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On a new generalization of some Hilbert-type inequalities

  • Minghui You EMAIL logo , Wei Song and Xiaoyu Wang
Published/Copyright: July 9, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.

Keywords: Hilbert-type inequality; partial fraction expansion; Euler number; Bernoulli number
MSC 2010: 26D15; 41A17

1 Introduction

Let ∥ f ∥ p , μ denote the norm of a measurable function f : R + → R + with respect to a measurable weighted function μ : R + → R + , that is,

∥ f ∥ p , μ ≔ ∫ R + μ ( x ) f p ( x ) d x 1 p ,

where p ≥ 1 . Under the definition of ∥ f ∥ p , μ , we define a weighted measured function space L p , μ ( R + ) as follows:

L p , μ ( R + ) ≔ { f : R + → R + , ∥ f ∥ p , μ < ∞ } .

In particular, for μ ( x ) = 1 , we have the abbreviated form: ∥ f ∥ p , μ = ∥ f ∥ p and L p , μ ( R + ) = L p ( R + ) .

Consider two nonnegative real-valued functions f ∈ L p ( R + ) and g ∈ L q ( R + ) , where p > 1 and 1 p + 1 q = 1 . Then we have the following classical inequality [1]:

(1.1) ∫ 0 ∞ ∫ 0 ∞ f ( x ) g ( y ) x + y d x d y < π sin π p ∥ f ∥ p ∥ g ∥ q ,

where the constant factor π sin π p is the best possible. Inequality (1.1) is the so-called Hilbert inequality, which is one of the most important inequalities in analysis and its relevant applications. Additionally, we also have some inequalities similar to inequality (1.1), such as [1]

(1.2) ∫ 0 ∞ ∫ 0 ∞ log x y x − y f ( x ) g ( y ) d x d y < π sin π p 2 ∥ f ∥ p ∥ g ∥ q .

In general, such inequalities as inequality (1.2) are called Hilbert-type inequalities. Although (1.1) and (1.2) were put forward more than 100 years ago, mathematicians have always been interested in their extensions, refinements, analogies and high-dimensional generalizations. The following inequality is a classical extension of (1.1), which was established by Yang [2] in 2004, that is,

(1.3) ∫ 0 ∞ ∫ 0 ∞ f ( x ) g ( y ) x β + y β d x d y < π β sin π r ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where β > 0 , μ ( x ) = x p 1 − β r − 1 , ν ( y ) = y q 1 − β s − 1 and 1 r + 1 s = 1 , and the constant factor on the right-hand side of (1.3) is the best possible. For some other extensions of inequality (1.1), and the relevant research on the discrete and half-discrete cases corresponding to (1.1), we can refer to [3,4,5, 6,7,8, 9,10,11, 12,13,14, 15,16,17]. Furthermore, some analogical forms of inequality (1.1) can be found in [16], such as

(1.4) ∫ 0 ∞ ∫ 0 ∞ f ( x ) g ( y ) x 2 β + x β y β + y 2 β d x d y < 2 3 π 9 β ∥ f ∥ 2 , μ ∥ g ∥ 2 , ν

and

(1.5) ∫ 0 ∞ ∫ 0 ∞ f ( x ) g ( y ) x 2 β − x β y β + y 2 β d x d y < 4 3 π 9 β ∥ f ∥ 2 , μ ∥ g ∥ 2 , ν ,

where β > 0 , μ ( x ) = x 1 − 2 β and ν ( y ) = y 1 − 2 β .

Regarding inequality (1.2), Yang [18] gave an analogy as follows in 2008:

(1.6) ∫ 0 ∞ ∫ 0 ∞ log x y x + y f ( x ) g ( y ) d x d y < 8 c 0 ∥ f ∥ 2 ∥ g ∥ 2 ,

where c 0 = ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) 2 = 0.9159 6 + is the Catalan constant.

Some other inequalities related to (1.2) and (1.6), including the discrete and half-discrete cases, can be found in [7,16,19,20,21]. In addition, by the introduction of different new kernel functions, multiple parameters and special functions, a large number of new Hilbert-type inequalities were established in the past several decades (see [22,23,24, 25,26,27, 28,29,30, 31,32,33]).

In this work, we will establish the following Hilbert-type inequalities with the best constant factors:

(1.7) ∫ 0 ∞ ∫ 0 ∞ log x y 2 x 2 + x y + y 2 f ( x ) g ( y ) d x d y < 16 3 π 243 ∥ f ∥ 2 , μ ∥ g ∥ 2 , ν 1 ,

(1.8) ∫ 0 ∞ ∫ 0 ∞ log x y ( x − y ) ( x 2 + y 2 ) f ( x ) g ( y ) d x d y < 3 π 2 16 ∥ f ∥ 2 , μ ∥ g ∥ 2 , ν 2 ,

(1.9) ∫ 0 ∞ ∫ 0 ∞ log x y 2 x 2 − x y + y 2 f ( x ) g ( y ) d x d y < 43 3 π 36 ∥ f ∥ 2 , μ ∥ g ∥ 2 , ν 1 ,

where μ ( x ) = x − 1 , ν 1 ( y ) = y − 1 and ν 2 ( y ) = y − 3 .

More generally, we will construct a new kernel including both the homogeneous and non-homogeneous cases, and establish a new Hilbert-type inequality which is a unified extension of inequality (1.1)–(1.9). Furthermore, the equivalent Hardy-type inequality is also considered. The discussions will be closed with some corollaries addressing special Hilbert-type inequalities with the constant factors associated with the higher derivatives of trigonometric functions.

2 Definitions and lemmas

Lemma 2.1

Let n ∈ N + , δ ∈ { 1 , − 1 } and n β > λ > 0 . Let α be such that α > − 1 for δ n = 1 and α > 0 for δ n = − 1 . Define

(2.1) K ( t ) ≔ ∣ log t ∣ α ∣ 1 + δ t β + ⋯ + ( δ t β ) n ∣ .

Then ∫ 0 ∞ K ( t ) t λ − 1 d t converges.

Proof

For the case α > − 1 and δ n = 1 , let θ be such that 0 < θ < n β − λ . Then

lim t → + ∞ K ( t ) t λ − 1 t − n β + λ + θ − 1 = lim t → + ∞ ∣ log t ∣ α t λ − 1 ( δ t β ) − n t − n β + λ + θ − 1 = lim t → + ∞ ∣ log t ∣ α t θ = 0 .

Since − n β + λ + θ − 1 < − 1 , it follows that ∫ 2 ∞ K ( t ) t λ − 1 d t converges.

In addition, since n ∈ N + , δ ∈ { 1 , − 1 } and δ n = 1 , we can easily obtain

lim t → 1 ∣ 1 + δ t β + ⋯ + ( δ t β ) n ∣ = 1 + δ + δ 2 + ⋯ + δ n ≠ 0 .

Therefore,

(2.2) lim t → 1 K ( t ) t λ − 1 ∣ 1 − t ∣ α = 1 1 + δ + δ 2 + ⋯ + δ n lim t → 1 ∣ log t ∣ α ∣ 1 − t ∣ α = 1 1 + δ + δ 2 + ⋯ + δ n .

Hence, it follows from (2.2) that ∫ 1 2 2 K ( t ) t λ − 1 d t converges.

Finally, it is needed to prove that ∫ 0 1 2 K ( t ) t λ − 1 d t converges, and this is obvious according to the following equality, that is,

lim t → 0 + K ( t ) t λ − 1 t λ + θ − 1 = lim t → 0 + ∣ log t ∣ α t θ = 0 ,

where − λ < θ < 0 . Based on the above discussions, it is proved that ∫ 0 ∞ K ( t ) t λ − 1 d t converges under the condition α > − 1 and δ n = 1 . Similarly, it can also be proved that ∫ 0 ∞ K ( t ) t λ − 1 d t converges under the condition α > 0 and δ n = − 1 .□

Lemma 2.2

Let n ∈ N + , δ ∈ { 1 , − 1 } and β n > λ > 0 . Let α be such that α > − 1 for δ n = 1 and α > 0 for δ n = − 1 . Define

(2.3) C ( α , β , λ , δ , n , β 1 , β 2 ) ≔ ∑ k = 0 ∞ δ k ( n + 1 ) 1 ( k β ( n + 1 ) + λ ) α + 1 − δ ( k β ( n + 1 ) + β ( n + 1 ) − λ ) α + 1 + ∑ k = 0 ∞ δ k ( n + 1 ) 1 ( k β ( n + 1 ) + β n − λ ) α + 1 − δ ( k β ( n + 1 ) + λ + β ) α + 1 .

Then

(2.4) ∫ 0 ∞ K ( t ) t λ − 1 d t = Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) ,

where Γ ( u ) ≔ ∫ 0 ∞ x u − 1 e − x d x , ( u > 0 ) is the Γ -function [34], and Γ ( u ) = ( u − 1 ) ! for u ∈ N + .

Proof

(2.5) ∫ 0 ∞ K ( t ) t λ − 1 d t = ∫ 0 ∞ ∣ t λ − 1 − δ t λ + β − 1 ∣ ∣ 1 − ( δ t β ) n + 1 ∣ ∣ log t ∣ α d t = ∫ 0 1 ∣ t λ − 1 − δ t λ + β − 1 ∣ ∣ 1 − ( δ t β ) n + 1 ∣ ∣ log t ∣ α d t + ∫ 1 ∞ ∣ t λ − 1 − δ t λ + β − 1 ∣ ∣ 1 − ( δ t β ) n + 1 ∣ ∣ log t ∣ α d t ≔ I 1 + I 2 .

Observing that β > 0 , δ ∈ { 1 , − 1 } and 0 < t < 1 , we have

1 1 − ( δ t β ) n + 1 = ∑ k = 0 ∞ ( δ t β ) k ( n + 1 ) .

Therefore,

(2.6) I 1 = ∑ k = 0 ∞ δ k ( n + 1 ) ∫ 0 1 t k β ( n + 1 ) + λ − 1 ∣ log t ∣ α d t − δ ∫ 0 1 t k β ( n + 1 ) + λ + β − 1 ∣ log t ∣ α d t ≔ ∑ k = 0 ∞ δ k ( n + 1 ) ( I 11 − δ I 12 ) .

Setting ∣ log t ∣ = u k β ( n + 1 ) + λ , we can obtain

(2.7) I 11 = 1 ( k β ( n + 1 ) + λ ) α + 1 ∫ 0 ∞ e − u u α d u = Γ ( α + 1 ) ( k β ( n + 1 ) + λ ) α + 1 .

Similarly, we have

(2.8) I 12 = 1 ( k β ( n + 1 ) + λ + β ) α + 1 ∫ 0 ∞ e − u u α d u = Γ ( α + 1 ) ( k β ( n + 1 ) + λ + β ) α + 1 .

It follows from plugging (2.7) and (2.8) into (2.6) that

(2.9) I 1 = ∑ k = 0 ∞ δ k ( n + 1 ) Γ ( α + 1 ) ( k β ( n + 1 ) + λ ) α + 1 − δ Γ ( α + 1 ) ( k β ( n + 1 ) + λ + β ) α + 1 .

Similarly, we can deduce that

(2.10) I 2 = ∫ 0 1 ∣ t β n − λ − 1 − δ t β ( n + 1 ) − λ − 1 ∣ ∣ 1 − ( δ t β ) n + 1 ∣ ∣ log t ∣ α d t = ∑ k = 0 ∞ δ k ( n + 1 ) Γ ( α + 1 ) ( k β ( n + 1 ) + β n − λ ) α + 1 − δ Γ ( α + 1 ) ( k β ( n + 1 ) + β ( n + 1 ) − λ ) α + 1 .

Plugging (2.9) and (2.10) back into (2.5), and using (2.3), we obtain (2.4).□

Lemma 2.3

Let a , b > 0 , a + b = s and m ∈ N . Let φ 1 ( x ) = cot x . Then

(2.11) φ 1 ( 2 m ) a π s = ( 2 m ) ! s 2 m + 1 π 2 m + 1 ∑ k = 0 ∞ 1 ( s k + a ) 2 m + 1 − 1 ( s k + b ) 2 m + 1 ,

(2.12) φ 1 ( 2 m + 1 ) a π s = − ( 2 m + 1 ) ! s 2 m + 2 π 2 m + 2 ∑ k = 0 ∞ 1 ( s k + a ) 2 m + 2 + 1 ( s k + b ) 2 m + 2 .

Proof

We start the proof of Lemma 2.3 from the partial fraction expansion of φ 1 ( x ) = cot x as follows [34]:

φ 1 ( x ) = 1 x + ∑ k = 1 ∞ 1 x + k π + 1 x − k π .

Taking the 2 m th derivative of φ 1 ( x ) , we have

(2.13) φ 1 ( 2 m ) ( x ) = ( 2 m ) ! lim n → ∞ ∑ k = 0 n 1 ( x + k π ) 2 m + 1 + ∑ k = 1 n 1 ( x − k π ) 2 m + 1 .

Setting x = a π s in (2.13), in view of a + b = s , it follows that

φ 1 ( 2 m ) a π s = ( 2 m ) ! s 2 m + 1 π 2 m + 1 lim n → ∞ ∑ k = 0 n 1 ( s k + a ) 2 m + 1 + ∑ k = 1 n 1 ( a − s k ) 2 m + 1 = ( 2 m ) ! s 2 m + 1 π 2 m + 1 lim n → ∞ ∑ k = 0 n 1 ( s k + a ) 2 m + 1 − ∑ k = 0 n − 1 1 ( s k + b ) 2 m + 1 = ( 2 m ) ! s 2 m + 1 π 2 m + 1 ∑ k = 0 ∞ 1 ( s k + a ) 2 m + 1 − 1 ( s k + b ) 2 m + 1 .

Therefore, the proof of (2.11) is completed. Similarly, we can obtain (2.12).□

Lemma 2.4

Let a , b > 0 , a + b = s and m ∈ N . Let φ 2 ( x ) = csc x . Then

(2.14) φ 2 ( 2 m ) a π s = ( 2 m ) ! s 2 m + 1 π 2 m + 1 ∑ k = 0 ∞ ( − 1 ) k 1 ( s k + a ) 2 m + 1 + 1 ( s k + b ) 2 m + 1 .

Proof

Take the 2 m th derivative the partial fraction expansion of φ 2 ( x ) = csc x as follows [34]:

(2.15) φ 2 ( x ) = 1 x + ∑ k = 1 ∞ ( − 1 ) k 1 x + k π + 1 x − k π .

Then

(2.16) φ 2 ( 2 m ) ( x ) = ( 2 m ) ! lim n → ∞ ∑ k = 0 n ( − 1 ) k ( x + k π ) 2 m + 1 + ∑ k = 1 n ( − 1 ) k ( x − k π ) 2 m + 1 = ( 2 m ) ! lim n → ∞ ∑ k = 0 n ( − 1 ) k ( k π + x ) 2 m + 1 + ∑ k = 0 n − 1 ( − 1 ) k ( k π + π − x ) 2 m + 1 .

Since a + b = s , then we can arrive at (2.14) by setting x = a π s in (2.16).□

Remark 2.5

Letting a = b = 1 in (2.12), we have

(2.17) φ 1 ( 2 m + 1 ) π 2 = − ( 2 m + 1 ) ! 2 2 m + 3 π 2 m + 2 ∑ k = 0 ∞ 1 ( 2 k + 1 ) 2 m + 2 .

Since [34]   ∑ k = 0 ∞ 1 k 2 m + 2 = ( 2 π ) 2 m + 2 2 ( 2 m + 2 ) ! B m + 1 , where B m + 1 is a Bernoulli number, B 1 = 1 6 , B 2 = 1 30 , B 3 = 1 42 , … , then we obtain

(2.18) ∑ k = 0 ∞ 1 ( 2 k + 1 ) 2 m + 2 = ∑ k = 1 ∞ 1 k 2 m + 2 − 1 ( 2 k ) 2 m + 2 = 2 2 m + 2 − 1 2 ( 2 m + 2 ) ! π 2 m + 2 B m + 1 .

Applying (2.18) to (2.17), we obtain

(2.19) φ 1 ( 2 m + 1 ) π 2 = − 1 m + 1 ( 2 2 m + 2 − 1 ) 2 2 m + 1 B m + 1 .

Similarly, letting a = 1 , b = 3 in (2.12), we can also obtain

(2.20) φ 1 ( 2 m + 1 ) π 4 = − 1 m + 1 ( 2 2 m + 2 − 1 ) 4 2 m + 1 B m + 1 .

In addition, letting a = 1 , b = 3 in (2.11) and a = b = 1 in (2.14), in view of [34] ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) 2 m + 1 = π 2 m + 1 2 2 m + 2 ( 2 m ) ! E m , where E m is an Euler number, E 0 = 1 , E 1 = 1 , E 2 = 5 , E 3 = 61 , … , we obtain

(2.21) E m = 1 4 m φ 1 ( 2 m ) π 4 = φ 2 ( 2 m ) π 2 .

3 Main results

Theorem 3.1

Let n ∈ N + , δ ∈ { 1 , − 1 } , β n > λ > 0 and β 1 β 2 ≠ 0 . Let α be such that α > − 1 for δ n = 1 and α > 0 for δ n = − 1 . Define μ ( x ) = x p ( 1 − λ β 1 ) − 1 and ν ( y ) = y q ( 1 − λ β 2 ) − 1 , where p > 1 and 1 p + 1 q = 1 . Suppose that f ( x ) , g ( x ) ≥ 0 and f ( x ) ∈ L p , μ ( R + ) , g ( x ) ∈ L q , ν ( R + ) . K ( t ) is defined via (2.1) and C ( α , β , λ , δ , n , β 1 , β 2 ) is defined via (2.3). Then

(3.1) ∫ 0 ∞ ∫ 0 ∞ K ( x β 1 y β 2 ) f ( x ) g ( y ) d x d y < ∣ β 1 ∣ − 1 q ∣ β 2 ∣ − 1 p Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where the constant factor ∣ β 1 ∣ − 1 q ∣ β 2 ∣ − 1 p Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) is the best possible.

Proof

By Hölder’s inequality and Fubini’s theorem, we have

(3.2) ∫ 0 ∞ ∫ 0 ∞ K ( x β 1 y β 2 ) f ( x ) g ( y ) d x d y = ∫ 0 ∞ ∫ 0 ∞ ( K ( x β 1 y β 2 ) ) 1 p x 1 − λ β 1 q y λ β 2 − 1 p f ( x ) ( K ( x β 1 y β 2 ) ) 1 q y 1 − λ β 2 p x λ β 1 − 1 q g ( y ) d x d y ≤ ∫ 0 ∞ ∫ 0 ∞ K ( x β 1 y β 2 ) y λ β 2 − 1 x p ( 1 − λ β 1 ) q f p ( x ) d x d y 1 p ∫ 0 ∞ ∫ 0 ∞ K ( x β 1 y β 2 ) x λ β 1 − 1 y q ( 1 − λ β 2 ) p g q ( y ) d x d y 1 q = ∫ 0 ∞ ω ( x ) x p ( 1 − λ β 1 ) q f p ( x ) d x 1 p ∫ 0 ∞ ϖ ( y ) y q ( 1 − λ β 2 ) p g q ( y ) d y 1 q ,

where ω ( x ) = ∫ 0 ∞ K ( x β 1 y β 2 ) y λ β 2 − 1 d y and ϖ ( y ) = ∫ 0 ∞ K ( x β 1 y β 2 ) x λ β 1 − 1 d x .

Setting x β 1 y β 2 = t , and using Lemma 2.2, we can obtain

(3.3) ω ( x ) = x − λ β 1 ∣ β 2 ∣ ∫ 0 ∞ K ( t ) t λ − 1 d t = 1 ∣ β 2 ∣ Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) x − λ β 1

and

(3.4) ϖ ( y ) = y − λ β 2 ∣ β 1 ∣ ∫ 0 ∞ K ( t ) t λ − 1 d t = 1 ∣ β 1 ∣ Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) y − λ β 2 .

Plugging (3.3) and (3.4) into (3.2), we obtain

(3.5) ∫ 0 ∞ ∫ 0 ∞ K ( x β 1 y β 2 ) f ( x ) g ( y ) d x d y ≤ ∣ β 1 ∣ − 1 q ∣ β 2 ∣ − 1 p Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) ∥ f ∥ p , μ ∥ g ∥ q , ν .

If (3.5) takes the form of an equation, then there must exist two constants C 1 and C 2 that are not both equal to zero, such that

C 1 K ( x β 1 y β 2 ) y λ β 2 − 1 x p ( 1 − λ β 1 ) q f p ( x ) = C 2 K ( x β 1 y β 2 ) x λ β 1 − 1 y q ( 1 − λ β 2 ) p g q ( y )

holds almost everywhere in the domain ( 0 , ∞ ) × ( 0 , ∞ ) . That is,

C 1 x p ( 1 − λ β 1 ) f p ( x ) = C 2 y q ( 1 − λ β 2 ) g q ( y ) .

Hence, there must be a constant C such that C 1 x p ( 1 − λ β 1 ) f p ( x ) = C and C 2 y q ( 1 − λ β 2 ) g q ( y ) = C hold almost everywhere in ( 0 , ∞ ) . Without loss of generality, assuming C 1 ≠ 0 , we can obtain x p ( 1 − λ β 1 ) − 1 f p ( x ) = C C 1 x holds almost everywhere in ( 0 , ∞ ) . This obviously contradicts the fact that f ( x ) ∈ L p , μ ( R + ) . Therefore, (3.5) keeps the form of a strict inequality, and (3.1) is obtained.

Finally, it will be proved that the constant factor in (3.1) is the best possible. If the constant factor ∣ β 1 ∣ − 1 q ∣ β 2 ∣ − 1 p Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) is not the best possible, then there will be a positive constant k < ∣ β 1 ∣ − 1 q ∣ β 2 ∣ − 1 p Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) , such that (3.1) still holds if ∣ β 1 ∣ − 1 q ∣ β 2 ∣ − 1 p Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) is replaced by k . That is,

(3.6) ∫ 0 ∞ ∫ 0 ∞ K ( x β 1 y β 2 ) f ( x ) g ( y ) d x d y < k ∥ f ∥ p , μ ∥ g ∥ q , ν .

Consider a sufficiently small positive number ε , and define the functions f ε ( x ) and g ε ( y ) as follows:

f ε ( x ) ≔ x p ( λ β 1 − 1 ) − β 1 ε p , x ∈ Ω 1 , 0 x ∈ R + ⧹ Ω 1 , g ε ( y ) ≔ y q ( λ β 2 − 1 ) + β 2 ε q , y ∈ Ω 2 , 0 , y ∈ R + ⧹ Ω 2 ,

where Ω 1 = x : x > 0 , x β 1 ∣ β 1 ∣ > 1 and Ω 2 = y : y > 0 , y β 2 ∣ β 2 ∣ < 1 .

Replacing f and g in (3.6) with f ε and g ε , respectively, we have

(3.7) ε ∫ 0 ∞ ∫ 0 ∞ K ( x β 1 y β 2 ) f ε ( x ) g ε ( y ) d x d y < ε k ∫ Ω 1 x − β 1 ε − 1 d x 1 p ∫ Ω 2 y β 2 ε − 1 d y 1 q = k ∣ β 1 ∣ − 1 p ∣ β 2 ∣ − 1 q .

On the other hand, setting x β 1 y β 2 = t , we have

(3.8) ε ∫ 0 ∞ ∫ 0 ∞ K ( x β 1 y β 2 ) f ε ( x ) g ε ( y ) d x d y = ε ∫ Ω 2 y q ( λ β 2 − 1 ) + β 2 ε q ∫ Ω 1 K ( x β 1 y β 2 ) x p ( λ β 1 − 1 ) − β 1 ε p d x d y = ε ∣ β 1 ∣ ∫ Ω 2 y β 2 ε − 1 ∫ y β 2 ∞ K ( t ) t λ − ε p − 1 d t d y = ε ∣ β 1 ∣ ∫ Ω 2 y β 2 ε − 1 ∫ 1 ∞ K ( t ) t λ − ε p − 1 d t d y + ε ∣ β 1 ∣ ∫ Ω 2 y β 2 ε − 1 ∫ y β 2 1 K ( t ) t λ − ε p − 1 d t d y = 1 ∣ β 1 β 2 ∣ ∫ 1 ∞ K ( t ) t λ − ε p − 1 d t + ε ∣ β 1 ∣ ∫ Ω 2 y β 2 ε − 1 ∫ y β 2 1 K ( t ) t λ − ε p − 1 d t d y .

No matter β 2 > 0 or β 2 < 0 , it follows from Fubini’s theorem that

(3.9) ∫ Ω 2 y β 2 ε − 1 ∫ y β 2 1 K ( t ) t λ − ε p − 1 d t d y = 1 ∣ β 2 ∣ ε ∫ 0 1 K ( t ) t λ + ε q − 1 d t .

Applying (3.9) to (3.8), we can obtain

(3.10) ε ∫ 0 ∞ ∫ 0 ∞ K ( x β 1 y β 2 ) f ε ( x ) g ε ( y ) d x d y = 1 ∣ β 1 β 2 ∣ ∫ 1 ∞ K ( t ) t λ − ε p − 1 d t + ∫ 0 1 K ( t ) t λ + ε q − 1 d t .

Letting ε → 0 + and using (2.4), we arrive at

(3.11) ε ∫ 0 ∞ ∫ 0 ∞ K ( x β 1 y β 2 ) f ε ( x ) g ε ( y ) d x d y = 1 ∣ β 1 β 2 ∣ Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) + o ( 1 ) .

Combining (3.7) and (3.11), and letting ε → 0 + , then we have ∣ β 1 ∣ − 1 q ∣ β 2 ∣ − 1 p Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) ≤ k , which obviously contradicts the hypothesis. Therefore, the constant factor in (3.1) is the best possible.□

By convention, we will establish the following equivalent form of Theorem 3.1, which is usually called the Hardy-Hilbert-type inequality.

Theorem 3.2

Let n ∈ N + , δ ∈ { 1 , − 1 } , β n > λ > 0 and β 1 β 2 ≠ 0 . Let α be such that α > − 1 for δ n = 1 and α > 0 for δ n = − 1 . Define μ ( x ) = x p ( 1 − λ β 1 ) − 1 and ν ( y ) = y q ( 1 − λ β 2 ) − 1 , where p > 1 and 1 p + 1 q = 1 . Suppose that f ( x ) ≥ 0 with f ( x ) ∈ L p , μ ( R + ) . K ( t ) is defined via (2.1) and C ( α , β , λ , δ , n , β 1 , β 2 ) is defined via (2.3). Then

(3.12) ∫ 0 ∞ y p λ β 2 − 1 ∫ 0 ∞ K ( x β 1 y β 2 ) f ( x ) d x p d y < ∣ β 1 ∣ − 1 q ∣ β 2 ∣ − 1 p Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) p ( ∥ f ∥ p , μ ) p ,

where the constant factor ∣ β 1 ∣ − 1 q ∣ β 2 ∣ − 1 p Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) p is the best possible, and (3.12) is equivalent to (3.1).

Proof

Consider g ( y ) ≔ y p λ β 2 − 1 ∫ 0 ∞ K ( x β 1 y β 2 ) f ( x ) d x p − 1 . It follows from Theorem 3.1 that

(3.13) 0 < ( ∥ g ∥ q , ν ) p q = ∫ 0 ∞ y q ( 1 − λ β 2 ) − 1 g q ( y ) d y p = ∫ 0 ∞ y p λ β 2 − 1 ∫ 0 ∞ K ( x β 1 y β 2 ) f ( x ) d x p d y p = ∫ 0 ∞ ∫ 0 ∞ K ( x β 1 y β 2 ) f ( x ) g ( y ) d x d y p ≤ ∣ β 1 ∣ − 1 q ∣ β 2 ∣ − 1 p Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) p ( ∥ f ∥ p , μ ) p ( ∥ g ∥ q , ν ) p .

Therefore,

(3.14) 0 < ( ∥ g ∥ q , ν ) q = ∫ 0 ∞ y p λ β 2 − 1 ∫ 0 ∞ K ( x β 1 y β 2 ) f ( x ) d x p d y ≤ ∣ β 1 ∣ − 1 q ∣ β 2 ∣ − 1 p Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) p ( ∥ f ∥ p , μ ) p .

Since f ( x ) ∈ L p , μ ( R + ) , it follows from (3.14) that g ( x ) ∈ L q , ν ( R + ) . By using Theorem 3.1 again, we can obtain that both (3.13) and (3.14) take the form of strict inequality, and therefore (3.12) is proved.

On the other hand, if (3.12) is valid, by Hölder’s inequality, we have

(3.15) ∫ 0 ∞ ∫ 0 ∞ K ( x β 1 y β 2 ) f ( x ) g ( y ) d x d y = ∫ 0 ∞ y λ β 2 − 1 + 1 q ∫ 0 ∞ K ( x β 1 y β 2 ) f ( x ) d x y 1 − λ β 2 − 1 q g ( y ) d y ≤ ∫ 0 ∞ y p λ β 2 − 1 ∫ 0 ∞ K ( x β 1 y β 2 ) f ( x ) d x p d y 1 p ∥ g ∥ q , ν .

Plugging (3.12) into (3.15), we can get (3.1). Therefore, (3.1) is equivalent to (3.12). According to the equivalence of (3.1) and (3.12), it can be shown that the constant factor ∣ β 1 ∣ − 1 q ∣ β 2 ∣ − 1 p Γ ( α + 1 ) C ( α , β , λ , δ , n , β 1 , β 2 ) p in (3.12) is the best possible. The proof of Theorem 3.2 is completed.□

4 Applications

Letting δ = 1 , α = 2 m ( m ∈ N ) in (2.3), and using (2.11), we can obtain

C ( α , β , λ , δ , n , β 1 , β 2 ) = π 2 m + 1 ( 2 m ) ! [ β ( n + 1 ) ] 2 m + 1 φ 1 ( 2 m ) λ π β ( n + 1 ) − φ 1 ( 2 m ) ( β + λ ) π β ( n + 1 ) .

Therefore, setting δ = 1 , α = 2 m ( m ∈ N ), β 1 = 1 and β 2 = − 1 in Theorem 3.1, and replacing g ( y ) y β n with g ( y ) , we can obtain the following corollary.

Corollary 4.1

Let n ∈ N + , m ∈ N and β n > λ > 0 . Let φ 1 ( x ) = cot x , μ ( x ) = x p ( 1 − λ ) − 1 and ν ( y ) = y q ( 1 + λ − β n ) − 1 , where p > 1 and 1 p + 1 q = 1 . Suppose that f ( x ) , g ( x ) ≥ 0 with f ( x ) ∈ L p , μ ( R + ) and g ( x ) ∈ L q , ν ( R + ) . Then

(4.1) ∫ 0 ∞ ∫ 0 ∞ log x y 2 m f ( x ) g ( y ) x β n + x β y β ( n − 1 ) + ⋯ + y β n d x d y < π 2 m + 1 [ β ( n + 1 ) ] 2 m + 1 φ 1 ( 2 m ) λ π β ( n + 1 ) − φ 1 ( 2 m ) ( β + λ ) π β ( n + 1 ) ∥ f ∥ p , μ ∥ g ∥ q , ν

Let n = 1 in (4.1), in view of

φ 1 ( 2 m ) λ π 2 β − φ 1 ( 2 m ) ( β + λ ) π 2 β = 2 2 m + 1 φ 2 ( 2 m ) λ π β ,

where φ 2 ( x ) = csc x , then (4.1) is transformed to

(4.2) ∫ 0 ∞ ∫ 0 ∞ log x y 2 m x β + y β f ( x ) g ( y ) d x d y < π β 2 m + 1 φ 2 ( 2 m ) λ π β ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where μ ( x ) = x p ( 1 − λ ) − 1 and ν ( y ) = y q ( 1 + λ − β ) − 1 . Let m = 0 , λ = β r in (4.2), where 1 r + 1 s = 1 , then (4.2) reduces to (1.3). Additionally, let λ = β 2 in (4.2), then it follows from (2.21) that

(4.3) ∫ 0 ∞ ∫ 0 ∞ log x y 2 m x β + y β f ( x ) g ( y ) d x d y < π β 2 m + 1 E m ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where μ ( x ) = x p 1 − β 2 − 1 and ν ( y ) = y q 1 − β 2 − 1 .

Let n = 2 in (4.1), then we have

(4.4) ∫ 0 ∞ ∫ 0 ∞ log x y 2 m x 2 β + x β y β + y 2 β f ( x ) g ( y ) d x d y < π 3 β 2 m + 1 φ 1 ( 2 m ) λ π 3 β − φ 1 ( 2 m ) ( β + λ ) π 3 β ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where μ ( x ) = x p ( 1 − λ ) − 1 and ν ( y ) = y q ( 1 + λ − 2 β ) − 1 . Letting m = 0 , λ = β and p = q = 2 in (4.4), we obtain (1.4). Letting m = 1 , λ = β = 1 and p = q = 2 in (4.4), we obtain (1.7). Letting λ = β 2 in (4.4), since φ 1 ( 2 m ) π 2 = 0 , we can also obtain

(4.5) ∫ 0 ∞ ∫ 0 ∞ log x y 2 m x 2 β + x β y β + y 2 β f ( x ) g ( y ) d x d y < π 3 β 2 m + 1 φ 1 ( 2 m ) π 6 ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where μ ( x ) = x p 1 − β 2 − 1 and ν ( y ) = y q 1 − 3 β 2 − 1 .

Let n = 3 and λ = β in (4.1). In view of φ 1 ( 2 m ) π 2 = 0 , and using (2.21), we have

(4.6) ∫ 0 ∞ ∫ 0 ∞ ( log ( x y ) ) 2 m ( x β + y β ) ( x 2 β + y 2 β ) f ( x ) g ( y ) d x d y < π β 2 m + 1 E m 4 m + 1 ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where μ ( x ) = x p ( 1 − β ) − 1 and ν ( y ) = y q ( 1 − 2 β ) − 1 .

Setting δ = 1 , α = 2 m ( m ∈ N ) and β 1 = β 2 = 1 in Theorem 3.1, we can have the following Hilbert-type inequality with a non-homogeneous kernel.

Corollary 4.2

Let n ∈ N + , m ∈ N and β n > λ > 0 . Let φ 1 ( x ) = cot x , μ ( x ) = x p ( 1 − λ ) − 1 and ν ( y ) = y q ( 1 − λ ) − 1 , where p > 1 and 1 p + 1 q = 1 . Suppose that f ( x ) , g ( x ) ≥ 0 with f ( x ) ∈ L p , μ ( R + ) and g ( x ) ∈ L q , ν ( R + ) . Then

(4.7) ∫ 0 ∞ ∫ 0 ∞ ( log ( x y ) ) 2 m f ( x ) g ( y ) 1 + ( x y ) β + ⋯ + ( x y ) β n d x d y < π 2 m + 1 [ β ( n + 1 ) ] 2 m + 1 φ 1 ( 2 m ) λ π β ( n + 1 ) − φ 1 ( 2 m ) ( β + λ ) π β ( n + 1 ) ∥ f ∥ p , μ ∥ g ∥ q , ν .

Let n = 1 in (4.7), then we have

(4.8) ∫ 0 ∞ ∫ 0 ∞ ( log ( x y ) ) 2 m 1 + ( x y ) β f ( x ) g ( y ) d x d y < π β 2 m + 1 φ 2 ( 2 m ) λ π β ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where φ 2 ( x ) = csc x , μ ( x ) = x p ( 1 − λ ) − 1 and ν ( y ) = y q ( 1 − λ ) − 1 .

Similarly, taking special values for the parameters in Corollary 4.2, we can obtain some other special cases of Corollary 4.2. For instance, setting n = 2 , λ = β and n = 3 , λ = β in (4.7), respectively, we obtain

(4.9) ∫ 0 ∞ ∫ 0 ∞ ( log ( x y ) ) 2 m 1 + ( x y ) β + ( x y ) 2 β f ( x ) g ( y ) d x d y < 2 π 3 β 2 m + 1 φ 1 ( 2 m ) π 3 ∥ f ∥ p , μ ∥ g ∥ q , ν ,

(4.10) ∫ 0 ∞ ∫ 0 ∞ ( log ( x y ) ) 2 m ( 1 + ( x y ) β ) ( 1 + ( x y ) 2 β ) f ( x ) g ( y ) d x d y < π β 2 m + 1 E m 4 m + 1 ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where μ ( x ) = x p ( 1 − β ) − 1 and ν ( y ) = y q ( 1 − β ) − 1 .

Let δ = n = α = 1 and λ = β 2 in (2.3), we can obtain

C ( α , β , λ , δ , n , β 1 , β 2 ) = 8 β 2 ∑ k = 0 ∞ 1 ( 4 k + 1 ) 2 − 1 ( 4 k + 3 ) 2 = 8 β 2 ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) 2 = 8 c 0 β 2 ,

where c 0 = Σ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) 2 = 0.9159 6 + is the Catalan constant. Therefore, setting δ = n = α = 1 , λ = β 2 , β 1 = 1 and β 2 = − 1 in Theorem 3.1, and replacing g ( y ) y β with g ( y ) , we can obtain another corollary.

Corollary 4.3

Let β > 0 , p > 1 and 1 p + 1 q = 1 . Let μ ( x ) = x p 1 − β 2 − 1 and ν ( y ) = y q 1 − β 2 − 1 . Suppose that f ( x ) , g ( x ) ≥ 0 with f ( x ) ∈ L p , μ ( R + ) and g ( x ) ∈ L q , ν ( R + ) . Then

(4.11) ∫ 0 ∞ ∫ 0 ∞ log x y x β + y β f ( x ) g ( y ) d x d y < 8 c 0 β 2 ∥ f ∥ p , μ ∥ g ∥ q , ν .

Let β = 1 and p = q = 2 , then (4.11) reduces to (1.6).

Furthermore, setting δ = α = 1 , n = 3 , λ = β , β 1 = 1 and β 2 = − 1 in Theorem 3.1, and replacing g ( y ) y 3 β with g ( y ) , we can obtain Corollary 4.4.

Corollary 4.4

Let β > 0 , p > 1 and 1 p + 1 q = 1 . Let μ ( x ) = x p ( 1 − β ) − 1 and ν ( y ) = y q ( 1 − 2 β ) − 1 . Suppose that f ( x ) , g ( x ) ≥ 0 with f ( x ) ∈ L p , μ ( R + ) and g ( x ) ∈ L q , ν ( R + ) . Then

(4.12) ∫ 0 ∞ ∫ 0 ∞ log x y ( x β + y β ) ( x 2 β + y 2 β ) f ( x ) g ( y ) d x d y < c 0 β 2 ∥ f ∥ p , μ ∥ g ∥ q , ν .

Remark 4.5

Corollaries 4.3 and 4.4 can be regarded as supplements to (4.2) and (4.6), respectively. It should be noted that if n takes natural numbers other than 1 and 3, the conclusion expressed by the Catalan constant like Corollaries 4.3 and 4.4 cannot be obtained.

Let δ = − 1 , n = 2 l − 1 ( l ∈ N + ) and α = 2 m + 1 ( m ∈ N ) in (2.3). By using (2.13), we can obtain

C ( α , β , λ , δ , n , β 1 , β 2 ) = π 2 m + 2 ( 2 m + 1 ) ! ( 2 β l ) 2 m + 2 φ 1 ( 2 m + 1 ) λ π 2 β l + φ 1 ( 2 m + 1 ) ( β + λ ) π 2 β l .

Therefore, setting δ = − 1 , n = 2 l − 1 ( l ∈ N + ) , α = 2 m + 1 ( m ∈ N ) , β 1 = 1 and β 2 = − 1 in Theorem 3.1, and replacing g ( y ) y ( 2 l − 1 ) β with g ( y ) , we can obtain Corollary 4.6.

Corollary 4.6

Let l ∈ N + , m ∈ N and ( 2 l − 1 ) β > λ > 0 . Let φ 1 ( x ) = cot x , μ ( x ) = x p ( 1 − λ ) − 1 and ν ( y ) = y q ( 1 + λ + β − 2 l β ) − 1 , where p > 1 and 1 p + 1 q = 1 . Suppose that f ( x ) , g ( x ) ≥ 0 with f ( x ) ∈ L p , μ ( R + ) and g ( x ) ∈ L q , ν ( R + ) . Then

(4.13) ∫ 0 ∞ ∫ 0 ∞ log x y 2 m + 1 ∣ x ( 2 l − 1 ) β − x β ( 2 l − 2 ) y β + ⋯ − y ( 2 l − 1 ) β ∣ f ( x ) g ( y ) d x d y < − π 2 m + 2 ( 2 l β ) 2 m + 2 φ 1 ( 2 m + 1 ) λ π 2 l β + φ 1 ( 2 m + 1 ) ( β + λ ) π 2 l β ∥ f ∥ p , μ ∥ g ∥ q , ν .

Let l = 1 in (4.13), in view of

φ 1 ( 2 m + 1 ) λ π 2 β + φ 1 ( 2 m + 1 ) ( β + λ ) π 2 β = 2 2 m + 2 φ 1 ( 2 m + 1 ) λ π β ,

then it follows from (4.13) that

(4.14) ∫ 0 ∞ ∫ 0 ∞ log x y 2 m + 1 x β − y β f ( x ) g ( y ) d x d y < − π β 2 m + 2 φ 1 ( 2 m + 1 ) λ π β ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where μ ( x ) = x p ( 1 − λ ) − 1 and ν ( y ) = y q ( 1 + λ − β ) − 1 . Let m = 0 , β = 1 and λ = 1 q in (4.14), then we obtain (1.2). Let λ = β 2 in (4.14), by using (2.19), then we obtain

(4.15) ∫ 0 ∞ ∫ 0 ∞ log x y 2 m + 1 x β − y β f ( x ) g ( y ) d x d y < π β 2 m + 2 2 2 m + 2 − 1 m + 1 2 2 m + 1 B m + 1 ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where μ ( x ) = x p 1 − β 2 − 1 and ν ( y ) = y q 1 − β 2 − 1 . Letting λ = β 4 in (4.14), and using (2.20), we obtain

(4.16) ∫ 0 ∞ ∫ 0 ∞ log x y 2 m + 1 x β − y β f ( x ) g ( y ) d x d y < π β 2 m + 2 2 2 m + 2 − 1 m + 1 4 2 m + 1 B m + 1 ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where μ ( x ) = x p 1 − β 4 − 1 and ν ( y ) = y q ( 1 − 3 β 4 ) − 1 .

Let l = 2 and β = λ in (4.13), then we have

(4.17) ∫ 0 ∞ ∫ 0 ∞ log x y 2 m + 1 f ( x ) g ( y ) ( x β − y β ) ( x 2 β + y 2 β ) d x d y < π 4 β 2 m + 2 2 2 m + 2 − 1 m + 1 ( 2 2 m + 1 + 4 2 m + 1 ) B m + 1 ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where μ ( x ) = x p ( 1 − β ) − 1 and ν ( y ) = y q ( 1 − 2 β ) − 1 . Let m = 0 , β = 1 and p = q = 2 , then we have (1.8).

Let δ = − 1 , n = 2 l ( l ∈ N + ) and α = 2 m ( m ∈ N ) in (2.3). By using (2.13), we can obtain

C ( α , β , λ , δ , n , β 1 , β 2 ) = π 2 m + 1 ( 2 m ) ! [ ( 2 l + 1 ) β ] 2 m + 1 φ 2 ( 2 m ) λ π ( 2 l + 1 ) β + φ 2 ( 2 m ) ( β + λ ) π ( 2 l + 1 ) β .

Therefore, setting δ = − 1 , n = 2 l ( l ∈ N + ) , α = 2 m ( m ∈ N ) , β 1 = 1 and β 2 = − 1 in Theorem 3.1, and replacing g ( y ) y 2 l β with g ( y ) , we can obtain the last corollary.

Corollary 4.7

Let l ∈ N + , m ∈ N and 2 l β > λ > 0 . Let φ 2 ( x ) = csc x , μ ( x ) = x p ( 1 − λ ) − 1 and ν ( y ) = y q ( 1 + λ − 2 l β ) − 1 , where p > 1 and 1 p + 1 q = 1 . Suppose that f ( x ) , g ( x ) ≥ 0 with f ( x ) ∈ L p , μ ( R + ) and g ( x ) ∈ L q , ν ( R + ) . Then

(4.18) ∫ 0 ∞ ∫ 0 ∞ log x y 2 m ∣ x 2 l β − x ( 2 l − 1 ) β y β + ⋯ + y 2 l β ∣ f ( x ) g ( y ) d x d y < π 2 m + 1 [ ( 2 l + 1 ) β ] 2 m + 1 φ 2 ( 2 m ) λ π ( 2 l + 1 ) β + φ 2 ( 2 m ) ( β + λ ) π ( 2 l + 1 ) β ∥ f ∥ p , μ ∥ g ∥ q , ν .

Let l = 1 in (4.18), then we have

(4.19) ∫ 0 ∞ ∫ 0 ∞ log x y 2 m f ( x ) g ( y ) x 2 β − x β y β + y 2 β d x d y < π 3 β 2 m + 1 φ 2 ( 2 m ) λ π 3 β + φ 2 ( 2 m ) ( β + λ ) π 3 β ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where μ ( x ) = x p ( 1 − λ ) − 1 and ν ( y ) = y q ( 1 + λ − 2 β ) − 1 . Let m = 0 , λ = β and p = q = 2 in (4.19), we can obtain (1.5). In addition, let m = 1 , λ = β = 1 and p = q = 2 in (4.19), then we can arrive at (1.9). At last, letting λ = β 2 in (4.19), and using (2.21), we have

(4.20) ∫ 0 ∞ ∫ 0 ∞ log x y 2 m x 2 β − x β y β + y 2 β f ( x ) g ( y ) d x d y < π 3 β 2 m + 1 φ 2 ( 2 m ) π 6 + E m ∥ f ∥ p , μ ∥ g ∥ q , ν ,

where μ ( x ) = x p 1 − β 2 − 1 and ν ( y ) = y q 1 − 3 β 2 − 1 .

  1. Funding information: This research was supported by the incubation foundation of Zhejiang Institute of Mechanical and Electrical Engineering (A-0271-21-206).

  2. Conflict of interest: Authors state no conflict of interest.

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Received: 2020-07-28
Revised: 2021-02-26
Accepted: 2021-03-01
Published Online: 2021-07-09

© 2021 Minghui You et al., published by De Gruyter

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

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  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
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  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0034
Keywords for this article
Hilbert-type inequality; partial fraction expansion; Euler number; Bernoulli number
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  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
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  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
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  89. Global optimization and applications to a variational inequality problem
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  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
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  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
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  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
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  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
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