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Article

Construction Theorem and Application of a Semi-Discrete Hilbert-Type Inequality Involving Partial Sums and Variable Upper Limit Integral Functions

1
Artificial Intelligence College, Guangzhou Huashang College, Guangzhou 511300, China
2
Department of Mathematics, Guangdong University of Education, Guangzhou 510303, China
*
Author to whom correspondence should be addressed.
Axioms 2025, 14(10), 755; https://doi.org/10.3390/axioms14100755
Submission received: 30 August 2025 / Revised: 2 October 2025 / Accepted: 3 October 2025 / Published: 7 October 2025

Abstract

By using the construction theorem of semi-discrete Hilbert-type inequalities with quasi-homogeneous kernels and real analysis techniques, this paper establishes a semi-discrete Hilbert-type inequality involving partial sums and variable upper limit integral functions, obtains the necessary and sufficient condition for constructing such an inequality, and, under certain conditions, derives the computational expression of the best constant factor. Finally, we discuss the boundedness and operator norm of the corresponding operator using the obtained results.

1. Introduction and Preliminary Knowledge

Let p > 1 , 1 p + 1 q = 1 , a ˜ = { a m } , and b ˜ = { b n } . Then, we have the following classical discrete Hilbert inequality [1]:
m = 1 n = 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 = π sin ( π / p ) a ˜ p b ˜ q ,
where the constant factor π sin ( π / p ) is the best possible. Correspondingly, integral and semi-discrete forms exist:
0 + 0 + f ( x ) g ( y ) x + y d x d y π sin ( π / p ) 0 + | f ( x ) | p d x 1 / p 0 + | g ( y ) | q d y 1 / q = π sin ( π / p ) f p g q ,
n = 1 0 + a n f ( x ) n + x d x π sin ( π / p ) n = 1 | a n | p 1 / p 0 + | f ( x ) | q d x 1 / q = π sin ( π / p ) a ˜ p f q ,
where the constant factors are also the best possible. Hilbert-type inequalities, due to their important applications in operator theory, have received extensive attention. For further research, it is necessary to generalize L p ( 0 , + ) and l p to weighted Lebesgue spaces and weighted Hilbert-type spaces:
L p α ( 0 , + ) = f ( x ) : f p , α = 0 + x α | f ( x ) | p d x 1 / p < + ,
l p α = a ˜ = { a n } : a ˜ p , α = n = 1 n α | a n | p 1 / p < + ,
where p > 1 and α > 0 . In weighted spaces, different authors have introduced many independent parameters and discussed Hilbert-type inequalities with homogeneous kernels, quasi-homogeneous kernels, and several non-homogeneous kernels such as
1 ( x + y ) λ , 1 m λ 1 + n λ 2 , ln ( x / y ) x λ y λ , | x λ 1 y λ 2 | min { x λ 1 , y λ 2 } , ( m λ 1 n λ 2 ) s ( m λ 1 + n λ 2 ) r
and the boundedness of their corresponding operators (see [2,3,4,5,6,7,8]). In [9] from 2016, the authors first discussed the problem of best matching parameters for Hilbert-type inequalities, solving the problem of parameter rules for Hilbert-type inequalities with best constant factors. In [10] from 2017, the authors further explored the construction conditions and best constant factor estimation for Hilbert-type inequalities. Since then, relevant discussions have continued and have been published in various academic journals (see [11,12,13]).
In [14] from 2019, the authors studied Hilbert-type inequalities from a new perspective and presented a Hilbert-type inequality involving partial sums. Afterwards, references [15,16,17,18,19,20] made various improvements and generalizations. For example, under certain parameter restrictions, the authors in [21] obtained obtained an inequality involving partial sums and multiple variable upper limit functions F m ( x ) :
0 + n = 1 a n f ( x ) ( x + n α ) λ d x Γ ( λ + m + 1 ) Γ ( λ ) 1 α K λ + m + 1 ( λ 2 + 1 ) 1 / p K λ + m + 1 ( λ 1 + m ) 1 / q
× 0 + x p ( λ ˜ 1 + m 1 ) 1 F m p ( x ) d x 1 / p n = 1 n q ( λ ˜ 2 + 1 ) 1 1 A n q 1 / q ,
where λ ˜ 1 = λ λ 2 p + λ 1 q , λ ˜ 2 = λ λ 1 q + λ 2 p , and Γ ( · ) is the Gamma function. It was proven that the constant factor in ( 1 ) is the best possible when α = 1 and λ 1 + λ 2 = λ .
The authors in [14] considered the discrete inequality with the homogeneous kernel 1 ( m + n ) λ , and the authors in [21] discussed the semi-discrete inequality with the quasi-homogeneous kernel 1 ( x + n α ) λ . The parameter relations in them are relatively cumbersome, and the necessity of these parameter relations is not discussed. This paper considers the parameter conditions and best constant factor problem of a semi-discrete Hilbert-type inequality involving partial sums and variable upper limit integral functions with kernel 1 ( n λ 1 + x λ 2 ) λ . The parameter condition relationships in the results are simple and clear, facilitating the application of the inequality. Meanwhile, by making full use of the construction theorem of semi-discrete Hilbert-type inequalities with quasi-homogeneous kernels, the proof process is optimized. Finally, its application in operator theory is discussed.

2. Notation Conventions and Preliminary Lemmas

To avoid unnecessary repetitions, this paper makes the following conventions. If a ˜ = { a n } and f ( x ) is integrable, denote
A n = i = 1 n a i ( n = 1 , 2 , ) , F ˜ ( f ) ( x ) = 0 x f ( t ) d t .
If K ( u , v ) is a binary measurable function, and λ 1 > 0 , λ 2 > 0 , denote
W 1 ( s ) = 0 + K ( 1 , t λ 2 ) t s d t , W 2 ( s ) = 0 + K ( t λ 1 , 1 ) t s d t .
Lemma 1 
([22]). Let p > 1 , 1 p + 1 q = 1 , σ , α , β R , λ 1 , λ 2 > 0 , and let K ( u , v ) be a homogeneous function of degree σ, that is, K ( t u , t v ) = t σ K ( u , v ) ( t > 0 ) . Suppose α λ 1 p + β λ 2 q 1 λ 1 q + 1 λ 2 p + σ = c , and both K ( t λ 1 , 1 ) t α + 1 p and K ( t λ 1 , 1 ) t α + 1 p + λ 1 c are decreasing on ( 0 , + ) , with W 1 β + 1 q < + . Then,
(i) There exists a constant M > 0 if and only if c 0 such that
n = 1 0 + K n λ 1 , x λ 2 a n f ( x ) d x M a ˜ p , α f q , β .
(ii) If c = 0 , i.e., α λ 1 p + β λ 2 q = σ + 1 λ 1 q + 1 λ 2 p , then the best constant factor in ( 2 ) is
inf { M } = λ 2 λ 1 1 / q W 1 β + 1 q = λ 1 λ 2 1 / p W 2 α + 1 p .
Lemma 2. 
Let p > 1 , 1 p + 1 q = 1 , λ , α , β R , λ 1 > 0 , λ 2 > 0 , λ > 0 , α + 1 λ 1 p + β + 1 λ 2 q + λ = c , α > max ( λ 1 1 ) p 1 , ( λ 1 1 ) p 1 + λ 1 p c , a n d ( λ + 1 ) λ 2 q 1 < β < 1 .
(i) There exists a constant M ¯ > 0 if and only if c 0 such that
n = 1 0 + n λ 1 1 x λ 2 1 n λ 1 + x λ 2 λ + 2 A n F ( x ) d x M ¯ A ˜ p , α F q , β ,
where A ˜ = { A n } .
(ii) When c = 0 , the best constant factor in ( 3 ) is
inf { M ¯ } = 1 λ 1 1 / q λ 2 1 / p Γ ( λ + 2 ) Γ 1 α + 1 λ 1 p Γ 1 β + 1 λ 2 q .
Proof. 
Let
K ( u , v ) = u 1 1 λ 1 v 1 1 λ 2 ( u + v ) λ + 2 , u > 0 , v > 0 .
Then, K ( u , v ) is a non-negative homogeneous function of order σ = 2 1 λ 1 1 λ 2 ( λ + 2 ) = λ 1 λ 1 1 λ 2 , and
K n λ 1 , x λ 2 = n λ 1 1 x λ 2 1 n λ 1 + x λ 2 λ + 2 .
Thus,
K t λ 1 , 1 t α + 1 p = 1 t λ 1 + 1 λ + 2 t λ 1 1 α + 1 p ,
K t λ 1 , 1 t α + 1 p + λ 1 c = 1 t λ 1 + 1 λ + 2 t λ 1 1 α + 1 p + λ 1 c .
Since α > ( λ 1 1 ) p 1 , and α > ( λ 1 1 ) p 1 + λ 1 p c , we have λ 1 1 α + 1 p < 0 , and λ 1 1 α + 1 p + λ 1 c < 0 . Also, since λ 1 > 0 , λ > 0 , both K ( t λ 1 , 1 ) t α + 1 p and K ( t λ 1 , 1 ) t α + 1 p + λ 1 c are decreasing on ( 0 , + ) .
From ( λ + 1 ) λ 2 q 1 < β < 1 < λ 2 q 1 , we have 1 β + 1 λ 2 q > 0 and λ + 2 1 β + 1 λ 2 q > 0 . Thus,
W 1 β + 1 q = 0 + K 1 , t λ 2 t β + 1 q d t = 0 + t λ 2 1 1 + t λ 2 λ + 2 t β + 1 q d t
= 1 λ 2 0 + 1 ( 1 + u ) λ + 2 u 1 β + 1 λ 2 q 1 d u = 1 λ 2 B 1 β + 1 λ 2 q , λ + 2 1 β + 1 λ 2 q < + ,
where B ( · , · ) is the beta function. Since K ( u , v ) is a homogeneous function of degree σ = λ 1 λ 1 1 λ 2 , the condition α λ 1 p + β λ 2 q 1 λ 1 q + 1 λ 2 p + σ = c in Lemma 1 is transformed into α + 1 λ 1 p + β + 1 λ 2 q + λ = c .
When c = 0 , we have α + 1 λ 1 p + β + 1 λ 2 q + λ = 0 ; hence, λ + 2 1 β + 1 λ 2 q = 1 α + 1 λ 1 p . Thus,
λ 2 λ 1 1 / q W 1 β + 1 q = λ 2 λ 1 1 / q 1 λ 2 B 1 β + 1 λ 2 q , 1 α + 1 λ 1 p
= 1 λ 1 1 / q λ 2 1 / p Γ ( λ + 2 ) Γ 1 β + 1 λ 2 q Γ 1 α + 1 λ 1 p .
In summary, according to Lemma 1, Lemma 2 holds. □
Lemma 3. 
Let t > 0 , 0 < λ 1 1 , A n = k = 1 n a k ( n = 1 , 2 , ) , and lim n + e t n λ 1 A n = 0 . Then,
n = 1 e t n λ 1 a n λ 1 t n = 1 n λ 1 1 e t n λ 1 A n .
Proof. 
By Abel’s summation formula and differential mean value theorem, we have
n = 1 N e t n λ 1 a n = e t N λ 1 A N n = 1 N 1 A n e t ( n + 1 ) λ 1 e t n λ 1
= e t N λ 1 A N + λ 1 t n = 1 N 1 ξ n λ 1 1 e t ξ n λ 1 A n ,
where n < ξ n < n + 1 . Since 0 < λ 1 1 , we have ξ n λ 1 1 e t ξ n λ 1 < n λ 1 1 e t n λ 1 , and thus
n = 1 N e t n λ 1 a n e t N λ 1 A N + λ 1 t n = 1 N 1 n λ 1 1 e t n λ 1 A n .
Let N + , and note that lim n + e t n λ 1 A n = 0 . Then, ( 4 ) is obtained. □
Lemma 4. 
Let t > 0 and lim x + e t x λ 2 F ˜ ( f ) ( x ) = 0 . Then,
0 + e t x λ 2 f ( x ) d x = λ 2 t 0 + x λ 2 1 e t x λ 2 F ˜ ( f ) ( x ) d x .
Proof. 
Since d d x F ˜ ( f ) ( x ) = f ( x ) and F ˜ ( f ) ( 0 ) = 0 , by integration by parts, we have
0 + e t x λ 2 f ( x ) d x = 0 + e t x λ 2 d F ˜ ( f ) ( x )
= lim x + e t x λ 2 F ˜ ( f ) ( x ) + λ 2 t 0 + x λ 2 1 e t x λ 2 F ˜ ( f ) ( x ) d x
= λ 2 t 0 + x λ 2 1 e t x λ 2 F ˜ ( f ) ( x ) d x .

3. Main Results

Theorem 1. 
Let p > 1 , 1 p + 1 q = 1 , λ , α , β R , satisfying 0 < λ 1 1 , λ 2 > 0 , λ > 0 , and α + 1 λ 1 p + β + 1 λ 2 q + λ = c , with max ( λ 1 1 ) p 1 , ( λ 1 1 ) p 1 + λ 1 p c < α < 1 , ( λ + 1 ) λ 2 q < β < 1 ,   lim n + e t n λ 1 A n = 0 ( t > 0 ) , and lim x + e t x λ 2 F ˜ ( f ) ( x ) = 0 (t > 0).
( i ) There exists a constant M > 0 if and only if c 0 , i.e., α + 1 λ 1 p + β + 1 λ 2 q + λ 0 , such that
n = 1 0 + a n f ( x ) ( n λ 1 + x λ 2 ) λ d x M A ˜ p , α F ˜ ( f ) q , β ,
where A ˜ = { A n } = k = 1 n a k .
( i i ) When c = 0 , i.e., α + 1 λ 1 p + β + 1 λ 2 q + λ = 0 , the best constant factor in ( 5 ) is
M 0 = λ 1 1 / p λ 2 1 / q 1 Γ ( λ ) Γ 1 α + 1 λ 1 p Γ 1 β + 1 λ 2 q .
Proof. 
( i ) According to the Gamma function definition, we obtain
1 ( n λ 1 + x λ 2 ) λ = 1 Γ ( λ ) 0 + t λ 1 e t ( n λ 1 + x λ 2 ) d t .
Thus, by Lemmas 3 and 4, we have
n = 1 0 + a n f ( x ) ( n λ 1 + x λ 2 ) λ d x = 1 Γ ( λ ) n = 1 0 + a n f ( x ) 0 + t λ 1 e t ( n λ 1 + x λ 2 ) d t d x = 1 Γ ( λ ) 0 + t λ 1 n = 1 e t n λ 1 a n 0 + e t x λ 2 f ( x ) d x d t λ 1 λ 2 Γ ( λ ) 0 + t ( λ + 2 ) 1 n = 1 n λ 1 1 e t n λ 1 A n 0 + x λ 2 1 e t x λ 2 F ˜ ( f ) ( x ) d x d t = λ 1 λ 2 Γ ( λ ) 0 + t ( λ + 2 ) 1 n = 1 0 + n λ 1 1 x λ 2 1 e t ( n λ 1 + x λ 2 ) A n F ˜ ( f ) ( x ) d x d t = λ 1 λ 2 Γ ( λ ) n = 1 0 + n λ 1 1 x λ 2 1 A n F ˜ ( f ) ( x ) 0 + t ( λ + 2 ) 1 e t ( n λ 1 + x λ 2 ) d t d x = λ 1 λ 2 Γ ( λ ) Γ ( λ + 2 ) n = 1 0 + n λ 1 1 x λ 2 1 ( n λ 1 + x λ 2 ) λ + 2 A n F ˜ ( f ) ( x ) d x .
By Lemma 2, when c 0 , there exists a constant M ¯ > 0 such that
n = 1 0 + n λ 1 1 x λ 2 1 ( n λ 1 + x λ 2 ) λ + 2 A n F ˜ ( f ) ( x ) d x M ¯ A ˜ p , α F ˜ ( f ) q , β .
Let M = λ 1 λ 2 Γ ( λ ) Γ ( λ + 2 ) M ¯ , then ( 5 ) can be obtained from ( 6 ) .
Conversely, suppose ( 5 ) holds. If c < 0 , take a sufficiently small ε > 0 , and let
a n = n α + 1 + λ 1 ε p 1 , n = 1 , 2 ,
f ( x ) = x β + 1 + λ 2 ε q 1 , x 1 , 0 , 0 < x < 1 .
Since α > p ( λ 1 1 ) > p 1 , we have α + 1 p 1 < 0 . And since ε > 0 is sufficiently small, α + 1 + λ 1 ε p 1 < 0 . Also, from α < 1 , and ε > 0 being sufficiently small, we get α + 1 + λ 1 ε p > 0 . Hence,
A n = k = 1 n k α + 1 + λ 1 ε p 1 < 0 n t α + 1 + λ 1 ε p 1 d t = p α + 1 + λ 1 ε n α + 1 + λ 1 ε p .
Since β < 1 and ε > 0 is sufficiently small, we have β + 1 + λ 2 ε q > 0 . Thus, when 0 < x < 1 , we obtain F ˜ ( f ) ( x ) = 0 . When x 1 , we have
F ˜ ( f ) ( x ) = 1 x t β + 1 + λ 2 ε q 1 d t = q β + 1 + λ 2 ε x β + 1 + λ 2 ε q .
From (7) and (8), we have
A ˜ p , α F ˜ ( f ) q , β = n = 1 n α | A n | p 1 / p 0 + x β | F ˜ ( f ) ( x ) | q d x 1 / q p q ( α + 1 + λ 1 ε ) ( β + 1 + λ 2 ε ) n = 1 n 1 λ 1 ε 1 / p 1 + x 1 λ 2 ε d x 1 / q p q ( α + 1 + λ 1 ε ) ( β + 1 + λ 2 ε ) 1 + 1 + t 1 λ 1 ε d t 1 / p 1 + x 1 λ 2 ε d x 1 / q = p q λ 1 1 / p λ 2 1 / q ε ( α + 1 + λ 1 ε ) ( β + 1 + λ 2 ε ) ( 1 + λ 1 ε ) 1 / p ,
n = 1 0 + a n f ( x ) ( n λ 1 + x λ 2 ) λ d x = n = 1 n α + 1 + λ 1 ε p 1 1 + 1 ( n λ 1 + x λ 2 ) λ x β + 1 + λ 2 ε q 1 d x = n = 1 n α + 1 + λ 1 ε p 1 λ λ 1 1 + 1 ( 1 + n λ 1 x λ 2 ) λ x β + 1 + λ 2 ε q 1 d x = 1 λ 2 n = 1 n α + 1 + λ 1 ε p 1 λ λ 1 λ 1 λ 2 β + 1 + λ 2 ε q + 1 + λ 1 λ 2 n λ 1 + 1 ( 1 + t ) λ t β + 1 + λ 2 ε λ 2 q 1 d t = 1 λ 2 n = 1 n 1 λ 1 ( c + ε ) n λ 1 + 1 ( 1 + t ) λ t β + 1 + λ 2 ε λ 2 q 1 d t 1 λ 2 n = 1 n 1 λ 1 ( c + ε ) 1 + 1 ( 1 + t ) λ t β + 1 + λ 2 ε λ 2 q 1 d t .
Thus, we obtain
1 λ 2 n = 1 n 1 λ 1 ( c + ε ) 1 + 1 ( 1 + t ) λ t β + 1 + λ 2 ε λ 2 q 1 d t M p q λ 1 1 / p λ 2 1 / q ε ( α + 1 + λ 1 ε ) ( β + 1 + λ 2 ε ) ( 1 + λ 1 ε ) 1 / p < + .
Since c < 0 and ε > 0 is sufficiently small, 1 + λ 1 ( c + ε ) < 1 , we have n = 1 n 1 λ 1 ( c + ε ) = + , which contradicts ( 9 ) . Hence, c < 0 does not hold; in other words, c 0 .
(ii) When c = 0 , by ( 6 ) and Lemma 2, we have
n = 1 0 + a n f ( x ) ( n λ 1 + x λ 2 ) λ d x λ 1 λ 2 Γ ( λ ) Γ ( λ + 2 ) n = 1 0 + n λ 1 1 x λ 2 1 ( n λ 1 + x λ 2 ) λ A n F ˜ ( f ) ( x ) d x λ 1 λ 2 Γ ( λ ) Γ ( λ + 2 ) 1 λ 1 1 / q λ 2 1 / p Γ ( λ + 2 ) Γ 1 α + 1 λ 1 p Γ 1 β + 1 λ 2 q A ˜ p , α F ˜ ( f ) q , β = λ 1 1 / p λ 2 1 / q Γ ( λ ) Γ 1 α + 1 λ 1 p Γ 1 β + 1 λ 2 q A ˜ p , α F ˜ ( f ) q , β = M 0 A ˜ p , α F ˜ ( f ) q , β .
If M 0 is not the best constant factor in ( 5 ) , then there exists a constant M 1 < M 0 such that
n = 1 0 + a n f ˜ ( x ) n λ 1 + x λ 2 λ d x M 1 A ˜ p , α F ˜ ( f ) q , β .
Take a sufficiently large N > 0 and a sufficiently small ε > 0 , and let
a n = n α + 1 + λ 1 ε p 1 , n = N , N + 1 , 0 , n = 1 , 2 , , N 1 , f ˜ ( x ) = x β + 1 + λ 2 ε q 1 , x 1 , 0 , 0 < x < 1 .
Thus, when n = 1 , 2 , , N 1 , we have A n = 0 . When n = N , N + 1 , , we obtain
A n = k = N n k α + 1 + λ 1 ε p 1 < N 1 n t α + 1 + λ 1 ε p 1 d t 0 n t α + 1 + λ 1 ε p 1 d t = p α + 1 + λ 1 ε n α + 1 + λ 1 ε p .
When 0 < x < 1 , F ˜ ( f ) ( x ) = 0 ; when x 1 , we have
F ˜ ( f ) ( x ) = 1 x t β + 1 + λ 2 ε q 1 d t < 0 x t β + 1 + λ 2 ε q 1 d t = q β + 1 + λ 2 ε x β + 1 + λ 2 ε q .
Thus, we have
A ˜ p , α F ˜ ( f ) q , β = n = N n α | A n | p 1 / p 0 + x β | F ˜ ( f ) ( x ) | q d x 1 / q p q ( α + 1 + λ 1 ε ) ( β + 1 + λ 2 ε ) n = N n 1 λ 1 ε 1 / p 1 + x 1 λ 2 ε d x 1 / q p q ( α + 1 + λ 1 ε ) ( β + 1 + λ 2 ε ) N 1 + t 1 λ 1 ε d t 1 / p 1 + x 1 λ 2 ε d x 1 / q = p q ( α + 1 + λ 1 ε ) ( β + 1 + λ 2 ε ) · 1 ε λ 1 1 / p λ 2 1 / q ( N 1 ) λ 1 ε p .
Similarly, we have
n = N 0 + a n f ( x ) n λ 1 + x λ 2 λ d x = n = N n α + 1 + λ 1 ε p 1 1 + 1 n λ 1 + x λ 2 λ x β + 1 + λ 2 ε q d x = 1 λ 2 n = N n 1 λ 1 ε n λ 1 + 1 ( 1 + t ) λ t β + 1 + λ 2 ε λ 2 q 1 d t 1 λ 2 N + t 1 λ 1 ε d t N λ 1 + 1 ( 1 + t ) λ t β + 1 + λ 2 ε λ 2 q 1 d t = 1 λ 1 λ 2 ε N λ 1 ε N λ 1 + 1 ( 1 + t ) λ t β + 1 + λ 2 ε λ 2 q 1 d t .
From (10)–(12), we obtain
1 λ 1 λ 2 ε N λ 1 ε N λ 1 + 1 ( 1 + t ) λ t β + 1 + λ 2 ε λ 2 q 1 d t M 1 p q ( α + 1 + λ 1 ε ) ( β + 1 + λ 2 ε ) 1 ε λ 1 1 / p λ 2 1 / q ( N 1 ) λ 1 ε p .
Thus,
( α + 1 + λ 1 ε ) ( β + 1 + λ 2 ε ) λ 1 1 / q λ 2 1 / p p q N λ 1 ε N λ 1 + 1 ( 1 + t ) λ t β + 1 + λ 2 ε λ 2 q 1 d t M 1 ( N 1 ) λ 1 ε p .
Let ε 0 + , we get
( α + 1 ) ( β + 1 ) λ 1 1 / q λ 2 1 / p p q N λ 1 + 1 ( 1 + t ) λ t β + 1 λ 2 q 1 d t M 1 .
Further, letting N + , we obtain
( α + 1 ) ( β + 1 ) λ 1 1 / q λ 2 1 / p p q 0 + 1 ( 1 + t ) λ t β + 1 λ 2 q 1 d t M 1 .
By the properties of the Beta function and Gamma function, when α < 1 and β < 1 , α + 1 λ 1 p > 0 and β + 1 λ 2 q > 0 ; thus,
M 1 ( α + 1 ) ( β + 1 ) λ 1 1 / q λ 2 1 / p p q B β + 1 λ 2 q , λ + β + 1 λ 2 q = ( α + 1 ) ( β + 1 ) λ 1 1 / q λ 2 1 / p p q B α + 1 λ 1 p , β + 1 λ 2 q = 1 λ 1 1 / q λ 2 1 / p Γ ( λ ) ( α + 1 ) ( β + 1 ) p q Γ α + 1 λ 1 p Γ β + 1 λ 2 q = λ 1 1 / p λ 2 1 / q 1 Γ ( λ ) Γ 1 α + 1 λ 1 p Γ 1 β + 1 λ 2 q = M 0 .
This contradicts M 1 < M 0 ; hence, M 0 is the best constant factor in ( 5 ) . □
Similar to the proof of Theorem 1, we can also obtain the following theorem:
Theorem 2. 
Let p > 1 , 1 p + 1 q = 1 , λ , α , β R , 0 < λ 1 1 , λ 2 > 0 , λ > 0 , α + 1 λ 1 p + β + 1 λ 2 q + λ 1 λ 2 = c , max ( λ 1 1 ) p 1 , ( λ 1 1 ) p 1 + λ 1 p c < α < 1 , q 1 ( λ + 1 ) λ 2 q < β < q 1 , and lim n + e t n λ 1 A n = 0 ( t > 0 ) .
(i) There exists a constant M > 0 if and only if c 0 , i.e., α + 1 λ 1 p + β + 1 λ 2 q + λ 1 λ 2 0 , such that
n = 1 0 + a n f ( x ) n λ 1 + x λ 2 λ d x M A ˜ p , α f q , β ,
where A ˜ = { A n } = k = 1 n a k ( a k 0 ).
(ii) When c = 0 , i.e., α + 1 λ 1 p + β + 1 λ 2 q + λ 1 λ 2 = 0 , the best constant factor in ( 13 ) is
M 0 = λ 1 λ 2 1 / p 1 Γ ( λ ) Γ 1 α + 1 λ 1 p Γ 1 λ 2 β + 1 λ 2 q .

4. Applications

Let p > 1 , and define
S λ 1 l p α = a ˜ = { a n } ( a n 0 ) : A ˜ p , α = n = 1 n α | A n | p 1 / p < + , lim n + e t n λ 1 A n = 0 ( t > 0 ) .
Below, we discuss the boundedness of the discrete operator from S λ 1 l p α to L p γ ( 0 , + ) and the problem of operator norms.
Theorem 3. 
Let 1 p + 1 q = 1 ( p > 1 ) , α , β R , 0 < λ 1 1 , λ 2 > 0 , λ > 0 , α + 1 λ 1 p + β + 1 λ 2 q + λ 1 λ 2 = c , max ( λ 1 1 ) p 1 , ( λ 1 1 ) p 1 + λ 1 p c < α < 1 , q 1 ( λ + 1 ) λ 2 q < β < q 1 , and define the operator T as
T ( a ˜ ) ( x ) = n = 1 a n n λ 1 + x λ 2 λ , a ˜ = { a n } S λ 1 l p α .
(i) T is a bounded operator from S λ 1 l p α to L p β ( p 1 ) ( 0 , + ) if and only if c 0 , i.e., α + 1 λ 1 p + β + 1 λ 2 q + λ 1 λ 1 0 . Then, there exists a constant M > 0 such that
T ( a ˜ ) p , β ( p 1 ) M A ˜ p , α .
(ii) When c = 0 , i.e., α + 1 λ 1 p + β + 1 λ 2 q + λ 1 λ 1 = 0 , the norm of operator T is
T = λ 1 λ 2 1 / p 1 Γ ( λ ) Γ 1 α + 1 λ 1 p Γ 1 λ 2 β + 1 λ 2 q .
Proof. 
We first prove that when a n 0 , ( 14 ) is equivalent to ( 13 ) .
If ( 13 ) holds, let
f ( x ) = x β ( p 1 ) | T ( a ˜ ) ( x ) | p 1 = x β ( p 1 ) n = 1 a n n λ 1 + x λ 2 λ p 1 .
Then,
T ( a ˜ ) p , β ( p 1 ) p = 0 + x β ( p 1 ) | T ( a ˜ ) ( x ) | p d x = 0 + f ( x ) n = 1 a n n λ 1 + x λ 2 λ d x = n = 1 0 + a n f ( x ) n λ 1 + x λ 2 λ d x M A ˜ p , α f q , β = M A ˜ p , α 0 + x β x β ( 1 p ) | T ( a ˜ ) ( x ) | p 1 q d x 1 / q = M A ˜ p , α 0 + x β ( 1 p ) | T ( a ˜ ) ( x ) | p d x ( p 1 ) / p = M A ˜ p , α T ( a ˜ ) p , β ( 1 p ) p 1 ,
from which ( 14 ) can be derived.
Conversely, if ( 14 ) holds, by Hölder’s inequality, we have
n = 1 0 + a n f ( x ) n λ 1 + x λ 2 λ d x = 0 + f ( x ) T ( a ˜ ) ( x ) d x = 0 + x β q T ( a ˜ ) ( x ) x β q f ( x ) d x 0 + x β p q | T ( a ˜ ) ( x ) | p d x 1 / p 0 + x β | f ( x ) | q d x 1 / q = 0 + x β ( p 1 ) | T ( a ˜ ) ( x ) | p d x 1 / p f q , β = T ( a ˜ ) p , β ( p 1 ) f q , β M A ˜ p , α f q , β ,
Thus, ( 13 ) holds. Since ( 14 ) is equivalent to ( 13 ) , Theorem 3 holds by Theorem 2. □
In Theorem 3, taking λ 1 = λ 2 = 1 2 and λ = 2 , we can obtain
Corollary 1. 
Let 1 p + 1 q = 1 ( p > 1 ) , α , β R , α + 1 p + β + 1 q = 0 , p 2 1 < α < 1 , and q 2 1 < β < q 1 . Then, the operator T,
T ( a ˜ ) ( x ) = n = 1 a n ( n + x ) 2
is a bounded operator from S 1 2 l p α to L p β ( p 1 ) ( 0 , + ) , and the operator norm of T is
T = Γ 1 2 α + 1 p Γ 2 1 β + 1 q .

5. Conclusions

In this paper, we discuss the more general semi-discrete Hilbert-type inequality involving the partial sum A n and the variable upper limit integral function F ˜ ( f ) ( x ) , where the kernel is the quasi-homogeneous kernel 1 ( n λ 1 + x λ 2 ) λ . We obtain the necessary and sufficient conditions for such inequalities to hold and the expressions of the best constant factors. Due to the full use of the construction theorem of quasi-homogeneous semi-discrete Hilbert-type inequalities, the proof process is optimized; because the parameter structure of the obtained results is simple and clear, our results are more convenient to apply. Finally, we also discuss the application of the results in operator theory, highlighting the application value of the obtained results. In the future, we will also explore high-dimensional cases.

Author Contributions

Validation, Q.Z.; Writing—original draft, Y.H.; Writing—review & editing, B.H. All authors have read and agreed to the published version of the manuscript.

Funding

Supported by Guangzhou Huashang College Research Team Project (No. 2021HSKT03).

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

  1. Hardy, G.H.; Littlewood, J.E.; Polya, G. Inequalities; Cambridge University Press: Cambridge, UK, 1934. [Google Scholar]
  2. Yang, B.C. On Hilbert’s integral inequality. J. Math. Anal. 1998, 220, 778–785. [Google Scholar]
  3. Kuang, J.C. On new extension of Hilbert’s integral inequality. J. Math. Anal. Appl. 1999, 235, 608–614. [Google Scholar]
  4. Krnic, M.; Pecaric, J. General Hilbert’s and Hardy’s inequalities. Math. Inequal. Appl. 2005, 8, 29–51. [Google Scholar] [CrossRef]
  5. Chen, Q.; Yang, B.C. Half-discrete Hardy-Hilbert’s inequality with two internal variables. J. Inequal. Appl. 2013, 2013, 485. [Google Scholar] [CrossRef]
  6. Krnic, M.; Pecaric, J. Extension of Hilbert’s inequality. J. Math. Anal. Appl. 2006, 324, 150–160. [Google Scholar] [CrossRef]
  7. Yang, B.C. The Norm of Operator and Hilbert-Type Inequalities; Science Press: Beijing, China, 2009. [Google Scholar]
  8. Xie, Z.T. A new half-discrete Hilbert’s inequality with the homogeneous kernel of degree −4 μ. J. Zhanjiang Norm. Univ. 2011, 32, 13–19. [Google Scholar]
  9. Hong, Y.; Wen, Y.M. A necessary and sufficient condition of that Hilbert type series inequality with homogeneous kernel has the best constant factor. Ann. Math. 2016, 37A, 329–336. [Google Scholar]
  10. Hong, Y. On the structure character of Hilbert’s type integral inequality with homogeneous kernel and application. J. Jilin Univ. (Sci. Ed.) 2017, 55, 189–194. [Google Scholar]
  11. Wang, A.; Yang, B.C. Equivalent statements of a Hilbert-type integral inequality with the extended Hurwitz Zeta function in the whole plane. J. Math. Inequal. 2020, 14, 1029–1054. [Google Scholar] [CrossRef]
  12. Rassias, M.T.; Yang, B.C. Equivalent conditions of a Hardy-type integral inequality related to the extended Riemann zeta function. Adv. Oper. Theory 2017, 2, 237–256. [Google Scholar]
  13. Rassias, M.T.; Yang, B.C. On a equivalent property of a reverse Hilbert-type integral inequality related to the extended Hurwitz-zeta function. J. Math. Inequal. 2019, 13, 315–334. [Google Scholar] [CrossRef]
  14. Adiyasuren, V.; Batbold, T.; Azar, L.E. A new discrete Hilbert-type inequality involving partial sums. J. Inequal. Appl. 2019, 2019, 127. [Google Scholar] [CrossRef]
  15. Huang, X.Y.; Luo, R.C.; Yang, B.C.; Huang, X.S. A new reverse Mulholland’s inequality with one partial sum in the kernel. J. Inequal. Appl. 2024, 2024, 9. [Google Scholar] [CrossRef]
  16. Peng, L.; Yang, B.C. A new extended Mulholland’s inequality involving one partial Sums. Open Math. 2024, 22, 20240039. [Google Scholar] [CrossRef]
  17. Yang, B.C.; Wu, S.H. An improved version of the parameterized Hardy-Hilbert inequality involving two partial sums. Mathematics 2025, 13, 1331. [Google Scholar] [CrossRef]
  18. Wang, A.Z.; Yang, B.C. A new half-discrete Hilbert -type inequality involving higher-order derivative function and partial sum. J. Jilin Univ. (Sci. Ed.) 2023, 61, 1296–1304. [Google Scholar]
  19. Huang, X.Y.; Yang, B.C. On a more accurate half-discrete Mulholland-type inequality involving one multiple upper limit function. J. Funct. Spaces 2021, 2021, 6970158. [Google Scholar] [CrossRef]
  20. Yang, B.C.; Wu, S.H. A weighted generalization of Hardy-Hilbert-type inequality involving two partial sums. Mathematics 2023, 11, 3212. [Google Scholar] [CrossRef]
  21. Wang, A.Z.; Hong, Y.; Yang, B.C. On a new half-discrete Hilbert-type inequality with the multiple upper limit function and the partial sums. J. Appl. Anal. Comput. 2022, 12, 814–830. [Google Scholar] [CrossRef] [PubMed]
  22. Hong, Y.; He, B. Theory and Applications of Hilbert-Type Inequalities; Science Press: Beijing, China, 2023; pp. 347–353. [Google Scholar]
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MDPI and ACS Style

Hong, Y.; He, B.; Zhao, Q. Construction Theorem and Application of a Semi-Discrete Hilbert-Type Inequality Involving Partial Sums and Variable Upper Limit Integral Functions. Axioms 2025, 14, 755. https://doi.org/10.3390/axioms14100755

AMA Style

Hong Y, He B, Zhao Q. Construction Theorem and Application of a Semi-Discrete Hilbert-Type Inequality Involving Partial Sums and Variable Upper Limit Integral Functions. Axioms. 2025; 14(10):755. https://doi.org/10.3390/axioms14100755

Chicago/Turabian Style

Hong, Yong, Bing He, and Qian Zhao. 2025. "Construction Theorem and Application of a Semi-Discrete Hilbert-Type Inequality Involving Partial Sums and Variable Upper Limit Integral Functions" Axioms 14, no. 10: 755. https://doi.org/10.3390/axioms14100755

APA Style

Hong, Y., He, B., & Zhao, Q. (2025). Construction Theorem and Application of a Semi-Discrete Hilbert-Type Inequality Involving Partial Sums and Variable Upper Limit Integral Functions. Axioms, 14(10), 755. https://doi.org/10.3390/axioms14100755

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