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A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order

  • Jianquan Liao and Bicheng Yang EMAIL logo
Published/Copyright: November 21, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, a new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order is obtained, by using the weight functions, the mid-value theorem, and the techniques of real analysis. A few equivalent statements of the best possible constant factor related to several parameters are considered. As applications, the equivalent forms and some particular inequalities are provided.

Keywords: half-discrete inequality; partial sums; weight function; best possible constant factor; parameter; reverse
MSC 2010: 26D15

1 Introduction

Assuming that p > 1 , 1 p + 1 q = 1 , a m , b n ≥ 0 , 0 < ∑ m = 1 ∞ a m p < ∞ , and 0 < ∑ n = 1 ∞ b n q < ∞ , we have the well-known Hardy-Hilbert’s inequality with the best possible constant factor π sin π p as follows (cf. [1], Theorem 315):

(1) ∑ n = 1 ∞ ∑ m = 1 ∞ a m b n m + n < π sin π p ∑ m = 1 ∞ a m p 1 p ∑ n = 1 ∞ b n q 1 q .

In 2006, by means of Euler-Maclaurin’s summation formula, Krnić and Pečarić (cf. [2]) gave an extension of equation (1) as follows:

(2) ∑ n = 1 ∞ ∑ m = 1 ∞ a m b n ( m + n ) λ < B ( λ 1 , λ 2 ) ∑ m = 1 ∞ m p ( 1 − λ 1 ) − 1 a m p 1 p ∑ n = 1 ∞ n q ( 1 − λ 2 ) − 1 b n q 1 q ,

where, λ i ∈ ( 0 , 2 ] ( i = 1 , 2 ) , λ 1 + λ 2 = λ ∈ ( 0 , 4 ] , B ( λ 1 , λ 2 ) is the best possible constant factor, and

B ( u , v ) = ∫ 0 ∞ t u − 1 ( 1 + t ) u + v d t ( u , v > 0 )

is the beta function. For λ = 1 , λ 1 = 1 q , and λ 2 = 1 p in equation (2), we have equation (1); for p = q = 2 and λ 1 = λ 2 = λ 2 , equation (2) reduces to the inequality in Yang’s article [3].

We still had a half-discrete Hilbert-type inequality with the nonhomogeneous kernel in 1934 as follows (cf. [1], Theorem 351): if K ( x ) is decreasing function, p > 1 , 1 p + 1 q = 1 , 0 < ϕ ( s ) = ∫ 0 ∞ K ( x ) x s − 1 d x < ∞ , f ( x ) ≥ 0 , and 0 < ∫ 0 ∞ f p ( x ) < ∞ , then

(3) ∑ m = 1 ∞ n p − 2 ∫ 0 ∞ K ( n x ) f ( x ) d x p < ϕ p 1 q ∫ 0 ∞ f p ( x ) d x .

Recently, some extensions of equation (3) were provided by Rassias and Yang [4] and Yang and Debnath [5]. In 2016, Hong and Wen [6] discussed the equivalent description of Hilbert-type inequality similar to equation (1) with the general homogeneous kernel related to some parameters and the optimal constant factors. In some studies by Hong and co-workers [7–15], further works are considered. In 2019, Adiyasuren et al. [16] gave an extension of equation (2) involving two partial sums. In 2023, Hong et al. [17] obtained a new more accurate half-discrete multidimensional Hilbert-type inequality involving one multiple upper limit function.

In this article, following the way of Hong et al. [17], by using the weight functions, the mid-value theorem, and the techniques of real analysis, a new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order is obtained, which is a new idea to extend the results of Adiyasuren et al. [16] into the field of reverses. The equivalent statements of the best possible constant factor related to several parameters are considered. As applications, the equivalent forms and some particular inequalities are provided.

2 Some lemmas

In what follows, we suppose that p < 0   ( 0 < q < 1 ) , 1 p + 1 q = 1 , N = { 1 , 2 , … } , m ∈ N ∪ { 0 } , λ > 0 , λ 1 ∈ ( 0 , λ ) , λ 2 ∈ ( 0 , 1 ] ∩ ( 0 , λ ) ,

k λ ( λ i ) ≔ B ( λ i , λ − λ i ) ( i = 1 , 2 ) ,

λ ^ 1 ≔ λ − λ 2 p + λ 1 q , λ ^ 2 ≔ λ − λ 1 q + λ 2 p , v ′ ( x ) > 0 , v ″ ( x ) ≤ 0   ( x > 0 ) , v ( 0 + ) = 0 , and v ( ∞ ) = ∞ . We also assume that f ( m ) ( x ) is a nonnegative continuous function unless at finite points in R + ≔ ( 0 , ∞ ) , for m ∈ N , k = 1 , … , m , f ( k − 1 ) ( x ) are differentiable functions with f ( k − 1 ) ( 0 + ) = 0 ,

f ( m ) ( x ) = o ( e t x ) ( t > 0 ; x → ∞ ) ,

and for a n ≥ 0 , A n = ∑ k = 1 n a k , e − t v ( n ) A n = o ( 1 )   ( n → ∞ ) , f ( m ) ( x ) ≥ 0 , satisfying

(4) 0 < ∫ 0 ∞ x p ( 1 − λ ^ 1 ) − 1 ( f ( m ) ( x ) ) p d x < ∞ , 0 < ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ ^ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q < ∞ .

Note. In view of the assumption, we observe that f ( k − 1 ) ( x ) are nonnegative increasing in R + and f ( k − 1 ) ( ∞ ) = ∞ (or constant) ( k = 1 , … , m ) . If f ( k − 1 ) ( ∞ ) = ⋯ = f ( m − 1 ) ( ∞ ) = ∞ , then we find

lim x → ∞ f ( k − 1 ) ( x ) e t x = ⋯ = 1 t m − k + 1 lim x → ∞ f ( m ) ( x ) e t x = 0 ( k = 1 , … , m ) ;

otherwise, there exists a least k 0 ∈ { k , … , m } such that f ( k 0 − 1 ) ( ∞ ) = constant and

lim x → ∞ f ( k − 1 ) ( x ) e t x = ⋯ = 1 t k 0 − k lim x → ∞ f ( k 0 − 1 ) ( x ) e t x = 0 .

Lemma 1

Define the following weight function:

(5) ω ˜ λ ( λ 2 , x ) ≔ x λ − λ 2 ∑ n = 1 ∞ v λ 2 − 1 ( n ) v ′ ( n ) ( x + v ( n ) ) λ ( x ∈ R + ) .

We have the following inequality:

(6) ω ˜ λ ( λ 2 , x ) < k λ ( λ 2 ) ( x ∈ R + ) .

Proof

Since v ′ ( x ) > 0 , v ″ ( x ) ≤ 0 , and v ( 0 + ) = 0 , it follows that v ( x )   ( x ∈ R + ) is positive increasing and v ′ ( x ) is decreasing. For fixed x > 0 , define the following positive function g x ( t ) ≔ v λ 2 − 1 ( t ) v ′ ( t ) ( x + v ( t ) ) λ ( t > 0 ) . By the assumption, g x ( t ) is strictly decreasing with respect to t ∈ R + . In view of the decreasingness property of series, we have the following inequality:

(7) ∑ n = 1 ∞ g x ( n ) < ∫ 0 ∞ g x ( t ) d t .

Setting u = v ( t ) x , we have

∫ 0 ∞ g x ( t ) d t = ∫ 0 ∞ v λ 2 − 1 ( t ) v ′ ( t ) ( x + v ( t ) ) λ d t = 1 x λ ∫ 0 ∞ ( x u ) λ 2 − 1 ( 1 + u ) λ x d u = 1 x λ − λ 2 ∫ 0 ∞ u λ 2 − 1 ( 1 + u ) λ d u = 1 x λ − λ 2 k λ ( λ 2 ) ,

and then,

ω ˜ λ ( λ 2 , x ) = x λ − λ 2 ∑ n = 1 ∞ g x ( n ) < x λ − λ 2 ∫ 0 ∞ g x ( t ) d t = k λ ( λ 2 ) .

Then, inequality (6) follows.

The lemma is proved.□

Lemma 2

We have the following reverse Hilbert-type inequality:

(8) I λ ≔ ∫ 0 ∞ ∑ n = 1 ∞ a n f ( m ) ( x ) d x ( x + v ( n ) ) λ > ( k λ ( λ 2 ) ) 1 p ( k λ ( λ 1 ) ) 1 q × ∫ 0 ∞ x p ( 1 − λ ^ 1 ) − 1 ( f ( m ) ( x ) ) p d x 1 p ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ ^ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q 1 q .

Proof

Setting u = x ⁄ v ( n ) , for 0 < λ 1 < λ , we obtain the following expression of another weight function:

(9) ω λ ( λ 1 , n ) ≔ v λ − λ 1 ( n ) ∫ 0 ∞ x λ 1 − 1 ( x + v ( n ) ) λ d x = ∫ 0 ∞ u λ 1 − 1 ( u + 1 ) λ d u = k λ ( λ 1 ) , n ∈ N .

By the reverse Hölder’s inequality (cf. [18]) and Lebesgue term by term integration theorem (cf. [18]), we have

(10) I λ = ∫ 0 ∞ ∑ n = 1 ∞ 1 ( x + v ( n ) ) λ x 1 − λ 1 q ( v ′ ( n ) ) 1 p ( v ( n ) ) 1 − λ 2 p f ( m ) ( x ) × ( v ( n ) ) 1 − λ 2 p a n x 1 − λ 1 q ( v ′ ( n ) ) 1 p d x ≥ ∫ 0 ∞ ∑ n = 1 ∞ 1 ( x + v ( n ) ) λ x ( 1 − λ 1 ) ( p − 1 ) v ′ ( n ) ( v ( n ) ) 1 − λ 2 ( f ( m ) ( x ) ) p d x 1 p × ∑ n = 1 ∞ ∫ 0 ∞ 1 ( x + v ( n ) ) λ ( v ( n ) ) ( 1 − λ 2 ) ( q − 1 ) x 1 − λ 1 ( v ′ ( n ) ) q − 1 d x a n q 1 q = ∫ 0 ∞ ω ˜ λ ( λ 2 , x ) x p ( 1 − λ ^ 1 ) − 1 ( f ( m ) ( x ) ) p d x ∑ n = 1 ∞ ω λ ( λ 1 , n ) ( v ( n ) ) q ( 1 − λ ^ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q 1 q .

Then, by equations (6) and (9), equation (8) follows.

The lemma is proved.□

Lemma 3

For t > 0 and m ∈ N ∪ { 0 } , we have the following expression:

(11) ∫ 0 ∞ e − t x f ( x ) d x = 1 t m ∫ 0 ∞ e − t x f ( m ) ( x ) d x .

Proof

For m = 0 , equation (11) is naturally valid. For m ∈ N , by the assumption and the Note, for k = 1 , … , m , we find e − t x f ( k − 1 ) ( x ) ∣ 0 ∞ = 0 , and

∫ 0 ∞ e − t x f ( k ) ( x ) d x = ∫ 0 ∞ e − t x d f ( k − 1 ) ( x ) = e − t x f ( k − 1 ) ( x ) ∣ 0 ∞ − ∫ 0 ∞ f ( k − 1 ) ( x ) d e − t x = t ∫ 0 ∞ e − t x f ( k − 1 ) ( x ) d x .

By substitution of k = 1 , … , m in the above expression, we obtain equation (11).

The lemma is proved.□

Lemma 4

For t > 0 , we have the following inequality:

(12) ∑ n = 1 ∞ e − t v ( n ) v ′ ( n ) A n ≥ 1 t ∑ n = 1 ∞ e − t v ( n ) a n .

Proof

For x ∈ [ n , n + 1 ]   ( n ∈ N ) , and setting g ( x ) = e − t v ( x ) , we obtain g ′ ( x ) = − t v ′ ( x ) e − t v ( x ) . Since v ″ ( x ) ≤ 0 , then v ′ ( x ) is decreasing in [ n , n + 1 ] . In view of e − t v ( n ) A n = o ( 1 )   ( n → ∞ ) and by Abel’s summation formula and the mid-value theorem, we have

∑ n = 1 ∞ e − t v ( n ) a n = lim n → ∞ e − t v ( n ) A n + ∑ n = 1 ∞ ( e − t v ( n ) − e − t v ( n + 1 ) ) A n = − ∑ n = 1 ∞ ( e − t v ( n + 1 ) − e − t v ( n ) ) A n = − ∑ n = 1 ∞ ( g ( n + 1 ) − g ( n ) ) A n = − ∑ n = 1 ∞ g ′ ( n + θ ) A n = t ∑ n = 1 ∞ v ′ ( n + θ ) e − t v ( n + θ ) A n ≤ t ∑ n = 1 ∞ e − t v ( n ) v ′ ( n ) A n ( 0 < θ < 1 ) .

Hence, we have equation (12).

The lemma is proved.□

3 Main results

Theorem 1

For m ∈ N ∪ { 0 } , we have the following reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of m-order:

(13) I ≔ ∫ 0 ∞ ∑ n = 1 ∞ v ′ ( n ) A n f ( x ) d x ( x + v ( n ) ) λ + m + 1 > Γ ( λ ) ( k λ ( λ 2 ) ) 1 p Γ ( λ + m + 1 ) ( k λ ( λ 1 ) ) 1 q × ∫ 0 ∞ x p ( 1 − λ ^ 1 ) − 1 ( f ( m ) ( x ) ) p d x 1 p ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ ^ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q 1 q .

In particular, for λ 1 + λ 2 = λ , we have

0 < ∫ 0 ∞ x p ( 1 − λ 1 ) − 1 ( f ( m ) ( x ) ) p d x < ∞ , 0 < ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q < ∞ ,

and the following reverse inequality:

(14) I = ∫ 0 ∞ ∑ n = 1 ∞ v ′ ( n ) A n f ( x ) ( x + v ( n ) ) λ + m + 1 d x > Γ ( λ ) B ( λ 1 , λ 2 ) Γ ( λ + m + 1 ) × ∫ 0 ∞ x p ( 1 − λ 1 ) − 1 ( f ( m ) ( x ) ) p d x 1 p ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q 1 q .

Proof

Since

1 ( x + v ( n ) ) λ + m + 1 = 1 Γ ( λ + m + 1 ) ∫ 0 ∞ t λ + m e − ( x + v ( n ) ) t d t ,

and by Lebesgue term-by-term integration theorem (cf. [19]) and (11), we have

I = 1 Γ ( λ + m + 1 ) ∫ 0 ∞ ∑ n = 1 ∞ v ′ ( n ) A n f ( x ) ∫ 0 ∞ t λ + m e − ( x + v ( n ) ) t d t d x = 1 Γ ( λ + m + 1 ) ∫ 0 ∞ t λ + m ∫ 0 ∞ e − x t f ( x ) d x ∑ n = 1 ∞ e − v ( n ) t v ′ ( n ) A n d t ≥ 1 Γ ( λ + m + 1 ) ∫ 0 ∞ t λ − 1 ∫ 0 ∞ e − x t f ( m ) ( x ) d x ∑ n = 1 ∞ e − v ( n ) t a n d t = 1 Γ ( λ + m + 1 ) ∫ 0 ∞ ∑ n = 1 ∞ a n f ( m ) ( x ) ∫ 0 ∞ t λ − 1 e − ( x + v ( n ) ) t d t d x = Γ ( λ ) Γ ( λ + m + 1 ) ∫ 0 ∞ ∑ n = 1 ∞ a n f ( m ) ( x ) ( x + v ( n ) ) λ d x .

Then, by equation (10), we have equation (13). For λ 1 + λ 2 = λ , equation (13) reduces to equation (14).

The theorem is proved.□

Theorem 2

If λ 1 + λ 2 = λ , then the constant factor

Γ ( λ ) Γ ( λ + m + 1 ) ( k λ ( λ 2 ) ) 1 p ( k λ ( λ 1 ) ) 1 q

in equation (13) is the best possible. On the other hand, if the same constant factor in equation (13) is the best possible, then for

λ − λ 1 − λ 2 ∈ ( − q λ 2 , q ( λ − λ 2 ) ) ∩ ( − q λ 2 , q ( 1 − λ 2 ) ] ,

we have λ 1 + λ 2 = λ .

Proof

If λ 1 + λ 2 = λ , then, equation (13) reduces to equation (14).

For any 0 < ε < min { ∣ p ∣ λ 2 , q λ 2 } , we set a ˜ n ≔ ( v ( n ) ) λ 2 − ε q − 1 v ′ ( n ) , n ∈ N ,

f ˜ ( m ) ( x ) = 0 , 0 < x ≤ 1 , ∏ i = 0 m − 1 λ 1 + i − ε p x λ 1 − ε p − 1 , x > 1 , f ˜ ( m − k ) ( x ) = ∫ 0 x ∫ 0 t k ⋯ ∫ 0 t 2 f ˜ ( m ) ( t 1 ) d t 1 ⋯ d t k − 1 d k ≥ 0 ( k = 1 , … , m ) ,

where we indicate that ∏ i = 0 m − 1 λ 1 + i − ε p = 1 for m = 0 . We find

A ˜ n ≔ ∑ k = 1 n a ˜ k = ∑ k = 1 n ( v ( k ) ) λ 2 − ε q − 1 v ′ ( k ) < ∫ 0 n ( v ( x ) ) λ 2 − ε q − 1 v ′ ( x ) d x = ( v ( n ) ) λ 2 − ε q λ 2 − ε q = o ( e t v ( n ) ) ( t > 0 ; n → ∞ ) .

We obtain that for m ∈ N , f ˜ ( k − 1 ) ( x ) are differentiable functions with f ˜ ( k − 1 ) ( 0 + ) = 0   ( k = 1 , … , m ) , f ( m ) ( x ) = o ( e t x ) ( t > 0 ; x → ∞ ) , and f ˜ ( x ) = 0   ( 0 < x ≤ 1 ) ,

f ˜ ( x ) = ∏ i = 0 m − 1 λ 1 + i − ε p ∫ 1 x ∫ 1 t m ⋯ ∫ 1 t 2 t 1 λ 1 − ε p − 1 d t 1 ⋯ d t m − 1 d t m ≤ ∏ i = 0 m − 1 λ 1 + i − ε p ∫ 0 x ∫ 0 t m ⋯ ∫ 0 t 2 t 1 λ 1 − ε p − 1 d t 1 ⋯ d t m − 1 d t m = x λ 1 − ε p + m − 1 ( x > 1 ) .

If there exists a constant M ≥ Γ ( λ ) B ( λ 1 , λ 2 ) Γ ( λ + m + 1 ) such that equation (14) is valid, when we replace Γ ( λ ) Γ ( λ + m + 1 ) B ( λ 1 , λ 2 ) by M , then in particular, we have

(15) I ˜ ≔ ∫ 0 ∞ ∑ n = 1 ∞ v ′ ( n ) A ˜ n f ˜ ( x ) ( x + v ( n ) ) λ + m + 1 d x > M ∫ 0 ∞ x p ( 1 − λ 1 ) − 1 ( f ˜ ( m ) ( x ) ) p d x 1 p ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ 2 ) − 1 ( v ′ ( n ) ) q − 1 a ˜ n q 1 q .

By the decreasingness property of series, we have

I ˜ > M ∏ i = 0 m − 1 λ 1 + i − ε p ∫ 1 ∞ x p ( 1 − λ 1 ) − 1 x p λ 1 − ε − p d x 1 p × ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ 2 ) − 1 ( v ( n ) ) q λ 2 − ε − q v ′ ( n ) 1 q = M ∏ i = 0 m − 1 λ 1 + i − ε p ∫ 1 ∞ x − ε − 1 d x 1 p ∑ n = 1 ∞ ( v ( n ) ) − ε − 1 v ′ ( n ) 1 q ≥ M ∏ i = 0 m − 1 λ 1 + i − ε p ∫ 1 ∞ x − ε − 1 d x 1 p ∫ 1 ∞ ( v ( y ) ) − ε − 1 v ′ ( y ) d y 1 q = M ε ∏ i = 0 m − 1 λ 1 + i − ε p ( v ( 1 ) ) − ε q .

Replacing λ by λ + m + 1 , and setting λ ˜ 1 ≔ λ 1 + m − ε p ∈ ( 0 , λ + m ) and λ ˜ 2 ≔ λ 2 + 1 + ε p ∈ ( 0 , λ + 1 ) in equation (5), we have

I ˜ = ∫ 0 ∞ ∑ n = 1 ∞ v ′ ( n ) A ˜ n f ˜ ( x ) d x ( x + v ( n ) ) λ + m + 1 ≤ 1 λ 2 − ε q ∑ n = 1 ∞ ∫ 1 ∞ ( v ( n ) ) λ 2 − ε q v ′ ( n ) x λ 1 − ε p + m − 1 d x ( x + v ( n ) ) λ + m + 1 = 1 λ 2 − ε q ∑ n = 1 ∞ ( v ( n ) ) λ + m + 1 − λ ˜ 1 ∫ 1 ∞ x λ ˜ 1 − 1 d x ( x + v ( n ) ) λ + m + 1 ( v ( n ) ) − ε − 1 v ′ ( n ) .

Hence, by equation (6), we obtain

I ˜ ≤ 1 λ 2 − ε q ∑ n = 1 ∞ ω λ + m + 1 ( n , λ ˜ 1 ) ( v ( n ) ) − ε − 1 v ′ ( n ) 0 = k λ + m + 1 ( λ ˜ 1 ) λ 2 − ε q ( v ( 1 ) ) − ε − 1 v ′ ( 1 ) + ∑ n = 2 ∞ ( v ( n ) ) − ε − 1 v ′ ( n ) < k λ + m + 1 ( λ ˜ 1 ) λ 2 − ε q ( v ( 1 ) ) − ε − 1 v ′ ( 1 ) + ∫ 1 ∞ ( v ( y ) ) − ε − 1 v ′ ( y ) d y = k λ + m + 1 ( λ ˜ 1 ) ε ( λ 2 − ε q ) [ ε ( v ( 1 ) ) − ε − 1 v ′ ( 1 ) + ( v ( 1 ) ) − ε ] .

Based on the above results, we have

k λ + m + 1 ( λ ˜ 1 ) λ 2 − ε q [ ε ( v ( 1 ) ) − ε − 1 v ′ ( 1 ) + ( v ( 1 ) ) − ε ] > ε I ˜ > M ∏ i = 0 m − 1 λ 1 + i − ε p ( v ( 1 ) ) − ε q .

For ε → 0 + , in view of the continuity of the beta function, we obtain

1 λ 2 B ( λ 1 + m , λ 2 + 1 ) ≥ M ∏ i = 0 m − 1 ( λ 1 + i ) ,

namely,

Γ ( λ ) B ( λ 1 , λ 2 ) Γ ( λ + m + 1 ) = Γ ( λ 1 ) Γ ( λ 2 ) Γ ( λ + m + 1 ) = Γ ( λ 1 + m ) Γ ( λ 2 ) Π i = 0 m − 1 ( λ 1 + i ) Γ ( λ + m + 1 ) = B ( λ 1 + m , λ 2 + 1 ) λ 2 Π i = 0 m − 1 ( λ 1 + i ) ≥ M .

Hence, M = Γ ( λ ) B ( λ 1 , λ 2 ) Γ ( λ + m + 1 ) is the best possible constant factor in equation (14).

On the other hand, for λ ^ 1 = λ − λ 2 p + λ 1 q , λ ^ 2 = λ − λ 1 q + λ 2 p = λ 2 + λ − λ 1 − λ 2 q , we have

λ ^ 1 + λ ^ 2 = λ − λ 2 p + λ 1 q + λ − λ 1 q + λ 2 p = λ .

and then, For λ − λ 1 − λ 2 ∈ ( − q λ 2 , q ( λ − λ 2 ) ) , we have 0 < λ ^ 2 < λ , 0 < λ ^ 1 = λ − λ ^ 2 < λ , and Γ ( λ ) B ( λ ^ 1 , λ ^ 2 ) Γ ( λ + m + 1 ) ∈ R + . For λ − λ 1 − λ 2 ≤ q ( 1 − λ 2 ) , we still have λ ^ 2 ≤ 1 . By substitution of λ ^ i = λ i ( i = 1 , 2 ) in equation (14), we have

(16) I = ∫ 0 ∞ ∑ n = 1 ∞ v ′ ( n ) A n ( f ( x ) ) d x ( x + v ( n ) ) λ + m > Γ ( λ ) B ( λ ^ 1 , λ ^ 2 ) Γ ( λ + m + 1 ) × ∫ 0 ∞ x p ( 1 − λ ^ 1 ) − 1 ( f ( m ) ( x ) ) p d x 1 p ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ ^ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q 1 q .

By the reverse Hölder’s inequality (cf. [18]), we find

(17) B ( λ ^ 1 , λ ^ 2 ) = k λ λ − λ 2 p + λ 1 q = ∫ 0 ∞ u λ − λ 2 p + λ 1 q − 1 ( 1 + u ) λ d u = ∫ 0 ∞ 1 ( 1 + u ) λ u λ − λ 2 − 1 p u λ 1 − 1 q d u ≥ ∫ 0 ∞ u λ − λ 2 − 1 ( 1 + u ) λ d u 1 p ∫ 0 ∞ u λ 1 − 1 ( 1 + u ) λ d u 1 q = ∫ 0 ∞ v λ 2 − 1 ( v + 1 ) λ d v 1 p ∫ 0 ∞ u λ 1 − 1 ( 1 + u ) λ d u 1 q = ( k λ ( λ 2 ) ) 1 p ( k λ ( λ 1 ) ) 1 q .

Since Γ ( λ ) Γ ( λ + m + 1 ) ( k λ ( λ 2 ) ) 1 p ( k λ ( λ 1 ) ) 1 q is the best possible constant factor in equation (13), by equation (16), we have

Γ ( λ ) ( k λ ( λ 2 ) ) 1 p Γ ( λ + m + 1 ) ( k λ ( λ 1 ) ) 1 q ≥ Γ ( λ ) B ( λ ^ 1 , λ ^ 2 ) Γ ( λ + m + 1 ) ( ∈ R + ) ,

namely, B ( λ ^ 1 , λ ^ 2 ) ≤ ( k λ ( λ 2 ) ) 1 p ( k λ ( λ 1 ) ) 1 q . It follows that equation (17) keeps the form of equality.

We observe that equation (17) keeps the form of equality if and only if there exist constants A and B (cf. [18]), such that they are not both zero and A u λ − λ 2 = B u λ 1 a.e. in R + . Assuming that A ≠ 0 , we find u λ − λ 1 − λ 2 = B A a.e. in R + , namely, λ − λ 1 − λ 2 = 0 . Hence, we have λ 1 + λ 2 = λ .

The theorem is proved.□

4 Equivalent forms and some particular inequalities

For m = 0 in equation (13), we have

(18) I = ∫ 0 ∞ ∑ n = 1 ∞ v ′ ( n ) A n f ( x ) ( x + v ( n ) ) λ + 1 d x > 1 λ ( k λ ( λ 2 ) ) 1 p ( k λ ( λ 1 ) ) 1 q × ∫ 0 ∞ x p ( 1 − λ ^ 1 ) − 1 f p ( x ) d x 1 p ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ ^ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q 1 q .

In particular, for λ 1 + λ 2 = λ , we have

(19) I = ∫ 0 ∞ ∑ n = 1 ∞ v ′ ( n ) A n f ( x ) ( x + v ( n ) ) λ + 1 d x > 1 λ B ( λ 1 , λ 2 ) × ∫ 0 ∞ x p ( 1 − λ 1 ) − 1 f p ( x ) d x 1 p ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q 1 q .

Theorem 3

We have the following reverse half-discrete Hilbert-type inequality equivalent to equation (18):

(20) J ≔ ∫ 0 ∞ x q λ ^ 1 − 1 ∑ n = 1 ∞ v ′ ( n ) A n ( x + v ( n ) ) λ + 1 q d x 1 q > 1 λ ( k λ ( λ 2 ) ) 1 p ( k λ ( λ 1 ) ) 1 q ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ ^ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q 1 q .

In particular, for λ 1 + λ 2 = λ , we have the following inequality equivalent to equation (19):

(21) ∫ 0 ∞ x q λ 1 − 1 ∑ n = 1 ∞ v ′ ( n ) A n ( x + v ( n ) ) λ + 1 q d x 1 q > 1 λ B ( λ 1 , λ 2 ) ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q 1 q .

Proof

Suppose that equation (20) is valid. By the reverse Hölder’s inequality (cf. [18]), we have

(22) I = ∫ 0 ∞ ( x 1 q − λ ^ 1 f ( x ) ) x − 1 q + λ ^ 1 ∑ n = 1 ∞ v ′ ( n ) A n ( x + v ( n ) ) λ + 1 d x > ∫ 0 ∞ x p ( 1 − λ ^ 1 ) − 1 f p ( x ) d x 1 p J ,

Then, by equation (20), we have equation (18).

On the other hand, assuming that equation (18) is valid, we set

f ( x ) ≔ x q λ ^ 1 − 1 ∑ n = 1 ∞ v ′ ( n ) A n ( x + v ( n ) ) λ + 1 q − 1 , x ∈ R + .

It follows that

(23) ∫ 0 ∞ x p ( 1 − λ ^ 1 ) − 1 f p ( x ) d x = J q = I .

If J = ∞ , then equation (20) is naturally valid; if J = 0 , then it is impossible to make equation (20) valid, namely, J > 0 . Suppose that 0 < J < ∞ . By equation (18), we have

∞ > ∫ 0 ∞ x p ( 1 − λ ^ 1 ) − 1 f p ( x ) d x = J q = I > 1 λ ( k λ ( λ 2 ) ) 1 p ( k λ ( λ 1 ) ) 1 q ∫ 0 ∞ x p ( 1 − λ ^ 1 ) − 1 f p ( x ) d x 1 p ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ ^ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q 1 q , J = ∫ 0 ∞ x p ( 1 − λ ^ 1 ) − 1 f p ( x ) d x 1 q > 1 λ ( k λ ( λ 2 ) ) 1 p ( k λ ( λ 1 ) ) 1 q ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ ^ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q 1 q ,

namely, equation (20) follows, which is equivalent to equation (18).

The theorem is proved.□

Theorem 4

If λ 1 + λ 2 = λ , then the constant factor

1 λ ( k λ ( λ 2 ) ) 1 p ( k λ ( λ 1 ) ) 1 q

in equation (20) is the best possible. On the other hand, if the same constant factor in equation (20) is the best possible, then for

λ − λ 1 − λ 2 ∈ ( − q λ 2 , q ( λ − λ 2 ) ) ∩ ( − q λ 2 , q ( 1 − λ 2 ) ] ,

we have λ 1 + λ 2 = λ .

Proof

If λ 1 + λ 2 = λ , then by Theorem 2, for m = 0 , the constant factor

1 λ ( k λ ( λ 2 ) ) 1 p ( k λ ( λ 1 ) ) 1 q

in equation (18) is the best possible. The constant factor in equation (20) is still the best possible. Otherwise, by equation (22), we would reach a contradiction that the constant in equation (18) is not the best possible.

On the other hand, if the same constant factor in equation (20) is the best possible, then, by the equivalency of equations (20) and (18), in view of J q = I (see the proof of Theorem 3), we can still show that the constant factor in equation (18) is the best possible. By the assumption and Theorem 2 (for m = 0 ), we have λ 1 + λ 2 = λ .

The theorem is proved.□

Replacing x by 1 x in equations (19) and (21), and setting g ( x ) = x λ − 1 f 1 x , we have 0 < ∫ 0 ∞ x − p λ 2 − 1 g p ( x ) d x < ∞ and Corollary 1 as follows:

Corollary 1

The following equivalent inequalities with the nonhomogeneous kernel are valid:

(24) ∫ 0 ∞ ∑ n = 1 ∞ v ′ ( n ) A n g ( x ) ( 1 + x v ( n ) ) λ + 1 d x > 1 λ B ( λ 1 , λ 2 ) ∫ 0 ∞ x − p λ 2 − 1 g p ( x ) d x 1 p ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q 1 q ,

(25) ∫ 0 ∞ x q ( λ 2 + 1 ) − 1 ∑ n = 1 ∞ v ′ ( n ) A n ( 1 + x v ( n ) ) λ + 1 q d x 1 q > 1 λ B ( λ 1 , λ 2 ) ∑ n = 1 ∞ ( v ( n ) ) q ( 1 − λ 2 ) − 1 ( v ′ ( n ) ) q − 1 a n q 1 q ,

where the constant factor 1 λ B ( λ 1 , λ 2 ) is the best possible.

Remark 2

For α ∈ ( 0 , 1 ] , and setting v ( t ) = ln α ( t + 1 ) , t ∈ R + , we find v ′ ( t ) = α ln α − 1 ( t + 1 ) t + 1 > 0 and v ″ ( t ) = α ln α − 2 ( t + 1 ) ( t + 1 ) 2 [ α − 1 − ln ( t + 1 ) ] < 0 , t ∈ R + . By equation (14), we have

(26) ∫ 0 ∞ ∑ n = 1 ∞ ln α − 1 ( n + 1 ) A n f ( x ) [ x + ln α ( n + 1 ) ] λ + m + 1 ( n + 1 ) d x > Γ ( λ ) B ( λ 1 , λ 2 ) α 1 + 1 p Γ ( λ + m + 1 ) ∫ 0 ∞ x p ( 1 − m − λ 1 ) − 1 F m p ( x ) d x 1 p ∑ n = 1 ∞ ln q ( 1 − α λ 2 ) − 1 ( n + 1 ) ( n + 1 ) 1 − q a n q 1 q ,

where, the constant factor Γ ( λ ) B ( λ 1 , λ 2 ) α 1 + 1 p Γ ( λ + m + 1 ) is the best possible. In particular, for m = 0 , we have the following equivalent inequalities with the best possible constant factor B ( λ 1 , λ 2 ) α 1 + 1 p λ :

(27) ∫ 0 ∞ ∑ n = 1 ∞ ln α − 1 ( n + 1 ) A n f ( x ) [ x + ln α ( n + 1 ) ] λ + 1 ( n + 1 ) d x > B ( λ 1 , λ 2 ) α 1 + 1 p λ ∫ 0 ∞ x p ( 1 − λ 1 ) − 1 F m p ( x ) d x 1 p ∑ n = 1 ∞ ln q ( 1 − α λ 2 ) − 1 ( n + 1 ) ( n + 1 ) 1 − q a n q 1 q ,

(28) ∫ 0 ∞ x q λ 1 − 1 ln α − 1 ( n + 1 ) A n [ x + ln α ( n + 1 ) ] λ + 1 ( n + 1 ) q d x 1 q > B ( λ 1 , λ 2 ) α 1 + 1 p λ ∑ n = 1 ∞ ln q ( 1 − α λ 2 ) − 1 ( n + 1 ) ( n + 1 ) 1 − q a n q 1 q .

By Corollary 1, we still have the following equivalent inequalities with the best possible constant factor B ( λ 1 , λ 2 ) α 1 + 1 p λ :

(29) ∫ 0 ∞ ∑ n = 1 ∞ ln α − 1 ( n + 1 ) A n g ( x ) [ 1 + x ln α ( n + 1 ) ] λ + 1 ( n + 1 ) d x > B ( λ 1 , λ 2 ) α 1 + 1 p λ ∫ 0 ∞ x − p λ 2 − 1 g p ( x ) d x 1 p ∑ n = 1 ∞ ln q ( 1 − α λ 2 ) − 1 ( n + 1 ) ( n + 1 ) 1 − q a n q 1 q ,

(30) ∫ 0 ∞ x q ( λ 2 + 1 ) − 1 ∑ n = 1 ∞ ln α − 1 ( n + 1 ) A n [ 1 + x ln α ( n + 1 ) ] λ + 1 ( n + 1 ) q d x 1 q > B ( λ 1 , λ 2 ) α 1 + 1 p λ ∑ n = 1 ∞ ln q ( 1 − α λ 2 ) − 1 ( n + 1 ) ( n + 1 ) 1 − q a n q 1 q .

5 Conclusion

In this article, a new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order is obtained, by using the weight functions, the mid-value theorem, and the techniques of real analysis in Theorem 1. The equivalent statements of the best possible constant factor related to some parameters are considered in Theorem 2. As applications, the equivalent forms are provided in Theorems 3 and 4 and Corollary 1, and some particular inequalities are deduced in Remark 2. The lemmas and theorems provide an extensive account of this type of inequalities.

Acknowledgments

The authors thank the referees for their useful proposal to revise the article.

  1. Funding information: This work was supported by the National Natural Science Foundation (No. 61772140), the Key Construction Discipline Scientific Research Ability Promotion Project of Guangdong Province (No 2021ZDJS056) and Guangzhou Basic and Applied Basic Research Project (No. 20220101181-7). We are grateful for this help.

  2. Author contributions: B.Y. carried out the mathematical studies, participated in the sequence alignment, and drafted the manuscript. J.L. participated in the design of the study and performed the numerical analysis. All authors read and approved the final manuscript.

  3. Conflict of interest: The authors declare that they have no conflict of interest.

  4. Data availability statement: We declare that the data and material in this article can be used publicly.

References

[1] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1934. Search in Google Scholar

[2] M. Krnić and J. Pečarić, Extension of Hilbert’s inequality, J. Math. Anal. Appl. 324 (2006), no. 1, 150–160. 10.1016/j.jmaa.2005.11.069Search in Google Scholar

[3] B. C. Yang, On a generalization of Hilbert double series theorem, Math. Inequal. Appl. 5 (2002), no. 2, 197–204. 10.7153/mia-05-21Search in Google Scholar

[4] M. Th. Rassias and B. C. Yang, On half-discrete Hilbert’s inequality, Appl. Math. Comput. 220 (2013), 75–93. 10.1016/j.amc.2013.06.010Search in Google Scholar

[5] B. C. Yang and L. Debnath, Half-discrete Hilbert-type Inequalities, World Scientific Pub. Co., Singapore, 2014. 10.1142/8799Search in Google Scholar

[6] Y. Hong and Y. M. Wen, A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernal has the best constant factor, Ann. Math. 37 (2016), 329–336. Search in Google Scholar

[7] B. He, Y. Hong, and Z. Li, Conditions for the validity of a class of optimal Hilbert type multiple integral inequalities with non-homogeneous, J. Inequal. Appl. 2021 (2021), 64. 10.1186/s13660-021-02593-zSearch in Google Scholar

[8] Y. Hong, On the structure character of Hilbert’s type integral inequality with homogeneous kernal and applications, J. Jilin Univ. Sci. 55 (2017), no. 2, 189–194. 10.1186/s13660-017-1592-8Search in Google Scholar

[9] Q. Chen, B. He, Y. Hong, and Z. Li, Equivalent parameter conditions for the validity of half-discrete Hilbert-type multiple integral inequality with generalized homogeneous kernel, J. Funct. Spaces 2020 (2020), 7414861. 10.1155/2020/7414861Search in Google Scholar

[10] B. He, Y. Hong, and Q. Chen, The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of non-homogeneous kernels and their applications, Open Math. 19 (2021), 400–411. 10.1515/math-2021-0023Search in Google Scholar

[11] Y. Hong, Q. L. Huang, and Q. Chen, The parameter conditions for the existence of the Hilbert-type multiple integral inequality and its best constant factor, Ann. Funct. Anal. 12 (2021), 7, DOI: https://doi.org/10.1007/s43034-020-00087-5. 10.1007/s43034-020-00087-5Search in Google Scholar

[12] Y. Hong and Q. Chen, Equivalent parameter conditions for the construction of Hilbert-type integral inequalities with a class of non?homogeneous kernels, J. South China Normal Univ. Natur. Sci. Ed. 52 (2020), no. 5, 124–128. Search in Google Scholar

[13] Y. Hong and Q. Chen, Research progress and applications of Hilbert-type series inequalities, J. Jilin Univ. Sci. 59 (2021), no. 5, 1131–1140. Search in Google Scholar

[14] Y. Hong and B. He, The optimal matching parameter of half-discrete Hilbert-type multiple integral inequalities with non-homogeneous kernels and applications, Chinese Quart. J. Math. 36 (2021), no. 3, 252–262. Search in Google Scholar

[15] Y. Hong, Q. Chen, and C. Y. Wu, The best matching parameters for semi-discrete Hilbert-type inequality with quasi-homogeneous kernel, Math. Appl. 34 (2021), no. 3, 779–785. Search in Google Scholar

[16] V. Adiyasuren, T. Batbold, and L. E. Azar, A new discrete Hilbert-type inequality involving partial sums, J. Inequal. Appl. 2019 (2019), 127. 10.1186/s13660-019-2087-6Search in Google Scholar

[17] Y. Hong, Y. R. Zhong, and B. C. Yang, A more accurate half-discrete multidimensional Hilbert-type inequality involving one multiple upper limit function, Axioms 12 (2023), no. 2, 211, DOI: https://doi.org/10.3390/axioms12020211. 10.3390/axioms12020211Search in Google Scholar

[18] J. C. Kuang, Applied Inequalities, Shangdong Science and Technology Press, Jinan, 2004. Search in Google Scholar

[19] J. C. Kuang, Introduction to Real Analysis, Hunan Education Press, Changsha, 1996. Search in Google Scholar
Received: 2023-08-11
Revised: 2023-09-28
Accepted: 2023-10-02
Published Online: 2023-11-21

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2023-0139
Keywords for this article
half-discrete inequality; partial sums; weight function; best possible constant factor; parameter; reverse
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  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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