem 3.2, we shall be able to determine the norm of such operators.

Remark 4.1.Regarding the notations as in Theorem 3.1, we define the half-discrete Mulholland operatorT :Lp,Φ(1 +α,∞)→l_p,Ψ^1−p in the following way: If f ∈Lp,Φ(1 +α,∞),thenTf ∈l_p,Ψ^1−p is defined as

(Tf)_n= Z ∞

1+α

1

ln^λ(x−α)(n−α)f(x)dx, n∈N\{1}.

By virtue of (10), the operatorT is well-defined and, moreover, it follows that ||Tf||_p,Ψ^1−p ≤ B(λ₁, λ₂)||f||_p,Φ. This means that T is bounded, that is,

||T || ≤B(λ1, λ2),since

kT k= sup

f∈Lp,Φ(1+α,∞),

||f||p,Φ6=0

||T f||_p,Ψ1−p

||f||_p,Φ .

In addition, since the constant factor B(λ₁, λ₂) is the best possible in (10), we also have ||T ||=B(λ₁, λ₂).

Similarly, by virtue of (11), we also define the Mulholland operator Te : l_q,Ψ→L_q,Φ^1−q(1 +α,∞) as

Tea(x) =

∞

X

n=2

1

ln^λ(x−α)(n−α)a_n, x∈ h1 +α,∞i.

As in the first case, inequality (11) asserts that||Tea||_q,Φ1−q ≤B(λ1, λ2)||a||_q,Ψ, that is,Teis a bounded operator with||T || ≤e B(λ₁, λ₂).Moreover, Theorem 3.2 yields the norm of this operator i.e. ||T ||e =B(λ₁, λ₂).

We conclude this paper with the half-discrete versions of inequalities (9), (10), and (11) in some particular settings concerning parameters p,q, λ₁, λ₂, λ, and α.

Remark 4.2.Let p=q = 2, λ₁ =λ₂ = 1/2, and λ= 1. If α = 1/2, then the inequalities (9), (10), and (11), in the expanded forms, respectively read

Z ∞

3 2

f(x)

∞

X

n=2

an

ln(x−¹₂)(n−¹₂)dx < π (Z ∞

3 2

x− 1

2

f²(x)dx

∞

X

n=2

n−1

2

a²_n )¹

2

,

∞

X

n=2

1 n−¹₂

"

Z ∞

3 2

f(x)dx ln(x−¹₂)(n−¹₂)

#2

< π² Z ∞

3 2

x− 1

2

f²(x)dx, Z ∞

3 2

1 x− ¹₂

"_∞ X

n=2

an

ln(x−¹₂)(n−¹₂)

#2

dx < π²

∞

X

n=2

n−1

2

a²_n,

while for α= 0 we obtain Z ∞

1

f(x)

∞

X

n=2

a_n

lnxndx < π (Z ∞

1

xf²(x)dx

∞

X

n=2

na²_n )¹₂

,

∞

X

n=2

1 n

Z ∞

1

f(x) lnxndx

2

< π² Z ∞

1

xf²(x)dx, Z ∞

1

1 x

"_∞ X

n=2

a_n lnxn

#2

dx < π²

∞

X

n=2

na²_n.

Of course, both series represent the equivalent inequalities with the best possible constant factors on the right-hand sides.

Acknowledgments.This research was supported by the Natural Science Foundation of Guangdong, under Research Grant 7004344 (first author), and the Croatian Ministry of Science, Education, and Sports, under Research Grant 036–1170889–1054 (second author).

REFERENCES

[1] G.H. Hardy, J.E. Littlewood and G. P`olya,Inequalities. Cambridge Univ. Press, 1934.

[2] J. Jin and L. Debnath, On a Hilbert-type linear series operator and its applications.

J. Math. Anal. Appl. 371(2010), 691–704.

[3] M. Krni´c and J. Peˇcari´c,Hilbert’s inequalities and their reverses. Publ. Math. Debrecen 67(2005), 315–331.

[4] M. Krni´c and J. Peˇcari´c, General Hilbert’s and Hardy’s inequalities. Math. Inequal.

Appl. 8(2005), 29–51.

[5] M. Krni´c and J. Peˇcari´c,Extension of Hilbert’s Inequality. J. Math. Anal. Appl. 324 (2006), 150–160.

[6] J. Kuang,Introduction to real analysis. Hunan Education Press, Chansha, China, 1996.

[7] Y. Li and B. He,On inequalities of Hilbert’s type. Bull. Aust. Math. Soc. 76(2007), 1–13.

[8] D.S. Mitrinovi´c, J.E. Peˇcari´c and A.M. Fink,Inequalities involving functions and their integrals and derivatives. Kluwer Academic Publishers, Boston, 1991.

[9] B. Yang,On Hilbert’s integral inequality. J. Math. Anal. Appl. 220(1998), 778–785.

[10] B. Yang and T. Rassias,On the way of weight coefficient and research for Hilbert-type inequalities. Math. Ineqal. Appl. 6(2003), 625–658.

[11] B. Yang, I. Brneti´c, M. Krni´c and J. Peˇcari´c,Generalization of Hilbert and Hardy-Hilbert integral inequalities. Math. Inequal Appl. 8(2005), 259–272.

[12] B. Yang,A mixed Hilbert-type inequality with a best constant factor. Int. J. Pure Appl.

Math. 20(2005), 319–328.

[13] B. Yang,Hilbert-type integral inequalities. Bentham Science Publishers Ltd., 2009.

[14] B. Yang,An extension of Mulholland’s inequality. Jordan J. Math. Statistics3(2010), 151–157.

[15] B. Yang,Discrete Hilbert-type inequalities. Bentham Science Publishers Ltd., 2011.

[16] B. Yang,A half-discrete Hilbert’s inequality. J. Guangdong University of Education31 (2011), 1–7.

Received 14 December 2011 Guangdong Education Institute, Department of Mathematics, Guangzhou, Guangdong 510303,

China bcyang@gdei.edu.cn University of Zagreb,

Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb,

Croatia mario.krnic@fer.hr