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A new extended Mulholland's inequality involving one partial sum

  • Ling Peng and Bicheng Yang EMAIL logo
Published/Copyright: September 25, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

By using the weight coefficients and the techniques of real analysis, a new extended Mulholland’s inequality with multi-parameters involving one partial sum is given. The equivalent statements of the best value related to several parameters are provided. The equivalent forms, some inequalities in particular parameters, and the operator expressions are obtained.

Keywords: beta function; differentiation mid-value theorem; Mulholland’s inequality; parameter; partial sum
MSC 2010: 26D15; 26D10; 47A05

1 Introduction

Suppose that p > 1 , 1 p + 1 q = 1 , a m , b n ≥ 0 , 0 < ∑ m = 1 ∞ a m p < ∞ , and 0 < ∑ n = 1 ∞ b n q < ∞ . The following Hardy-Hilbert’s inequality with the best value π sin ( π ⁄ p ) was provided (cf. [1], Theorem 315):

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

With regard to a similar assumption, the following Mulholland’s inequality with the same best value was obtained (cf. [1], Theorem 343):

(2) ∑ m = 2 ∞ ∑ n = 2 ∞ a m b n ln m n < π sin ( π ⁄ p ) ∑ m = 2 ∞ a m p m 1 − p 1 p ∑ n = 2 ∞ b n q n 1 − q 1 q .

In 2006, an extension of (1) was given by Krnić and Pečarić [2] as follows:

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

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

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

is the beta function. For p = q = 2 , λ 1 = λ 2 = λ 2 , (3) deduces to Yang’s result in [3], which was published in 2001. In 2019, following the idea of (3), Adiyasuren et al. [4] gave an extension of (3) involving two partial sums as follows: For λ ∈ ( 0 , 2 ] , λ i ∈ ( 0 , 1 ] ∩ ( 0 , λ ) ( i = 1 , 2 ) , λ 1 + λ 2 = λ , a m , b n ≥ 0 ,

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

where λ 1 λ 2 B ( λ 1 , λ 2 ) is the best value, and A m = ∑ i = 1 m a i and B n = ∑ k = 1 n b k ( m , n ∈ { 1 , 2 , … } ) , satisfy A m = o ( e t m ) , B n = o ( e t n ) ( t > 0 ; m , n → ∞ ) ,

0 < ∑ m = 1 ∞ m − p λ 1 − 1 A m p < ∞ and 0 < ∑ n = 1 ∞ n − q λ 2 − 1 B n q < ∞ .

We can see some new extensions and applications of (1) and (2) with their integral analogs in [5–14]. In 2016, Hong and Wen [15] considered some equivalent conditions of the best value related to some parameters in the general form of (1). The further results were provided by [16–21]. In 2021, Gu and Yang [22] gave an extension of (4) with the kernel as 1 ( m α + n β ) λ , where α , β ∈ ( 0 , 1 ] . But the constant factor is not the best possible unless at α = β = 1 . In 2023, Yang et al. [23] gave an improvement of [22] and proved that the constant factor in the new inequality is the best possible for α , β ∈ ( 0 , 1 ] . Recently, Huang et al. [24,25] provided the same kinds of applications in Hilbert-type inequalities.

In this article, following the way of [15], by the use of the weight coefficients and the techniques of real analysis, a new extended Mulholland’s inequality with multi-parameters involving one partial sum is given. The equivalent conditions of the best value related to several parameters are provided. The equivalent forms, a few inequalities for particular parameters, and the operator expressions are obtained.

2 Some lemmas

In this section, it is necessary to introduce the following conditions on the parameters that we use in the following results.

Assumption 1

Assume that p > 1 ( q > 1 ) , 1 p + 1 q = 1 , λ > 0 , λ 1 ∈ ( 0 , 2 ] ∩ ( 0 , λ + 1 ) , λ 2 ∈ ( 0 , 1 ] ∩ ( 0 , λ ) , λ ˆ 1 ≔ λ − λ 2 p + λ 1 q , λ ˆ 2 ≔ λ − λ 1 q + λ 2 p , N = { 1 , 2 , … } , a m , b n ≥ 0 , B n ≔ ∑ k = 2 n b k ( m , n ∈ N 2 ≔ N \ { 1 } ) , B n = o ( e t ln n ) ( t > 0 ; n → ∞ ) , and

(5) 0 < ∑ m = 2 ∞ ln p ( 1 − λ ˆ 1 ) − 1 m m 1 − p a m p < ∞ and 0 < ∑ n = 2 ∞ B n q n ln q λ ˆ 2 + 1 n < ∞ .

Lemma 1

(cf. [5], (2.2.3)) (i) Assuming that ( − 1 ) d i d t i g ( t ) > 0 , t ∈ [ k , ∞ ) ( k ∈ N ) , g ( i ) ( ∞ ) = 0 ( i = 0 , 1 , 2 , 3 ) , P i ( t ) , B i ( i ∈ N ) are the Bernoulli functions and Bernoulli numbers of i-order, we have

(6) ∫ k ∞ P 2 q − 1 ( t ) g ( t ) d t = − ε q B 2 q 2 q g ( k ) ( 0 < ε q < 1 ; q = 1 , 2 , … ) .

In particular, for q = 1 , B 2 = 1 6 , it follows that

(7) − 1 12 g ( k ) < ∫ k ∞ P 1 ( t ) g ( t ) d t < 0 .

(ii) If ( − 1 ) i d i d t i g ( t ) > 0 , t ∈ [ k , ∞ ) ( k ∈ N ) , g ( i ) ( ∞ ) = 0 ( i = 0 , 1 ), then we obtain (cf. [5], (2.2.13))

(8) − 1 8 g ( k ) < ∫ k ∞ P 1 ( t ) g ( t ) d t < 0 .

(iii) (cf. [5], (2.3.2)) Assuming that f ( t ) ( ≥ 0 ) ∈ C [ k , ∞ ) ( k ∈ N ) , f ( i ) ( ∞ ) = 0 ( i = 0 , 1 ) , the following expression is valid:

(9) ∑ i = k ∞ f ( i ) = ∫ k ∞ f ( t ) d t + 1 2 f ( k ) + ∫ k ∞ P 1 ( t ) f ′ ( t ) d t .

Lemma 2

Let s > 0 , s 2 ∈ ( 0 , 2 ] ∩ ( 0 , s ) , k s ( s 2 ) = B ( s 2 , s − s 2 ) . Define the weight coefficient as follows:

(10) ϖ s ( s 2 , m ) ≔ ( ln m ) s − s 2 ∑ n = 2 ∞ ( ln n ) s 2 − 1 ( ln m n ) s n .

Then the following inequalities are valid:

(11) 0 < k s ( s 2 ) 1 − O 1 ln s 2 m < ϖ s ( s 2 , m ) < k s ( s 2 ) ( m ∈ N 2 ) ,

where we indicate that O 1 ln s 2 m ≔ 1 k s ( s 2 ) ∫ 0 ln 2 ln m u s 2 − 1 ( 1 + u ) s d u ( > 0 ) .

Proof

For fixed m ∈ N 2 , we define a positive function g m ( t ) as follows:

(12) g m ( t ) ≔ ln s 2 − 1 t ( ln m + ln t ) s t ( t > 1 ) .

(i) For s 2 ∈ ( 0 , 1 ] ∩ ( 0 , s ) , we find

g m ′ ( t ) = − ( 1 − s 2 ) ln s 2 − 2 t ( ln m + ln t ) s t 2 − s ln s 2 − 1 t ( ln m + ln t ) s + 1 t 2 − ln s 2 − 1 t ( ln m + ln t ) s t 2 < 0 .

By means of the decreasing property of series, it follows that

(13) ∫ 2 ∞ g m ( t ) d t < ∑ n = 2 ∞ g m ( n ) < ∫ 1 ∞ g m ( t ) d t .

(ii) For s 2 ∈ ( 1 , 2 ] ∩ ( 0 , s ) , by (9), we have

∑ n = 2 ∞ g m ( n ) = ∫ 2 ∞ g m ( t ) d t + 1 2 g m ( 2 ) + ∫ 2 ∞ P 1 ( t ) g m ′ ( t ) d t = ∫ 1 ∞ g m ( t ) d t − h ( m ) , h ( m ) ≔ ∫ 1 2 g m ( t ) d t − 1 2 g m ( 2 ) − ∫ 2 ∞ P 1 ( t ) g m ′ ( t ) d t .

We obtain − 1 2 g m ( 2 ) = − ln s 2 − 1 2 4 ( ln m + ln 2 ) s . Setting u = ln t , by integration by parts, we find

∫ 1 2 g m ( t ) d t = ∫ 0 ln 2 u s 2 − 1 ( ln m + u ) s d u = ∫ 0 ln 2 d u s 2 s 2 ( ln m + u ) s = u s 2 s 2 ( ln m + u ) s 0 ln 2 − ∫ 0 ln 2 u s 2 s 2 d 1 ( ln m + u ) s = ln s 2 2 s 2 ( ln m + ln 2 ) s + s s 2 ( s 2 + 1 ) ∫ 0 ln 2 d u s 2 + 1 ( ln m + u ) s + 1 > ln s 2 2 s 2 ( ln m + ln 2 ) s + s s 2 ( s 2 + 1 ) ln s 2 + 1 2 ( ln m + ln 2 ) s + 1 .

Since ln t t 2 > 0 , ln t t 2 ′ = 1 − 2 ln t t 3 < 0 ( t > 2 ) , by (8), we have

g m ′ ( t ) = ( s 2 − 1 ) ln s 2 − 2 t ( ln m + ln t ) s t 2 − s ln s 2 t ( ln m + ln t ) s + 1 ln − 1 t t 2 − ln s 2 − 2 t ( ln m + ln t ) s ln t t 2 ,

− ∫ 2 ∞ P 1 ( t ) ( s 2 − 1 ) ln s 2 − 2 t ( ln m + ln t ) s t 2 d t = ( 1 − s 2 ) ∫ 2 ∞ P 1 ( t ) ln s 2 − 2 t ( ln m + ln t ) s t 2 d t > 0 ,

∫ 2 ∞ P 1 ( t ) s ln s 2 t ( ln m + ln t ) s + 1 ln − 1 t t 2 + ln s 2 − 2 t ( ln m + ln t ) s ln t t 2 d t > − 1 8 s ln s 2 − 2 2 ( ln m + ln 2 ) s + 1 ln − 1 2 2 2 + ln s 2 − 2 2 ( ln m + ln 2 ) s ln 2 2 2 ,

− ∫ 2 ∞ P 1 ( t ) g m ′ ( t ) d t > − s ln s 2 − 1 2 8 ( ln m + ln 2 ) s + 1 2 2 − ln s 2 − 1 2 8 ( ln m + ln 2 ) s 2 2 .

Then we obtain

h ( m ) > ln s 2 2 s 2 ( ln m + ln 2 ) s + s s 2 ( s 2 + 1 ) ln s 2 + 1 2 ( ln m + ln 2 ) s + 1 − ln s 2 − 1 2 4 ( ln m + ln 2 ) s − s ln s 2 2 8 ( ln m + ln 2 ) s + 1 2 2 − ln s 2 − 1 2 8 ( ln m + ln 2 ) s 2 2 = ln s 2 − 1 2 ( ln m + ln 2 ) s ln 2 s 2 − 1 4 − 1 32 + s ln s 2 − 1 2 ( ln m + ln 2 ) s + 1 ln 2 2 s 2 ( s 2 + 1 ) − 1 32 ≥ ln s 2 − 1 2 ( ln m + ln 2 ) s ln 2 2 − 9 32 + s ln s 2 − 1 2 ( ln m + ln 2 ) s + 1 ln 2 2 6 − 1 32 > 0 .

By (9), we still have

∑ n = 2 ∞ g m ( n ) = ∫ 2 ∞ g m ( t ) d t + 1 2 g m ( 2 ) + ∫ 2 ∞ P 1 ( t ) g m ′ ( t ) d t = ∫ 2 ∞ g m ( t ) d t + h 1 ( m ) , h 1 ( m ) ≔ 1 2 g m ( 2 ) + ∫ 2 ∞ P 1 ( t ) g m ′ ( t ) d t .

For s 2 ∈ ( 1 , 2 ] ∩ ( 0 , s ) , in view of (8), we find

∫ 2 ∞ P 1 ( t ) ( s 2 − 1 ) s 2 − 2 t ( ln m + ln t ) s t 2 d t = ( s 2 − 1 ) ∫ 2 ∞ P 1 ( t ) ln s 2 − 2 t ( ln m + ln t ) s t 2 d t > − ( s 2 − 1 ) ln s 2 − 2 2 32 ( ln m + ln 2 ) s , − ∫ 2 ∞ P 1 ( t ) s ln s 2 − 2 t ( ln m + ln t ) s + 1 ln t t 2 + ln s 2 − 2 t ( ln m + ln t ) s ln t t 2 d t > 0 , ∫ 2 ∞ P 1 ( t ) g m ′ ( t ) d t > − ( s 2 − 1 ) ln s 2 − 2 2 32 ( ln m + ln 2 ) s , h 1 ( m ) > ln s 2 − 1 2 4 ( ln m + ln 2 ) s − ( s 2 − 1 ) ln s 2 − 2 2 32 ( ln m + ln 2 ) s ≥ ln s 2 − 2 2 4 ( ln m + ln 2 ) s ln 2 − 1 8 > 0 .

Hence, we also have (13).

In a word, for s 2 ∈ ( 0 , 2 ] ∩ ( 0 , s ) , by (13), setting u = ln t ln m , we find

ϖ s ( s 2 , m ) = ln s − s 2 m ∑ n = 2 ∞ g m ( n ) < ln s − s 2 m ∫ 1 ∞ g m ( t ) d t = ∫ 0 ∞ u s 2 − 1 d u ( 1 + u ) s = B ( s 2 , s − s 2 ) = k s ( s 2 ) ,

ϖ s ( s 2 , m ) > ln s − s 2 m ∫ 2 ∞ g m ( t ) d t = ∫ ln 2 ln m ∞ u s 2 − 1 d u ( 1 + u ) s = k s ( s 2 ) 1 − O 1 ln s 2 m > 0 ,

where we indicate that O 1 ln s 2 m = 1 k s ( s 2 ) ∫ 0 ln 2 ln m u s 2 − 1 ( 1 + u ) s d u , satisfying

0 < ∫ 0 ln 2 ln m u s 2 − 1 ( 1 + u ) s d u ≤ ∫ 0 ln 2 ln m u s 2 − 1 d u = 1 s 2 ln 2 ln m s 2 .

Therefore, inequalities (11) follow.

This proves the lemma.□

The following extended Mulholland’s inequality is valid.

Lemma 3

Let Assumption 1 hold. Let us define

k λ + 1 ( λ 2 + 1 ) = B ( λ 2 + 1 , λ − λ 2 ) ,

k λ + 1 ( λ 1 ) = B ( λ 1 , λ + 1 − λ 1 ) . Then the following inequality is valid:

(14) ∑ n = 2 ∞ ∑ m = 2 ∞ a m B n ( ln m n ) λ + 1 n < ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 q ∑ m = 2 ∞ ln p ( 1 − λ ^ 1 ) − 1 m m 1 − p a m p 1 p ∑ n = 2 ∞ B n q n ln q λ ^ 2 + 1 n 1 q .

Proof

By the symmetry, for s > 0 , s 1 ∈ ( 0 , 2 ] ∩ ( 0 , s ) , the following estimations for the next weight coefficient are valid:

(15) 0 < k s ( s 1 ) 1 − O 1 ln s 1 n < ω s ( s 1 , n ) ≔ ( ln n ) s − s 1 ∑ m = 2 ∞ ( ln n ) s 1 − 1 ( ln m n ) s m < k s ( s 1 ) ≔ B ( s 1 , s − s 1 ) ( n ∈ N 2 ) ,

where O 1 ln s 1 n ≔ 1 k s ( s 1 ) ∫ 0 ln 2 ln n u s 1 − 1 ( 1 + u ) s d u > 0 .

Let us denote I λ + 1 ≔ ∑ n = 2 ∞ ∑ m = 2 ∞ a m B n ( ln m n ) λ + 1 n . By Hölder’s inequality (cf. [26]),

I λ + 1 ≔ ∑ n = 2 ∞ ∑ m = 2 ∞ 1 ( ln m n ) λ + 1 m 1 ⁄ q ln λ 2 ⁄ p n n 1 ⁄ p ln ( λ 1 − 1 ) ⁄ q m a m × ln ( λ 1 − 1 ) ⁄ q m m 1 ⁄ q ln λ 2 ⁄ p n 1 n 1 ⁄ q B n ≤ ∑ n = 2 ∞ ∑ m = 2 ∞ 1 ( ln m n ) λ + 1 m p − 1 ln λ 2 n n ln ( λ 1 − 1 ) ⁄ ( p − 1 ) m a m p 1 p × ∑ n = 2 ∞ ∑ m = 2 ∞ 1 ( ln m n ) λ + 1 ln λ 1 − 1 m m ln λ 2 ( q − 1 ) n B n q n 1 q = ∑ m = 2 ∞ ϖ λ + 1 ( λ 2 + 1 , m ) ln p ( 1 − λ ^ 1 ) − 1 m 1 − p a m p 1 p × ∑ n = 2 ∞ ω λ + 1 ( λ 1 , n ) B n q n ln q λ ^ 2 + 1 n 1 q .

Then by (11) and (15), for s = λ + 1 > 1 , s 1 = λ 1 ∈ ( 0 , 2 ] ∩ ( 0 , λ + 1 ) ,

s 2 = λ 2 + 1 ∈ ( 0 , 2 ] ∩ ( 0 , λ + 1 ) ,

in view of (5), we obtain (14).

This proves the lemma.□

Lemma 4

Let b n , B n , n ∈ N 2 be as in Assumption 1. If t > 0 , then the following inequality is valid:

(16) ∑ n = 2 ∞ e − t ln n b n ≤ t ∑ n = 2 ∞ e − t ln n 1 n B n .

Proof

Since B n e − t ln n = o ( 1 ) ( n → ∞ ) , by Abel’s summation by parts formula, it follows that

∑ n = 2 ∞ e − t ln n b n = lim n → ∞ B n e − t ln n + ∑ n = 2 ∞ B n [ e − t ln n − e − t ln ( n + 1 ) ] = ∑ n = 2 ∞ B n [ e − t ln n − e − t ln ( n + 1 ) ] .

We set a function f ( x ) = e − t ln x , x ∈ [ n , n + 1 ] . Then we find f ′ ( x ) = − t x e − t ln x = − h ( x ) , where h ( x ) ≔ 1 x e − ln x is decreasing in [ n , n + 1 ] . In view of the differentiation mid-value theorem, we obtain

∑ n = 2 ∞ e − t ln n b n = − ∑ n = 2 ∞ B n ( f ( n + 1 ) − f ( n ) ) = − ∑ n = 2 ∞ B n f ′ ( n + θ n ) = ∑ n = 2 ∞ h ( n + θ n ) B n ≤ ∑ n = 2 ∞ h ( n ) B n = t ∑ n = 2 ∞ e − t ln n 1 n B n ( θ n ∈ ( 0 , 1 ) ) ,

namely, (16) follows.

This proves the lemma.□

3 Main results

We have the following extended Mulholland’s inequality involving one partial sum.

Theorem 1

Let Assumption 1 hold. Let k λ + 1 ( λ 2 + 1 ) and k λ + 1 ( λ 1 ) be defined as in Lemma 3. Then the following inequality is valid:

(17) ∑ m = 2 ∞ ∑ n = 2 ∞ a m b n ( ln m n ) λ < λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 q × ∑ m = 2 ∞ ln p ( 1 − λ ˆ 1 ) − 1 m m 1 − p a m p 1 p ∑ n = 2 ∞ B n q n ln q λ ˆ 2 + 1 n 1 q .

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

0 < ∑ m = 2 ∞ ln p ( 1 − λ 1 ) − 1 m m 1 − p a m p < ∞ , 0 < ∑ n = 2 ∞ B n q n ln q λ 2 + 1 n < ∞ ,

and the following inequality:

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

Proof

Use the following expression:

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

where Γ is the gamma function. Let us denote I ≔ ∑ m = 2 ∞ ∑ n = 2 ∞ a m b n ( ln m n ) λ . By (16), it follows that

I = 1 Γ ( λ ) ∑ m = 2 ∞ ∑ n = 2 ∞ a m b n ∫ 0 ∞ t λ − 1 e − ( ln m + ln n ) t d t = 1 Γ ( λ ) ∫ 0 ∞ t λ − 1 ∑ m = 2 ∞ e − t ln m a m ∑ n = 2 ∞ e − t ln n b n d t ≤ 1 Γ ( λ ) ∫ 0 ∞ t λ − 1 ∑ m = 2 ∞ e − t ln m a m t ∑ n = 2 ∞ e − t ln n 1 n B n d t = 1 Γ ( λ ) ∑ m = 2 ∞ ∑ n = 2 ∞ a m 1 n B n ∫ 0 ∞ t ( λ + 1 ) − 1 e − ( ln m + ln n ) t d t = Γ ( λ + 1 ) Γ ( λ ) ∑ m = 2 ∞ ∑ n = 2 ∞ 1 ( ln m + ln n ) λ + 1 n a m B n = Γ ( λ + 1 ) Γ ( λ ) I λ + 1 ,

where I λ + 1 is defined as in (14). Then by (14), in view of Γ ( λ + 1 ) = λ Γ ( λ ) , we have (17).

This proves the theorem.□

Theorem 2

Suppose that λ > 0 , λ 1 ∈ ( 0 , 2 ] ∩ ( 0 , λ ) , λ 2 ∈ ( 0 , 1 ] ∩ ( 0 , λ ) . If λ 1 + λ 2 = λ , then the constant λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 q in (17) is the best value.

Proof

We now prove that the constant λ B ( λ 1 , λ 2 + 1 ) ( = λ 2 B ( λ 1 , λ 2 ) ) in (18) is the best value following the condition. For any number ε , 0 < ε < min { p λ 1 , q λ 2 } , we set

a ˜ m ≔ 1 m ln λ 1 − ε p − 1 m , b ˜ n ≔ 1 n ln λ 2 − ε q − 1 n ( m , n ∈ N 2 ) .

Since 0 < λ 2 − ε q < 1 , by means of the decreasingness property of series (cf. [26]), we obtain

B ˜ n ≔ ∑ k = 2 n b ˜ k < ∫ 1 n 1 y ln λ 2 − ε q − 1 y d y = 1 λ 2 − ε q ln λ 2 − ε q n ( n ∈ N 2 ) ,

and observe that B ˜ n = o ( e t ln n ) ( t > 0 ; n → ∞ ) .

If there exists a positive constant M ≤ λ 2 B ( λ 1 , λ 2 ) ( = λ B ( λ 1 , λ 2 + 1 ) ) , such that (18) is valid when we replace λ B ( λ 1 , λ 2 + 1 ) by M , then in particular, substitution of a m = a ˜ m , b n = b ˜ n and B n = B ˜ n in (18), we have

(19) I ˜ ≔ ∑ n = 2 ∞ ∑ m = 2 ∞ a ˜ m b ˜ n ( ln m n ) 2 < M ∑ m = 2 ∞ ln p ( 1 − λ 1 ) − 1 m 1 − p a ˜ m p 1 p ∑ n = 2 ∞ B ˜ n q n ln q λ 2 + 1 n 1 q .

By (19) (cf. [26]), we find

I ˜ < M λ 2 − ε q ∑ m = 2 ∞ ln − ε − 1 m m 1 p ∑ n = 2 ∞ ln − ε − 1 n n 1 q = M λ 2 − ε q ln − ε − 1 2 2 + ∑ m = 3 ∞ ln − ε − 1 m m < M λ 2 − ε q ln − ε − 1 2 2 + ∫ 2 ∞ ln − ε − 1 x x d x = M ε λ 2 − ε q ε ln − ε − 1 2 2 + ln − ε 2 .

In view of (15) (for s = λ > 0 , s 1 = λ 1 − ε p ∈ ( 0 , 2 ) ∩ ( 0 , λ ) ), we have

I ˜ = ∑ n = 2 ∞ ∑ m = 2 ∞ ln λ 1 − ε p − 1 m ( ln m n ) λ m n ln λ 2 − ε q − 1 n = ∑ n = 2 ∞ 1 n ln − ε − 1 n ln λ 2 + ε p n ∑ m = 2 ∞ ln λ 1 − ε p − 1 m ( ln m n ) λ m ≥ k λ λ 1 − ε p ∑ n = 2 ∞ 1 n ln − ε − 1 n 1 − O 1 ln λ 1 − ε p n = k λ λ 1 − ε p ∑ n = 2 ∞ 1 n ln − ε − 1 n − ∑ n = 2 ∞ 1 n O 1 ln λ 1 + ε q + 1 n > k λ λ 1 − ε p ∫ 2 ∞ 1 y ln − ε − 1 y d y − O ( 1 ) = 1 ε B λ 1 − ε p , λ 2 + ε p ( ln − ε 2 − ε O ( 1 ) ) .

Then we find

B λ 1 − ε p , λ 2 + ε p ( ln − ε 2 − ε O ( 1 ) ) < ε I ˜ < M λ 2 − ε q ε ln − ε − 1 2 2 + ln − ε 2 .

For ε → 0 + , we obtain B ( λ 1 , λ 2 ) ≤ M λ 2 , namely, λ 2 B ( λ 1 , λ 2 ) ≤ M .

Hence, M = λ 2 B ( λ 1 , λ 2 ) ( = λ B ( λ 1 , λ 2 + 1 ) ) is the best value in (18).

This proves the theorem.□

Theorem 3

Assume that λ 1 ∈ ( 0 , 2 ] ∩ ( 0 , λ ) , λ 2 ∈ ( 0 , 1 ] ∩ ( 0 , λ ) ( λ > 0 ) . If the constant λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 4 in (17) is the best value, then for

λ − λ 1 − λ 2 ≤ min { p ( 2 − λ 1 ) , q ( 1 − λ 2 ) } ,

we have λ 1 + λ 2 = λ .

Proof

For λ ˆ 1 = λ − λ 2 p + λ 1 q = λ − λ 1 − λ 2 p + λ 1 , λ ˆ 2 = λ − λ 1 q + λ 2 p = λ − λ 1 − λ 2 q + λ 2 , we find λ ˆ 1 + λ ˆ 2 = λ , 0 < λ ˆ 1 , λ ˆ 2 < λ p + λ q = λ , and B ( λ ˆ 1 , λ ˆ 2 + 1 ) ∈ ( 0 , ∞ ) . For λ − λ 1 − λ 2 ≤ p ( 2 − λ 1 ) , we have λ ˆ 1 ≤ 2 ; for λ − λ 1 − λ 2 ≤ q ( 1 − λ 2 ) , we have λ ˆ 2 ≤ 1 . Substitution of λ i = λ ˆ i ( i = 1 , 2 ) in (18), it follows that

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

By Hölder’s inequality with weight (cf. [26]), we obtain

(21) B ( λ ˆ 1 , λ ˆ 2 + 1 ) = k λ + 1 ( λ ˆ 1 ) = k λ + 1 λ − λ 2 p + λ 1 q = ∫ 0 ∞ u λ − λ 2 p + λ 1 q − 1 ( 1 + u ) λ + 1 d u = ∫ 0 ∞ 1 ( 1 + u ) λ + 1 u λ − λ 2 − 1 p u λ 1 − 1 q d u ≤ ∫ 0 ∞ 1 ( 1 + u ) λ + 1 u λ − λ 2 − 1 d u 1 p ∫ 0 ∞ 1 ( 1 + u ) λ + 1 u λ 1 − 1 d u 1 q = ∫ 0 ∞ 1 ( 1 + v ) λ + 1 v ( λ 2 + 1 ) − 1 d v 1 p ∫ 0 ∞ 1 ( 1 + u ) λ + 1 u λ 1 − 1 d u 1 q = ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 q .

If the constant λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 q in (17) is the best value, then compared with the constant factors in (17) and (20), we have

λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 q ≤ λ B ( λ ˆ 1 , λ ˆ 2 + 1 ) ( ∈ R + ) ,

which follows that

(22) B ( λ ˆ 1 , λ ˆ 2 + 1 ) ≥ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 q .

Hence, (21) keep the form of equality. We observe that (21) keep the form of equality if and only if there exist constants A and B , such that they are not both zero and (cf. [26]) A u λ − λ 2 − 1 = B u λ 1 − 1 a.e. in R + . Suppose that A ≠ 0 . We have u λ − λ 2 − λ 1 = B A a.e. in R + , and then λ − λ 2 − λ 1 = 0 . Hence, we have λ 1 + λ 2 = λ .

This proves the theorem.□

4 Equivalent forms, some particular inequalities, and the operator expressions

The following inequalities equivalent to (17) and (18), respectively, are valid.

Theorem 4

Let Assumption 1 hold. Let k λ + 1 ( λ 2 + 1 ) and k λ + 1 ( λ 1 ) be defined as in Lemma 3. Then inequality (23) is equivalent to (17):

(23) ∑ m = 2 ∞ ln q λ ˆ 1 − 1 m m ∑ n = 2 ∞ b n ( ln m n ) λ q 1 q < λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 q ∑ n = 2 ∞ B n q ln q λ ˆ 2 + 1 n 1 q .

In particular, for λ 1 + λ 2 = λ , inequality (24) is equivalent to (18):

(24) ∑ m = 2 ∞ ln q λ 1 − 1 m m ∑ n = 2 ∞ b n ( ln m n ) λ q 1 q < λ B ( λ 1 , λ 2 + 1 ) ∑ n = 2 ∞ B n q n ln q λ 2 + 1 n 1 q .

Proof

Assume that inequality (23) holds. Let us denote

J ≔ ∑ m = 2 ∞ ln q λ ˆ 1 − 1 m m ∑ n = 2 ∞ b n ( ln m n ) λ q 1 q .

Using notation I = ∑ m = 2 ∞ ∑ n = 2 ∞ a m b n ( ln m n ) λ from the proof of Theorem 1, with Hölder’s inequality, it follows that

(25) I = ∑ m = 2 ∞ ln − λ ˆ 1 − 1 q m m − 1 ⁄ q a m ln λ ˆ 1 − 1 q m m 1 ⁄ q ∑ n = 2 ∞ b n ( ln m m ) λ ≤ ∑ m = 2 ∞ ln p 1 − λ ˆ 1 − 1 m m 1 − p a m p 1 p J .

Then by (23), we have (17).

On the other hand, suppose that (17) is valid. We set

a m ≔ ln q λ 1 − 1 m m ∑ n = 2 ∞ b n ( ln m n ) λ q − 1 , m ∈ N 2 .

We find J q = I . If J = 0 , then (23) is naturally valid; if J = ∞ , then (23) is impossible valid, namely, J < ∞ . Suppose that 0 < J < ∞ . By (17), we have

0 < ∑ m = 2 ∞ ln p ( 1 − λ ˆ 1 ) − 1 m m 1 − p a m p = J q = I < λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 q J q − 1 ∑ n = 2 ∞ B n q n ln q λ ˆ 2 + 1 n 1 q < ∞ , J < λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 q ∑ n = 2 ∞ B n q n ln q λ ˆ 2 + 1 n 1 q .

Hence, we have (23), which is equivalent to (17).

This proves the theorem.□

Theorem 5

Assume that λ > 0 , λ 1 ∈ ( 0 , 2 ] ∩ ( 0 , λ ) , λ 2 ∈ ( 0 , 1 ] ∩ ( 0 , λ ) . If λ 1 + λ 2 = λ , then the constant λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 q in (23) is the best value. On the other hand, if the constant λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 9 in (23) is the best value, then for

λ − λ 1 − λ 2 ≤ min { p ( 2 − λ 1 ) , q ( 1 − λ 2 ) } ,

we have λ 1 + λ 2 = λ .

Proof

If λ 1 + λ 2 = λ , then the constant in (17) is the best value. We can show that the constant in (23) is the best value. Otherwise, by using (25), we can prove that the constant in (17) is not the best value. On the other hand, if the constant in (23) is the best value, then for the equivalency of (23) and (17), in view of J q = I (see the proof of Theorem 4.1), we can prove that the constant in (17) is the best value. Then by Theorem 3.3, for

λ − λ 1 − λ 2 ≤ min { p ( 2 − λ 1 ) , q ( 1 − λ 2 ) } ,

we have λ 1 + λ 2 = λ .

This proves the theorem.□

Remark 1

(i) For λ = 1 , λ 1 = 1 q , λ 2 = 1 p in (18) and (24), the following equivalent forms are valid:

(26) ∑ m = 2 ∞ ∑ n = 2 ∞ a m b n ln m n < π p sin ( π ⁄ p ) ∑ m = 2 ∞ a m p m 1 − p 1 p ∑ n = 2 ∞ B n q n ln q n 1 q ,

(27) ∑ m = 2 ∞ 1 m ∑ n = 2 ∞ b n ln m q 1 q < π p sin ( π ⁄ p ) ∑ n = 2 ∞ B n q n ln q n 1 q .

(ii) For λ = 1 , λ 1 = 1 p , λ 2 = 1 q in (18) and (24), the following equivalent dual forms of (26) and (27) are valid:

(28) ∑ m = 2 ∞ ∑ n = 2 ∞ a m b n ln m n < π q sin ( π ⁄ p ) ∑ m = 2 ∞ ln p − 2 m m 1 − p a m p 1 p ∑ n = 2 ∞ B n q n ln 2 n 1 q ,

(29) ∑ m = 2 ∞ ln q − 2 m m ∑ n = 2 ∞ b n ln m n q 1 q < π q sin ( π ⁄ p ) ∑ n = 2 ∞ B n q n ln 2 n 1 q .

In particular, for p = q = 2 , both (26) and (28) deduce to

(30) ∑ m = 2 ∞ ∑ n = 2 ∞ a m b n ln m n < π 2 ∑ m = 2 ∞ m a m 2 ∑ n = 2 ∞ B n 2 n ln 2 n 1 2 ,

and both (27) and (29) deduce to the following equivalent form of (30):

(31) ∑ m = 2 ∞ 1 m ∑ n = 2 ∞ b n ln m n 2 1 2 < π 2 ∑ n = 2 ∞ B n 2 n ln 2 n 1 2 .

The constants in the above inequalities are all the best values.

We set the following functions: φ ( m ) ≔ ln p ( 1 − λ ^ 1 ) − 1 m m 1 − p , ψ ( n ) ≔ ln q ( 1 − λ ^ 2 ) − 1 n n 1 − q , and Ψ ( n ) ≔ 1 n ln q λ ^ 2 + 1 n , wherefrom,

ψ 1 − q ( m ) ≔ ln q λ ^ 1 − 1 m m ( m , n ∈ N 2 ) .

Define the normed linear spaces as follows:

l p , φ ≔ a = { a m } m = 2 ∞ ; ‖ a ‖ p , φ = ∑ m = 2 ∞ φ ( m ) ∣ a m ∣ p 1 p < ∞ , l q , ψ ≔ b = { b n } n = 2 ∞ ; ‖ b ‖ q , ψ = ∑ n = 2 ∞ ψ ( n ) ∣ b n ∣ q 1 q < ∞ , l q , Ψ ≔ B = { B n } n = 2 ∞ ; ‖ B ‖ q , Ψ = ∑ n = 2 ∞ Ψ ( n ) ∣ B n ∣ q 1 q < ∞ , l q , φ 1 − q ≔ c = { c m } m = 2 ∞ ; ‖ c ‖ q , φ 1 − q = ∑ m = 2 ∞ φ 1 − q ( m ) ∣ c m ∣ q 1 q < ∞ , l ˜ q , ψ ≔ b = { b n } n = 2 ∞ ∈ l q , ψ ; B n = ∑ i = 2 n b i , B = { B n } n = 2 ∞ ∈ l q , Ψ .

For b = { b n } n = 2 ∞ ( ≥ 0 ) ∈ l ˜ q , ψ , setting c = { c m } m = 2 ∞ , c m ≔ ∑ n = 2 ∞ b n ( ln m n ) λ .

Inequality (23) can be rewritten as follows:

‖ c ‖ q , φ 1 − q ≤ λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 q ‖ B ‖ q , Ψ < ∞ ,

namely, c ∈ l q , φ 1 − q .

Definition 1

Define an extended Mulholland’s operator T : l ˜ q , ψ → l q , φ 1 − q as follows: For any b ∈ l ˜ q , ψ , there exists a unique representation c = T b ∈ l q , φ 1 − q , such that for any m ∈ N 2 , T b ( m ) = c m . Define the formal inner product of a ∈ l p , φ and T b , and the norm of T as follows:

( a , T b ) ≔ ∑ m = 2 ∞ a m ∑ n = 2 ∞ b n ( ln m n ) λ , ‖ T ‖ ≔ sup b ( ≠ 0 ) ∈ l ˜ q , ψ ‖ T b ‖ q , φ 1 − q ‖ B ‖ q , ψ .

Theorem 6

Assuming that b ( ≥ 0 ) ∈ l ˜ q , ψ , ‖ B ‖ q , Ψ > 0 , a ( ≥ 0 ) ∈ l p , φ , ‖ a ‖ p , φ > 0 , the following equivalent operator inequalities are valid:

(32) ( a , T b ) < λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 q ‖ a ‖ p , q ‖ B ‖ q , Ψ ,

(33) ‖ T b ‖ q , φ 1 − q < λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 4 ‖ B ‖ q , Ψ .

Moreover, assume that λ 1 ∈ ( 0 , 2 ] ∩ ( 0 , λ ) , λ 2 ∈ ( 0 , 1 ] ∩ ( 0 , λ ) ( λ > 0 ) . If λ 1 + λ 2 = λ , then the constant λ ( k λ + 1 ( λ 2 + 1 ) ) 1 p ( k λ + 1 ( λ 1 ) ) 1 4 in (32) and (33) is the best value, namely,

‖ T ‖ = λ 2 B ( λ 1 , λ 2 ) .

On the other hand, if the same constant in (32) (or (33)) is the best value, then for

λ − λ 1 − λ 2 ≤ min { p ( 2 − λ 1 ) , q ( 1 − λ 2 ) } ,

we have λ 1 + λ 2 = λ .

5 Conclusions

In this article, by using the techniques of real analysis, a new extended Mulholland’s inequality with multi-parameters involving one partial sum is given in Theorem 3.1. The equivalent conditions of the best possible value related to several parameters are provided by Theorems 3.2 and 3.3. The equivalent forms and some inequalities in particular parameters are obtained in Theorems 4.1, 4.2, and Remark 4.1. The operator expressions are considered in Theorem 4.3. The idea and the methods provide an extensive account of discrete Hilbert-type inequalities.

Acknowledgement

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

  1. Funding information: This work was supported by the National Natural Science Foundation (No. 61772140).

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

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

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2024-03-03
Revised: 2024-06-18
Accepted: 2024-07-22
Published Online: 2024-09-25

© 2024 the author(s), published by De Gruyter

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

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https://doi.org/10.1515/math-2024-0039
Keywords for this article
beta function; differentiation mid-value theorem; Mulholland’s inequality; parameter; partial sum
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  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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