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On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions

  • Minghui You EMAIL logo , Xia Sun and Xiansheng Fan
Published/Copyright: July 27, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

In this work, by the introduction of a new kernel function composed of exponent functions with several parameters, and using the method of weight coefficient, Hermite-Hadamard’s inequality, and some other techniques of real analysis, a more accurate half-discrete Hilbert-type inequality including both the homogeneous and non-homogeneous cases is established. Furthermore, by introducing the Bernoulli number and the rational fraction expansion of tangent function, some special and interesting Hilbert-type inequalities and their equivalent hardy-type inequalities are presented at the end of the paper.

Keywords: Hilbert-type inequality; hyperbolic function; Hermite-Hadamard’s inequality; rational fraction expansion; Bernoulli number
MSC 2010: 26D15; 41A17

1 Introduction

Suppose p > 1 , f ( x ) , ν ( x ) are measurable functions defined on Ω , and f ( x ) , ν ( x ) > 0 . Define a function space as follows:

L p , ν ( Ω ) ≔ f : ‖ f ‖ p , ν ≔ ∫ Ω f p ( x ) ν ( x ) d x 1 p < ∞ .

Particularly, if ν ( x ) = 1 , then we have the following abbreviated forms: ‖ f ‖ p ≔ ‖ f ‖ p , ν and L p ( Ω ) ≔ L p , ν ( Ω ) .

Suppose p > 1 , a n , ν n > 0 , and a = { a n } n = 0 ∞ . Define a sequence space as follows:

l p , ν ≔ a : ‖ a ‖ p , ν ≔ ∑ n = 0 ∞ a n p ν n 1 p < ∞ .

Particularly, we abbreviate ‖ a ‖ p , ν to ‖ a ‖ p and l p , ν to l p for ν n = 1 .

Assuming that f ( x ) , g ( x ) are two nonnegative real-valued functions, and f , g ∈ L 2 ( R + ) , we have [1]

(1.1) ∫ R + ∫ R + f ( x ) g ( y ) x + y d x d y < π ‖ f ‖ 2 ‖ g ‖ 2 .

Similarly, let a = { a m } m = 1 ∞ ∈ l 2 and b = { b n } n = 1 ∞ ∈ l 2 . Then

(1.2) ∑ n = 1 ∞ ∑ m = 1 ∞ a m b n m + n < π ‖ a ‖ 2 ‖ b ‖ 2 .

The constant factor π in (1.1) and (1.2) is the best possible, and inequalities (1.1) and (1.2) are usually named as Hilbert inequality. By introducing a pair of conjugate parameters ( p , q ) , p > 1 , and 1 p + 1 q = 1 , (1.1) and (1.2) can be extended to more general forms:

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

(1.4) ∫ R + ∫ R + 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. In addition, we also have some classical inequalities similar to inequalities (1.3) and (1.4), such as [1]

(1.5) ∑ n = 1 ∞ ∑ m = 1 ∞ a m b n max { m , n } < p q ‖ a ‖ p ‖ b ‖ q ,

(1.6) ∫ R + ∫ R + f ( x ) g ( y ) max { x , y } d x d y < p q ‖ f ‖ p ‖ g ‖ q .

In general, such inequalities as (1.3), (1.4), (1.5), and (1.6) are known as Hilbert-type inequalities. Although these classical inequalities were proposed for more than 100 years, considerable attention has been paid to their parameter extensions, strengthened forms, and higher dimensional generalizations by researchers all over the world in recent years, and some new valuable Hilbert-type inequalities were established. The following inequality is a classical extension of (1.3) established by Krnić and Pečarić [2]: If p > 1 , and 1 p + 1 q = 1 , 0 < β 1 , β 2 ≤ 2 , β 1 + β 2 = β , μ m = m p ( 1 − β 1 ) − 1 , and ν n = n q ( 1 − β 2 ) − 1 , then

(1.7) ∑ n = 1 ∞ ∑ m = 1 ∞ a m b n ( m + n ) β < B ( β 1 , β 2 ) ‖ a ‖ p , μ ‖ b ‖ q , ν ,

where B ( u , v ) is the beta function. In addition, Yang [3] gave an extension of (1.4) as follows:

(1.8) ∫ R + ∫ R + 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 .

For other extensions of classical discrete Hilbert-type inequalities, we can refer to [4,5,6, 7,8,9, 10,11,12], and some extended results of integral version can be found in [10,11,13,14,15, 16,17]. Furthermore, by introducing various new kernel functions, special constants, and special functions, and considering discrete and integral forms, many Hilbert-type inequalities with new kernel functions were established in the past 20 years (see [18,19,20, 21,22,23, 24,25,26]). In addition to such types of Hilbert-type inequalities mentioned above, some new results on time scales were also established in recent years (see [27,28]). In what follows, we present the following two integral Hilbert-type inequalities, which involve the kernels related to hyperbolic functions, and are closely related to our research in the present paper, that is [25,26],

(1.9) ∫ R + ∫ R + csch ( x δ y ) f ( x ) g ( y ) d x d y < π 2 4 ‖ f ‖ 2 , μ ‖ g ‖ 2 , ν ,

(1.10) ∫ R + ∫ R + ( coth ( x y ) − 1 ) f ( x ) g ( y ) d x d y < π 2 12 ‖ f ‖ 2 , ν ‖ g ‖ 2 , ν ,

where δ ∈ { 1 , − 1 } , μ ( x ) = x 1 − 4 δ , ν ( y ) = y − 3 .

Besides the integral and discrete Hilbert-type inequalities, it should be pointed out that Hilbert-type inequalities sometimes appear in half-discrete form, such as the following two [29,30]:

(1.11) ∫ R + f ( x ) ∑ n = 1 ∞ a n x + n d x < π sin π p ‖ f ‖ p ‖ a ‖ q ,

(1.12) ∫ R + f ( x ) ∑ n = 1 ∞ log x n x − n a n d x < π sin π p 2 ‖ f ‖ p ‖ a ‖ q .

With regard to some other half-discrete inequalities, we refer to [31,32, 33,34].

In this paper, by using techniques of real analysis, particularly Hermite-Hadamard’s inequality, we consider the half-discrete forms of (1.9) and (1.10), then the following Hilbert-type inequalities involving some hyperbolic functions will be established:

(1.13) ∫ 0 a f ( x ) ∑ n = 0 ∞ csch ( x ( 2 n + 1 ) 2 m ) a n d x < 2 − 1 p ( 2 2 m − 1 ) B m π 2 m ‖ f ‖ p , μ 1 ‖ a ‖ q , ν ( a ≥ 1 ) ,

(1.14) ∫ a ∞ f ( x ) ∑ n = 0 ∞ coth 2 n + 1 x 2 m − 1 a n d x < 2 − 1 p B m π 2 m ‖ f ‖ p , μ 2 ‖ a ‖ q , ν ( 0 ≤ a ≤ 1 ) ,

(1.15) ∫ 0 a f ( x ) ∑ n = s ∞ cosh ( x n 2 m ) csch ( 3 x n 2 m ) a n d x < 2 m π 2 m 6 2 m ψ ( 2 m − 1 ) π 6 ‖ f ‖ p , μ 1 ‖ a ‖ q , ν ¯ ( a ≥ 1 ) ,

where m , s ∈ N + , B m is the Bernoulli number, μ 1 ( x ) = 1 x , μ 2 ( x ) = x 2 p − 1 , ν n = 1 2 n + 1 , and ν ¯ n = 1 n .

More generally, a new kernel function in more general form is constructed in Section 2, and then a Hilbert-type inequality including both the homogeneous and non-homogeneous cases is established. It will be shown that the newly obtained inequality is a unified extension of inequalities (1.13), (1.14), and (1.15), and some other special cases of the newly obtained inequality are presented in Section 4.

2 Definitions and lemmas

Definition 2.1

For t > 0 , define Γ -function as follows:

Γ ( t ) ≔ ∫ R + z t − 1 e − z d z .

In particular, we have Γ ( t ) = ( t − 1 ) ! for t ∈ N + .

Lemma 2.2

Let − λ 3 ≤ λ 2 ≤ λ 1 < λ 3 , and λ > 1 . Define

(2.1) k ( z ) ≔ e λ 1 z + e λ 2 z e λ 3 z − e − λ 3 z , z > 0 ,

and

(2.2) C ( λ 1 , λ 2 , λ 3 , λ ) ≔ ∑ j = 0 ∞ 1 ( 2 λ 3 j − λ 1 + λ 3 ) λ + 1 ( 2 λ 3 j − λ 2 + λ 3 ) λ .

Then

(2.3) ∫ R + k ( z ) z λ − 1 d z = Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) .

Proof

It can be easy to show that λ 3 > 0 and

(2.4) k ( z ) = ∑ j = 0 ∞ ( e ( − 2 λ 3 j + λ 1 − λ 3 ) z + e ( − 2 λ 3 j + λ 2 − λ 3 ) z ) .

By Lebesgue term-by-term integration theorem, we obtain

(2.5) ∫ R + k ( z ) z λ − 1 d z = ∑ j = 0 ∞ ∫ R + e ( − 2 λ 3 j + λ 1 − λ 3 ) z z λ − 1 d z + ∫ R + e ( − 2 λ 3 j + λ 2 − λ 3 ) z z λ − 1 d z .

Setting ( 2 λ 3 j − λ 1 + λ 3 ) z = u , we obtain

(2.6) ∫ R + e ( − 2 λ 3 j + λ 1 − λ 3 ) z z λ − 1 d z = ∫ R + e − u u λ − 1 d u ( 2 λ 3 j − λ 1 + λ 3 ) λ = Γ ( λ ) ( 2 λ 3 j − λ 1 + λ 3 ) λ .

Similarly, setting ( 2 λ 3 j − λ 2 + λ 3 ) z = u , we obtain

(2.7) ∫ R + e ( − 2 λ 3 j + λ 2 − λ 3 ) z z λ − 1 d z = ∫ R + e − u u λ − 1 d u ( 2 λ 3 j − λ 2 + λ 3 ) λ = Γ ( λ ) ( 2 λ 3 j − λ 2 + λ 3 ) λ .

Plugging (2.6) and (2.7) back into (2.5), and using (2.2), we arrive at (2.3). Lemma 2.2 is proved.□

Lemma 2.3

Suppose that β ≠ 0 , b ≥ 1 2 , and − λ 3 ≤ λ 2 ≤ λ 1 < λ 3 . Let 0 ≤ a ≤ 1 , and Ω = ( a , ∞ ) when β < 0 . Let a ≥ 1 , and Ω = ( 0 , a ) when β > 0 . Assume that λ > 1 , 0 < γ ≤ 1 , and 0 < λ γ ≤ 1 . Define

(2.8) K ( x , y ) ≔ e λ 1 x β ( y + b ) γ + e λ 2 x β ( y + b ) γ e λ 3 x β ( y + b ) γ − e − λ 3 x β ( y + b ) γ , x > 0 , y ≥ 0 .

Then

(2.9) ω ( n ) ≔ ∫ Ω K ( x , n ) x β λ − 1 d x ≤ Γ ( λ ) ∣ β ∣ ( n + b ) γ λ C ( λ 1 , λ 2 , λ 3 , λ ) , n ∈ N + ,

(2.10) ϖ ( x ) ≔ ∑ n = 0 ∞ K ( x , n ) ( n + b ) γ λ − 1 < Γ ( λ ) γ x β λ C ( λ 1 , λ 2 , λ 3 , λ ) , x > 0 .

Proof

Setting x β ( n + b ) γ = z , and using (2.3), we obtain

(2.11) ∫ Ω K ( x , n ) x β λ − 1 d x = 1 ∣ β ∣ ( n + b ) γ λ ∫ R + k ( z ) z λ − 1 d z = Γ ( λ ) ∣ β ∣ ( n + b ) γ λ C ( λ 1 , λ 2 , λ 3 , λ ) .

It follows therefore that (2.9) holds true.

Since − λ 3 ≤ λ 2 ≤ λ 1 < λ 3 , it follows from (2.1) that

(2.12) d k d z = [ ( λ 1 − λ 3 ) e ( λ 1 + λ 3 ) z − ( λ 1 + λ 3 ) e ( λ 1 − λ 3 ) z + ( λ 2 − λ 3 ) e ( λ 2 + λ 3 ) z − ( λ 2 + λ 3 ) e ( λ 2 − λ 3 ) z ] ( e λ 3 z − e − λ 3 z ) − 2 < 0 .

In view of that

6 λ 3 2 − 2 λ 2 2 > 2 ( λ 3 − λ 2 ) ( λ 3 + λ 2 ) ≥ 0

and

6 λ 3 2 − 2 λ 1 2 > 2 ( λ 3 − λ 1 ) ( λ 3 + λ 1 ) ≥ 0 ,

we obtain

(2.13) d 2 k d z 2 = [ ( λ 1 − λ 3 ) 2 e ( λ 1 + 2 λ 3 ) z + ( λ 1 + λ 3 ) 2 e ( λ 1 − 2 λ 3 ) z + ( λ 2 − λ 3 ) 2 e ( λ 2 + 2 λ 3 ) z + ( λ 2 + λ 3 ) 2 e ( λ 2 − 2 λ 3 ) z + ( 6 λ 3 2 − 2 λ 1 2 ) e λ 1 z + ( 6 λ 3 2 − 2 λ 2 2 ) e λ 2 z ] ( e λ 3 z − e − λ 3 z ) − 3 > 0 .

For arbitrary x > 0 , let z ≔ x β ( y + b ) γ . In view of 0 < γ ≤ 1 , we obtain d z d y > 0 and d 2 z d y 2 ≤ 0 . It follows therefore that

(2.14) d K d y = d k d z d z d y < 0

and

(2.15) d 2 K d y 2 = d 2 k d z 2 d z d y 2 + d k d z d 2 z d y 2 > 0 .

Let

H ( y ) ≔ K ( x , y ) ( y + b ) γ λ − 1 ≔ K ( x , y ) h ( y ) .

Observing that 0 < γ λ ≤ 1 , and using (2.14) and (2.15), we have

d H d y = d K d y h ( y ) + d h d y K ( x , y ) < 0

and

d 2 H d y 2 = d 2 K d y 2 h ( y ) + 2 d K d y d h d y + d 2 h d y 2 K ( x , y ) > 0 .

Hence, by Hermite-Hadamard’s inequality (see [35,36]), we obtain

(2.16) ϖ ( x ) = ∑ n = 0 ∞ H ( n ) < ∑ n = 0 ∞ ∫ n − 1 2 n + 1 2 H ( y ) d y = ∫ − 1 2 ∞ H ( y ) d y ≤ ∫ − b ∞ H ( y ) d y .

Setting x β ( y + b ) γ = z , and using (2.3), we obtain

(2.17) ∫ − b ∞ H ( y ) d y = 1 γ x β λ ∫ R + k ( z ) z λ − 1 d z = Γ ( λ ) γ x β λ C ( λ 1 , λ 2 , λ 3 , λ ) .

Applying (2.17) to (2.16), we obtain (2.10), and the proof of Lemma 2.3 is completed.□

Lemma 2.4

Suppose that β ≠ 0 , b ≥ 1 2 , and − λ 3 ≤ λ 2 ≤ λ 1 < λ 3 . Let 0 ≤ a ≤ 1 , and Ω = ( a , ∞ ) when β < 0 . Let a ≥ 1 , and Ω = ( 0 , a ) when β > 0 . Assume that λ > 1 , 0 < γ ≤ 1 , and 0 < λ γ ≤ 1 . Let p > 1 , 1 p + 1 q = 1 , μ ( x ) = x p ( 1 − β λ ) − 1 , and ν n = ( n + b ) q ( 1 − γ λ ) − 1 . K ( x , y ) is defined via Lemma 2.3. For an arbitrary positive integer l , which is sufficiently large, set

(2.18) a ˜ ≔ { a ˜ n } n = 0 ∞ ≔ ( n + b ) γ λ − 1 − γ q l n = 0 ∞ ,

(2.19) f ˜ ( x ) ≔ x β λ − 1 + β p l , x ∈ E , 0 , x ∈ Ω ⧹ E ,

where E = { x : x > 0 , x sgn β < 1 } . Then

(2.20) J ˜ ≔ ∑ n = 0 ∞ a ˜ n ∫ E K ( x , n ) f ˜ ( x ) d x = ∫ E f ˜ ( x ) ∑ n = 0 ∞ K ( x , n ) a ˜ n d x > l ∣ β γ ∣ ∫ 1 ∞ k ( z ) z λ − 1 q l − 1 d z + b − γ l ∫ 0 1 k ( z ) z λ + 1 p l − 1 d z .

Proof

Observing that 0 < λ γ ≤ 1 , and using (2.14), we obtain

(2.21) J ˜ > ∫ E x β λ − 1 + β p l ∫ 0 ∞ K ( x , y ) ( y + b ) γ λ − 1 − γ q l d y d x .

Setting x β ( y + b ) γ = z , we obtain

(2.22) J ˜ > 1 γ ∫ E x β l − 1 ∫ x β b γ ∞ k ( z ) z λ − 1 q l − 1 d z d x = 1 γ ∫ E x β l − 1 ∫ 1 ∞ k ( z ) z λ − 1 q l − 1 d z d x + 1 γ ∫ E x β l − 1 ∫ x β b γ 1 k ( z ) z λ − 1 q l − 1 d z d x = l ∣ β γ ∣ ∫ 1 ∞ k ( z ) z λ − 1 q l − 1 d z + 1 γ ∫ E x β l − 1 ∫ x β b γ 1 k ( z ) z λ − 1 q l − 1 d z d x .

For β > 0 , by the use of Fubini’s theorem, we obtain

(2.23) ∫ E x β l − 1 ∫ x β b γ 1 k ( z ) z λ − 1 q l − 1 d z d x = ∫ 0 1 k ( z ) z λ − 1 q l − 1 ∫ 0 z 1 β b − γ β x β l − 1 d x d z = l b − γ l β ∫ 0 1 k ( z ) z λ + 1 p l − 1 d z .

Applying (2.23) to (2.22), it follows (2.20). Similarly, for β < 0 , by the use of Fubini’s theorem again, we have

(2.24) ∫ E x β l − 1 ∫ x β b γ 1 k ( z ) z λ − 1 q l − 1 d z d x = ∫ 0 1 k ( z ) z λ − 1 q l − 1 ∫ z 1 β b − γ β ∞ x β l − 1 d x d z = l b − γ l ∣ β ∣ ∫ 0 1 k ( z ) z λ + 1 p l − 1 d z .

Plug (2.24) back into (2.22), then (2.20) follows obviously. Lemma 2.4 is proved.□

Lemma 2.5

Let − 1 < u < 1 , ψ ( x ) = tan x , and m ∈ N + . Then

(2.25) ψ ( 2 m − 1 ) u π 2 = 2 2 m ( 2 m − 1 ) ! π 2 m ∑ j = 0 ∞ 1 ( 2 j + 1 − u ) 2 m + 1 ( 2 j + 1 + u ) 2 m .

Proof

Observing that ψ ( x ) = tan x can be expressed in the form of rational fraction expansion as follows [37]:

(2.26) ψ ( x ) = tan x = 2 ∑ j = 0 ∞ 1 ( 2 j + 1 ) π − 2 x − 1 ( 2 j + 1 ) π + 2 x .

Finding the ( 2 m − 1 ) th derivative of ψ ( x ) , we have

(2.27) ψ ( 2 m − 1 ) ( x ) = 2 2 m ( 2 m − 1 ) ! ∑ j = 0 ∞ 1 [ ( 2 j + 1 ) π − 2 x ] 2 m + 1 [ ( 2 j + 1 ) π + 2 x ] 2 m .

Setting x = u π 2 in (2.27), we arrive at (2.25). Lemma 2.5 is proved.□

3 Main results

Theorem 3.1

Suppose that β ≠ 0 , b ≥ 1 2 , and − λ 3 ≤ λ 2 ≤ λ 1 < λ 3 . Let 0 ≤ a ≤ 1 , and Ω = ( a , ∞ ) when β < 0 . Let a ≥ 1 , and Ω = ( 0 , a ) when β > 0 . Assume that λ > 1 , 0 < γ ≤ 1 , and 0 < λ γ ≤ 1 . Let p > 1 , 1 p + 1 q = 1 , μ ( x ) = x p ( 1 − β λ ) − 1 , and ν n = ( n + b ) q ( 1 − γ λ ) − 1 . Let f ( x ) , a n ≥ 0 with f ( x ) ∈ L p , μ ( Ω ) , and a = { a n } n = 0 ∞ ∈ l q , ν . C ( λ 1 , λ 2 , λ 3 , λ ) and K ( x , y ) are defined via Lemmas 2.2  and  2.3, respectively. Then the following three inequalities are equivalent:

(3.1) J 1 ≔ ∑ n = 0 ∞ ( n + b ) p γ λ − 1 ∫ Ω K ( x , n ) f ( x ) d x p < ∣ β ∣ − 1 q ∣ γ ∣ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) p ‖ f ‖ p , μ p ,

(3.2) J 2 ≔ ∫ Ω x q β λ − 1 ∑ n = 0 ∞ K ( x , n ) a n q d x < ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) q ‖ a ‖ q , ν q ,

(3.3) J ≔ ∑ n = 0 ∞ a n ∫ Ω K ( x , n ) f ( x ) d x = ∫ Ω f ( x ) ∑ n = 0 ∞ K ( x , n ) a n d x < ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) ‖ f ‖ p , μ ‖ a ‖ q , ν ,

where the constant ∣ β ∣ − 1 p γ − 1 q Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) in (3.1), (3.2), and (3.3) is the best possible.

Proof

By Hölder’s inequality, and using (2.9), we obtain

(3.4) ∫ Ω K ( x , n ) f ( x ) d x p = ∫ Ω K ( x , n ) x 1 − β λ q f ( x ) x β λ − 1 q d x p ≤ ∫ Ω K ( x , n ) x p ( 1 − β λ ) q f p ( x ) d x ∫ Ω K ( x , n ) x β λ − 1 d x p − 1 = [ ω ( n ) ] p − 1 ∫ Ω K ( x , n ) x p ( 1 − β λ ) q f p ( x ) d x ≤ Γ ( λ ) ∣ β ∣ ( n + b ) γ λ C ( λ 1 , λ 2 , λ 3 , λ ) p − 1 ∫ Ω K ( x , n ) x p ( 1 − β λ ) q f p ( x ) d x .

Plugging (3.4) back into the left hand side of (3.1), and using Lebesgue term-by-term integration theorem as well as inequality (2.10), we have

J 1 ≤ Γ ( λ ) ∣ β ∣ C ( λ 1 , λ 2 , λ 3 , λ ) p − 1 ∫ Ω f p ( x ) x p ( 1 − β λ ) q ∑ n = 0 ∞ K ( x , n ) ( n + b ) γ λ − 1 d x < ∣ β ∣ − 1 q ∣ γ ∣ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) p ‖ f ‖ p , μ p .

The proof of inequality (3.1) is completed. Similarly, by the use of H o ¨ lder’s inequality again and (2.10), we have

(3.5) ∑ n = 0 ∞ K ( x , n ) a n q = ∑ n = 0 ∞ K ( x , n ) ( n + b ) γ λ − 1 p ( n + b ) 1 − γ λ p a n q ≤ [ ϖ ( x ) ] q − 1 ∑ n = 0 ∞ K ( x , n ) ( n + b ) q ( 1 − γ λ ) p a n q < Γ ( λ ) γ x β λ C ( λ 1 , λ 2 , λ 3 , λ ) q − 1 ∑ n = 0 ∞ K ( x , n ) ( n + b ) q ( 1 − γ λ ) p a n q .

It follows from Lebesgue term-by-term integration theorem and (2.9) that

J 2 < Γ ( λ ) ∣ β ∣ C ( λ 1 , λ 2 , λ 3 , λ ) q − 1 ∑ n = 0 ∞ ( n + b ) q ( 1 − γ λ ) p a n q ∫ Ω K ( x , n ) x β λ − 1 d x < ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) q ‖ a ‖ q , ν q .

Inequality (3.2) is proved. Additionally, we will prove (3.3) by (3.1). In fact, we can first obtain two representations of J by Lebesgue term-by-term integration theorem, and then by the use of H o ¨ lder’s inequality, we obtain

(3.6) J = ∑ n = 0 ∞ ( n + b ) γ λ − 1 p ∫ Ω K ( x , n ) f ( x ) d x a n ( n + b ) − γ λ + 1 p ≤ J 1 1 p ∑ n = 0 ∞ a n q ( n + b ) q ( 1 − γ λ ) − 1 1 q = J 1 1 p ‖ a ‖ q , ν .

Applying (3.1) to (3.6), we have (3.3). On the contrary, we assume that (3.3) holds true, and let b = { b n } n = 0 ∞ , where

b n ≔ ( n + b ) p γ λ − 1 ∫ Ω K ( x , n ) f ( x ) d x p − 1 .

By the use of (3.3), it follows that

(3.7) J 1 = ∑ n = 0 ∞ b n ∫ Ω K ( x , n ) f ( x ) d x < ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) ‖ f ‖ p , μ ‖ b ‖ q , ν = ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) ‖ f ‖ p , μ J 1 1 q .

By (3.7), inequality (3.1) follows naturally. Hence, inequalities (3.1) and (3.3) are equivalent. In order to prove the equivalence of inequalities (3.1), (3.2), and (3.3), it suffices to prove that (3.2) and (3.3) are equivalent. In fact, if (3.2) is assumed to be true, then

(3.8) J = ∫ Ω x − β λ + 1 q f ( x ) x β γ − 1 q ∑ n = 0 ∞ K ( x , n ) a n d x ≤ ‖ f ‖ p , μ J 2 1 q .

Applying (3.2) to (3.8), we have (3.3). Conversely, assume (3.3) holds true, and let

g ( x ) ≔ x q β λ − 1 ∑ n = 0 ∞ K ( x , n ) a n q − 1 .

Then

(3.9) J 2 = ∫ Ω g ( x ) ∑ n = 0 ∞ K ( x , n ) a n d x < ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) ‖ g ‖ p , μ ‖ a ‖ q , ν = ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) ‖ a ‖ q , ν J 2 1 p .

Therefore, inequality (3.2) holds true, and the equivalence of inequalities (3.1), (3.2), and (3.3) is proved. Finally, it will be proved that the constant ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) in (3.1), (3.2), and (3.3) is optimal. Assuming that the constant factor ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) in (3.3) is not optimal, there must be a real number c satisfying

(3.10) 0 < c < ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) ,

so that (3.3) still holds if ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) is replaced by c , that is,

(3.11) ∑ n = 0 ∞ a n ∫ Ω K ( x , n ) f ( x ) d x = ∫ Ω f ( x ) ∑ n = 0 ∞ K ( x , n ) a n d x < c ‖ f ‖ p , μ ‖ a ‖ q , ν .

Replace f and a n in (3.11) with f ˜ and a ˜ n defined in Lemma 2.4, respectively. It implies that

(3.12) ∫ E f ˜ ( x ) ∑ n = 0 ∞ K ( x , n ) a ˜ n d x < c ‖ f ˜ ‖ p , μ ‖ a ˜ ‖ q , ν = c ∫ E x β l − 1 d x 1 p ∑ n = 0 ∞ ( n + b ) − γ l − 1 1 q = c l ∣ β ∣ 1 p b − γ l − 1 + ∑ n = 1 ∞ ( n + b ) − γ l − 1 1 q < c l ∣ β ∣ 1 p b − γ l − 1 + ∫ 0 ∞ ( y + b ) − γ l − 1 d y 1 q = c l ∣ β ∣ 1 p b − γ l − 1 + l b − γ l γ 1 q .

Combining (2.20) and (3.12), we have

(3.13) ∫ 1 ∞ k ( z ) z λ − 1 q l − 1 d z + b − γ l ∫ 0 1 k ( z ) z λ + 1 p l − 1 d z < c γ ∣ β ∣ 1 q b − γ l − 1 l + b − γ l γ 1 q .

Letting l → + ∞ in (3.13), and using (2.3), we have

(3.14) c ≥ ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) .

Inequalities (3.10) and (3.14) are apparently contradictory, and therefore ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) in (3.3) is the best possible. It can also be proved ∣ β ∣ − 1 q γ − 1 p Γ ( λ ) C ( λ 1 , λ 2 , λ 3 , λ ) in (3.1) and (3.2) is the best possible from the equivalence of (3.1), (3.2), and (3.3). Theorem 3.1 is proved.□

4 Some corollaries

Let λ 1 = λ 2 = 0 , λ 3 = ρ ( ρ > 0 ) , and λ = 2 m ( m ∈ N + ) in Theorem 3.1. In view of the equation [37]:

(4.1) ∑ j = 0 ∞ 2 ( 2 j + 1 ) 2 m = B m ( 2 m ) ! ( 2 2 m − 1 ) π 2 m , m ∈ N + ,

where B m is the Bernoulli number, B 1 = 1 6 , B 2 = 1 30 , B 3 = 1 42 , … , we obtain the following corollary.

Corollary 4.1

Suppose that ρ > 0 , β ≠ 0 , b ≥ 1 2 , and 0 < 2 m γ ≤ 1 ( m ∈ N + ) . Let 0 ≤ a ≤ 1 , and Ω = ( a , ∞ ) when β < 0 . Let a ≥ 1 , and Ω = ( 0 , a ) when β > 0 . Assume that p > 1 , 1 p + 1 q = 1 , μ ( x ) = x p ( 1 − 2 m β ) − 1 , and ν n = ( n + b ) q ( 1 − 2 m γ ) − 1 . Let f ( x ) , a n ≥ 0 with f ( x ) ∈ L p , μ ( Ω ) , and a = { a n } n = 0 ∞ ∈ l q , ν . Then the following three inequalities are equivalent:

(4.2) ∑ n = 0 ∞ ( n + b ) 2 m p γ − 1 ∫ Ω csch ( ρ x β ( n + b ) γ ) f ( x ) d x p < ∣ β ∣ − 1 q γ − 1 p ( 2 2 m − 1 ) B m 2 m π ρ 2 m p ‖ f ‖ p , μ p ,

(4.3) ∫ Ω x 2 m q β − 1 ∑ n = 0 ∞ csch ( ρ x β ( n + b ) γ ) a n q d x < ∣ β ∣ − 1 q γ − 1 p ( 2 2 m − 1 ) B m 2 m π ρ 2 m q ‖ a ‖ q , ν q ,

(4.4) ∫ Ω f ( x ) ∑ n = 0 ∞ csch ( ρ x β ( n + b ) γ ) a n d x < ∣ β ∣ − 1 q γ − 1 p ( 2 2 m − 1 ) B m 2 m π ρ 2 m ‖ f ‖ p , μ ‖ a ‖ q , ν ,

where the constant ∣ β ∣ − 1 q γ − 1 p ( 2 2 m − 1 ) B m 2 m π ρ 2 m in (4.2), (4.3), and (4.4) is the best possible.

Setting β = γ ( 0 < 2 m γ ≤ 1 ) , b = s ( s ∈ N + ) in Corollary 4.1, we obtain the following inequalities with a non-homogeneous kernel ( a ≥ 1 ):

(4.5) ∑ n = s ∞ n 2 m p γ − 1 ∫ 0 a csch ( ρ ( x n ) γ ) f ( x ) d x p < ( 2 2 m − 1 ) B m 2 m γ π ρ 2 m p ‖ f ‖ p , μ p ,

(4.6) ∫ 0 a x 2 m q γ − 1 ∑ n = s ∞ csch ( ρ ( x n ) γ ) a n q d x < ( 2 2 m − 1 ) B m 2 m γ π ρ 2 m q ‖ a ‖ q , ν q ,

(4.7) ∫ 0 a f ( x ) ∑ n = s ∞ csch ( ρ ( x n ) γ ) a n d x < ( 2 2 m − 1 ) B m 2 m γ π ρ 2 m ‖ f ‖ p , μ ‖ a ‖ q , ν ,

where μ ( x ) = x p ( 1 − 2 m γ ) − 1 and ν n = n q ( 1 − 2 m γ ) − 1 .

Setting β = − γ ( 0 < 2 m γ ≤ 1 ) , b = s ( s ∈ N + ) in Corollary 4.1, we obtain the following inequalities with a homogeneous kernel ( 0 ≤ a ≤ 1 ):

(4.8) ∑ n = s ∞ n 2 m p γ − 1 ∫ a ∞ csch ρ n x γ f ( x ) d x p < ( 2 2 m − 1 ) B m 2 m γ π ρ 2 m p ‖ f ‖ p , μ p ,

(4.9) ∫ a ∞ x − 2 m q γ − 1 ∑ n = s ∞ csch ρ n x γ a n q d x < ( 2 2 m − 1 ) B m 2 m γ π ρ 2 m q ‖ a ‖ q , ν q ,

(4.10) ∫ a ∞ f ( x ) ∑ n = s ∞ csch ρ n x γ a n d x < ( 2 2 m − 1 ) B m 2 m γ π ρ 2 m ‖ f ‖ p , μ ‖ a ‖ q , ν ,

where μ ( x ) = x p ( 1 + 2 m γ ) − 1 and ν n = n q ( 1 − 2 m γ ) − 1 .

Setting β = γ ( 0 < 2 m γ ≤ 1 ) , b = 1 2 , and replacing ρ with ρ 2 γ in Corollary 4.1, it follows that

(4.11) ∑ n = 0 ∞ ( 2 n + 1 ) 2 m p γ − 1 ∫ 0 a csch ( ρ ( x ( 2 n + 1 ) ) γ ) f ( x ) d x p < 2 − 1 ∕ p ( 2 2 m − 1 ) B m 2 m γ π ρ 2 m p ‖ f ‖ p , μ p ,

(4.12) ∫ 0 a x 2 m q γ − 1 ∑ n = 0 ∞ csch ( ρ ( x ( 2 n + 1 ) ) γ ) a n q d x < 2 − 1 ∕ p ( 2 2 m − 1 ) B m 2 m γ π ρ 2 m q ‖ a ‖ q , ν q ,

(4.13) ∫ 0 a f ( x ) ∑ n = 0 ∞ csch ( ρ ( x ( 2 n + 1 ) ) γ ) a n d x < 2 − 1 ∕ p ( 2 2 m − 1 ) B m 2 m γ π ρ 2 m ‖ f ‖ p , μ ‖ a ‖ q , ν ,

where a ≥ 1 , μ ( x ) = x p ( 1 − 2 m γ ) − 1 , and ν n = ( 2 n + 1 ) q ( 1 − 2 m γ ) − 1 . Letting γ = 1 2 m , ρ = 1 in (4.13), we obtain inequality (1.13).

Setting β = − γ ( 0 < 2 m γ ≤ 1 ) , b = 1 2 , and replacing ρ with ρ 2 γ in Corollary 4.1, we can obtain the homogeneous forms corresponding to (4.11), (4.12), and (4.13) with the same constant factors, such as

(4.14) ∫ a ∞ f ( x ) ∑ n = 0 ∞ csch ρ 2 n + 1 x γ a n d x < 2 − 1 ∕ p ( 2 2 m − 1 ) B m 2 m γ π ρ 2 m ‖ f ‖ p , μ ‖ a ‖ q , ν ,

where 0 < a ≤ 1 , μ ( x ) = x p ( 1 + 2 m γ ) − 1 , and ν n = ( 2 n + 1 ) q ( 1 − 2 m γ ) − 1 .

Let λ 1 = 0 , λ 2 = − 2 ρ , λ 3 = 2 ρ ( ρ > 0 ) , and λ = 2 m ( m ∈ N + ) in Theorem 3.1. It follows from (2.1) that

k ( t ) = 1 + e − 2 ρ t e 2 ρ t − e − 2 ρ t = 1 2 ( coth ( ρ t ) − 1 ) .

Additionally, by the equation [37]:

(4.15) ∑ j = 0 ∞ 2 ( j + 1 ) 2 m = B m ( 2 m ) ! ( 2 π ) 2 m , m ∈ N + ,

and (4.1), we have

C ( λ 1 , λ 2 , λ 3 , λ ) = 1 ρ 2 m ∑ j = 0 ∞ 1 ( 4 j + 4 ) 2 m + 1 ( 4 j + 2 ) 2 m = B m 2 ( 2 m ) ! π ρ 2 m .

Therefore, we obtain Corollary 4.2.

Corollary 4.2

Suppose that ρ > 0 , β ≠ 0 , b ≥ 1 2 , and 0 < 2 m γ ≤ 1 ( m ∈ N + ) . Let 0 ≤ a ≤ 1 , and Ω = ( a , ∞ ) when β < 0 . Let a ≥ 1 , and Ω = ( 0 , a ) when β > 0 . Let p > 1 , 1 p + 1 q = 1 , μ ( x ) = x p ( 1 − 2 m β ) − 1 , and ν n = ( n + b ) q ( 1 − 2 m γ ) − 1 . Suppose that f ( x ) , a n ≥ 0 with f ( x ) ∈ L p , μ ( Ω ) , and a = { a n } n = 0 ∞ ∈ l q , ν . Then the following three inequalities are equivalent:

(4.16) ∑ n = 0 ∞ ( n + b ) 2 m p γ − 1 ∫ Ω ( coth ( ρ x β ( n + b ) γ ) − 1 ) f ( x ) d x p < ∣ β ∣ − 1 q γ − 1 p B m 2 m π ρ 2 m p ‖ f ‖ p , μ p ,

(4.17) ∫ Ω x 2 m q β − 1 ∑ n = 0 ∞ ( coth ( ρ x β ( n + b ) γ ) − 1 ) a n q d x < ∣ β ∣ − 1 q γ − 1 p B m 2 m π ρ 2 m q ‖ a ‖ q , ν q ,

(4.18) ∫ Ω f ( x ) ∑ n = 0 ∞ ( coth ( ρ x β ( n + b ) γ ) − 1 ) a n d x < ∣ β ∣ − 1 q γ − 1 p B m 2 m π ρ 2 m ‖ f ‖ p , μ ‖ a ‖ q , ν ,

where the constant ∣ β ∣ − 1 q γ − 1 p B m 2 m π ρ 2 m in (4.16), (4.17), and (4.18) is the best possible.

Remark 4.3

Letting λ 1 = λ 2 = − λ 3 = − ρ ( ρ > 0 ) , and λ = 2 m ( m ∈ N + ) in Theorem 3.1, we can also obtain Corollary 4.2. In fact, by (2.2) and (4.15), we obtain

C ( λ 1 , λ 2 , λ 3 , λ ) = 1 2 ρ 2 m ∑ j = 0 ∞ 2 ( j + 1 ) 2 m = B m ( 2 m ) ! π ρ 2 m .

Therefore, Theorem 3.1 reduces to Corollary 4.2 obviously.

Setting β = γ , b = 1 2 , and replacing ρ with ρ 2 γ in Corollary 4.2, it follows that

(4.19) ∑ n = 0 ∞ ( 2 n + 1 ) 2 m p γ − 1 ∫ 0 a ( coth ( ρ ( x ( 2 n + 1 ) ) γ ) − 1 ) f ( x ) d x p < 2 − 1 ∕ p − 1 B m m γ π ρ 2 m p ‖ f ‖ p , μ p ,

(4.20) ∫ 0 a x 2 m q γ − 1 ∑ n = 0 ∞ ( coth ( ρ ( x ( 2 n + 1 ) ) γ ) − 1 ) a n q d x < 2 − 1 ∕ p − 1 B m m γ π ρ 2 m q ‖ a ‖ q , ν q ,

(4.21) ∫ 0 a f ( x ) ∑ n = 0 ∞ [ coth ( ρ ( x ( 2 n + 1 ) ) γ ) − 1 ] a n d x < 2 − 1 ∕ p − 1 B m m γ π ρ 2 m γ ‖ f ‖ p , μ ‖ a ‖ q , ν ,

where a ≥ 1 , μ ( x ) = x p ( 1 − 2 m γ ) − 1 , and ν n = ( 2 n + 1 ) q ( 1 − 2 m γ ) − 1 .

Setting β = − γ , b = 1 2 , and replacing ρ with ρ 2 γ in Corollary 4.2, we obtain the homogeneous forms corresponding to (4.19), (4.20), and (4.21) with the same constant factors, such as

(4.22) ∫ a ∞ f ( x ) ∑ n = 0 ∞ coth ρ 2 n + 1 x γ − 1 a n d x < 2 − 1 ∕ p − 1 B m m γ π ρ 2 m γ ‖ f ‖ p , μ ‖ a ‖ q , ν ,

where 0 ≤ a ≤ 1 , μ ( x ) = x p ( 1 + 2 m γ ) − 1 , and ν n = ( 2 n + 1 ) q ( 1 − 2 m γ ) − 1 . If we set γ = 1 2 m ρ = 1 in (4.22), then (4.22) reduces to (1.14).

Setting β = γ = 1 2 m , b = s ( m , s ∈ N + ) , and β = − 1 2 m , γ = 1 2 m , b = s ( m , s ∈ N + ) in Corollary 4.2, respectively, we obtain

(4.23) ∫ 0 a f ( x ) ∑ n = s ∞ [ coth ( ρ x n 2 m ) − 1 ] a n d x < B m π ρ 2 m ‖ f ‖ p , μ 1 ‖ a ‖ q , ν ( a ≥ 1 ) ,

(4.24) ∫ a ∞ f ( x ) ∑ n = s ∞ coth ρ n x 2 m − 1 a n d x < B m π ρ 2 m ‖ f ‖ p , μ 2 ‖ a ‖ q , ν ( 0 ≤ a ≤ 1 ) ,

where μ 1 ( x ) = 1 x , μ 2 ( x ) = x 2 p − 1 , ν n = 1 n .

Let λ 2 = − λ 1 , and λ = 2 m ( m ∈ N + ) in Theorem 3.1. By Lemma 2.5, we obtain the following corollary.

Corollary 4.4

Suppose that β ≠ 0 , b ≥ 1 2 , − λ 3 < 0 ≤ λ 1 < λ 3 , and 0 < 2 m γ ≤ 1 ( m ∈ N + ) . Let 0 ≤ a ≤ 1 , and Ω = ( a , ∞ ) when β < 0 . Let a ≥ 1 , and Ω = ( 0 , a ) when β > 0 . Let p > 1 , 1 p + 1 q = 1 , ψ ( x ) = tan x , μ ( x ) = x p ( 1 − 2 m β ) − 1 , and ν n = ( n + b ) q ( 1 − 2 m γ ) − 1 . Suppose that f ( x ) , a n ≥ 0 with f ( x ) ∈ L p , μ ( Ω ) , and a = { a n } n = 0 ∞ ∈ l q , ν . Then the following three inequalities are equivalent:

(4.25) ∑ n = 0 ∞ ( n + b ) 2 m p γ − 1 ∫ Ω cosh ( λ 1 x β ( n + b ) γ ) csch ( λ 3 x β ( n + b ) γ ) f ( x ) d x p < ∣ β ∣ − 1 q γ − 1 p π 2 λ 3 2 m ψ ( 2 m − 1 ) λ 1 π 2 λ 3 p ‖ f ‖ p , μ p ,

(4.26) ∫ Ω x 2 m q β − 1 ∑ n = 0 ∞ cosh ( λ 1 x β ( n + b ) γ ) csch ( λ 3 x β ( n + b ) γ ) a n q d x < ∣ β ∣ − 1 q γ − 1 p π 2 λ 3 2 m ψ ( 2 m − 1 ) λ 1 π 2 λ 3 q ‖ a ‖ q , ν q ,

(4.27) ∫ Ω f ( x ) ∑ n = 0 ∞ cosh ( λ 1 x β ( n + b ) γ ) csch ( λ 3 x β ( n + b ) γ ) a n d x < ∣ β ∣ − 1 q γ − 1 p π 2 λ 3 2 m ψ ( 2 m − 1 ) λ 1 π 2 λ 3 ‖ f ‖ p , μ ‖ a ‖ q , ν ,

where the constant ∣ β ∣ − 1 q γ − 1 p π 2 λ 3 2 m ψ ( 2 m − 1 ) λ 1 π 2 λ 3 in (4.25), (4.26), and (4.27) is the best possible.

In Corollary 4.4, let β = γ , b = 1 2 , and replace λ 1 and λ 3 with λ 1 2 γ and λ 3 2 γ , respectively, then we obtain

(4.28) ∑ n = 0 ∞ ( 2 n + 1 ) 2 m p γ − 1 ∫ 0 a cosh ( λ 1 ( x ( 2 n + 1 ) ) γ ) csch ( λ 3 ( x ( 2 n + 1 ) ) γ ) f ( x ) d x p < 1 γ 2 1 ∕ p π 2 λ 3 2 m ψ ( 2 m − 1 ) λ 1 π 2 λ 3 p ‖ f ‖ p , μ p ,

(4.29) ∫ 0 a x 2 m q γ − 1 ∑ n = 0 ∞ cosh ( λ 1 ( x ( 2 n + 1 ) ) γ ) csch ( λ 3 ( x ( 2 n + 1 ) ) γ ) a n q d x < 1 γ 2 1 ∕ p π 2 λ 3 2 m ψ ( 2 m − 1 ) λ 1 π 2 λ 3 q ‖ a ‖ q , ν q ,

(4.30) ∫ 0 a f ( x ) ∑ n = 0 ∞ cosh ( λ 1 ( x ( 2 n + 1 ) ) γ ) csch ( λ 3 ( x ( 2 n + 1 ) ) γ ) a n d x < 1 γ 2 1 ∕ p π 2 λ 3 2 m ψ ( 2 m − 1 ) λ 1 π 2 λ 3 ‖ f ‖ p , μ ‖ a ‖ q , ν ,

where a ≥ 1 , μ ( x ) = x p ( 1 − 2 m γ ) − 1 , and ν n = ( 2 n + 1 ) q ( 1 − 2 m γ ) − 1 .

In Corollary 4.4, let β = − γ , b = 1 2 , and replace λ 1 and λ 3 with λ 1 2 γ and λ 3 2 γ , respectively, then we can obtain the homogeneous forms corresponding to (4.28), (4.29), and (4.30).

In Corollary 4.4, let β = γ , b = s ( s ∈ N + ) , then we obtain

(4.31) ∑ n = s ∞ n 2 m p γ − 1 ∫ 0 a cosh ( λ 1 ( x n ) γ ) csch ( λ 3 ( x n ) γ ) f ( x ) d x p < 1 γ π 2 λ 3 2 m ψ ( 2 m − 1 ) λ 1 π 2 λ 3 p ‖ f ‖ p , μ p ,

(4.32) ∫ 0 a x 2 m q β − 1 ∑ n = s ∞ cosh ( λ 1 ( x n ) γ ) csch ( λ 3 ( x n ) γ ) a n q d x < 1 γ π 2 λ 3 2 m ψ ( 2 m − 1 ) λ 1 π 2 λ 3 q ‖ a ‖ q , ν q ,

(4.33) ∫ 0 a f ( x ) ∑ n = s ∞ cosh ( λ 1 ( x n ) γ ) csch ( λ 3 ( x n ) γ ) a n d x < 1 γ π 2 λ 3 2 m ψ ( 2 m − 1 ) λ 1 π 2 λ 3 ‖ f ‖ p , μ ‖ a ‖ q , ν ,

where a ≥ 1 , μ ( x ) = x p ( 1 − 2 m γ ) − 1 , and ν n = n q ( 1 − 2 m γ ) − 1 . Letting λ 1 = 1 , λ 3 = 3 , and γ = 1 2 m in (4.33), we obtain (1.15).

Remark 4.5

Setting λ 1 = ρ , λ 3 = 2 ρ ( ρ > 0 ) in Corollary 4.4, and observing that

cosh ( u ) csch ( 2 u ) = 1 2 csch ( u ) ,

it implies that

(4.34) ∑ n = 0 ∞ ( n + b ) 2 m p γ − 1 ∫ Ω csch ( ρ x β ( n + b ) γ ) f ( x ) d x p < ∣ β ∣ − 1 q γ − 1 p 2 1 − 4 m π ρ 2 m ψ ( 2 m − 1 ) π 4 p ‖ f ‖ p , μ p ,

(4.35) ∫ Ω x 2 m q β − 1 ∑ n = 0 ∞ csch ( ρ x β ( n + b ) γ ) a n q d x < ∣ β ∣ − 1 q γ − 1 p 2 1 − 4 m π ρ 2 m ψ ( 2 m − 1 ) π 4 q ‖ a ‖ q , ν q ,

(4.36) ∫ Ω f ( x ) ∑ n = 0 ∞ csch ( ρ x β ( n + b ) γ ) a n d x < ∣ β ∣ − 1 q γ − 1 p 2 1 − 4 m π ρ 2 m ψ ( 2 m − 1 ) π 4 ‖ f ‖ p , μ ‖ a ‖ q , ν .

Inequalities (4.2), (4.3), (4.4), (4.34), (4.35), and (4.36) are equivalent. In fact, by Lemma 2.5 and (4.1), we obtain

ψ ( 2 m − 1 ) π 4 = ( 2 m − 1 ) ! 4 π 2 m ∑ j = 0 ∞ 1 ( 4 j + 1 ) 2 m + 1 ( 4 j + 3 ) 2 m = 4 π 2 m ∑ j = 0 ∞ ( 2 m − 1 ) ! ( 2 j + 1 ) 2 m = B m m ( 2 2 m − 1 ) 2 4 m − 2 ,

and it follows therefore that (4.2), (4.3), (4.3), (4.34), (4.35), and (4.36) are equivalent.

Remark 4.6

Setting λ 1 = ρ , λ 3 = 4 ρ ( ρ > 0 ) in Corollary 4.4, and observing that

cosh ( u ) csch ( 4 u ) = 1 4 csch ( u ) sech ( 2 u ) ,

we obtain the following Hilbert-type inequalities involving hyperbolic secant and hyperbolic cosecant functions:

(4.37) ∑ n = 0 ∞ ( n + b ) 2 m p γ − 1 ∫ Ω csch ( ρ x β ( n + b ) γ ) sech ( 2 ρ x β ( n + b ) γ ) f ( x ) d x p < ∣ β ∣ − 1 q γ − 1 p 4 1 − 3 m π ρ 2 m ψ ( 2 m − 1 ) π 8 p ‖ f ‖ p , μ p ,

(4.38) ∫ Ω x 2 m q β − 1 ∑ n = 0 ∞ csch ( ρ x β ( n + b ) γ ) sech ( 2 ρ x β ( n + b ) γ ) a n q d x < ∣ β ∣ − 1 q γ − 1 p 4 1 − 3 m π ρ 2 m ψ ( 2 m − 1 ) π 8 q ‖ a ‖ q , ν q ,

(4.39) ∫ Ω f ( x ) ∑ n = 0 ∞ csch ( ρ x β ( n + b ) γ ) sech ( 2 ρ x β ( n + b ) γ ) a n d x < ∣ β ∣ − 1 q γ − 1 p 4 1 − 3 m π ρ 2 m ψ ( 2 m − 1 ) π 8 ‖ f ‖ p , μ ‖ a ‖ q , ν .

Letting β = γ , ρ = 1 , b = s ( s ∈ N + ) , and m = 1 in (4.39), we have

∫ 0 a f ( x ) ∑ n = s ∞ csch ( ( x n ) γ ) sech ( 2 ( x n ) γ ) a n d x < π 2 16 γ ψ π 8 ‖ f ‖ p , μ ‖ a ‖ q , ν ,

where a ≥ 1 , 0 < γ ≤ 1 2 , μ ( x ) = x p ( 1 − 2 γ ) − 1 , and ν n = n q ( 1 − 2 γ ) − 1 .

5 Conclusion

The main objective of this work is to establish some half-discrete Hilbert-type inequalities involving hyperbolic functions. This interest is mainly motivated by [22,25,26], where the authors provided some integral inequalities with hyperbolic functions. In order to do so, we first constructed a more general kernel function composed of several exponent functions with multiple parameters. By using the Hermite-Hadamard’s inequality, Hölder’s inequality as well as some other techniques of real analysis, we established a half-discrete Hilbert-type with the newly constructed kernel function. Second, by constructing a special sequence and a special function (Lemma 2.4), we proved that the constant factor of the established Hilbert-type inequality is the best possible. At last, by the introduction of the Bernoulli number and the rational fraction expansion of tangent function, some special examples and their equivalent forms were considered. We need to point out that some hyperbolic functions or their combinations are not included in our results, and it is worthy of further research.

Acknowledgements

The authors are indebted to the anonymous referees for their valuable suggestions and comments that helped improve the paper significantly.

  1. Funding information: This work was supported by scientific research project of Zhejiang Provincial Department of Education (Y202148139).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Not applicable.

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Received: 2021-11-29
Revised: 2022-04-01
Accepted: 2022-04-12
Published Online: 2022-07-27

© 2022 Minghui You et al., published by De Gruyter

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

Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0041
Keywords for this article
Hilbert-type inequality; hyperbolic function; Hermite-Hadamard’s inequality; rational fraction expansion; Bernoulli number
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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