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Extension of Fejér's inequality to the class of sub-biharmonic functions

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Published/Copyright: October 14, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

Fejér’s integral inequality is a weighted version of the Hermite-Hadamard inequality that holds for the class of convex functions. To derive his inequality, Fejér [Über die Fourierreihen, II, Math. Naturwiss, Anz. Ungar. Akad. Wiss. 24 (1906), 369–390] assumed that the weight function is symmetric w.r.t. the midpoint of the interval. In this study, without assuming any symmetry condition on the weight function, Fejér’s inequality is extended to the class of sub-biharmonic functions, namely, the set of functions f ∈ C 4 ( I ) satisfying f ″ ″ ≤ 0 , where I is an interval of R . In the special case when the weight function is symmetric w.r.t. the midpoint of some interval, and the function f is convex and sub-biharmonic, an interesting refinement of Fejér’s inequality is deduced. Moreover, this inequality is extended to a new class of functions defined on the plane, that we call the set of sub-biharmonic functions on the coordinates. Assuming in addition that the function is convex on the coordinates, a refinement of an inequality due to Dragomir [On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwan. J. Math. 5 (2001), 775–788] is obtained.

Keywords: Fejér’s inequality; the Hermite-Hadamard inequality; convex function; sub-biharmonic function
MSC 2010: 26A51; 26B25; 26D15

1 Introduction

Let I be an interval of R and υ ∈ C ( I ) be a nonnegative function. Assume that υ is symmetric w.r.t. α + β 2 , for some α , β ∈ I with α < β . If f is a convex function on I , then

(1.1) ∫ α β f ( s ) υ ( s ) d s ≤ f ( α ) + f ( β ) 2 ∫ α β υ ( s ) d s .

Inequality (1.1) has been derived by Fejér [1] in 1906. The Hermite-Hadamard inequality [2,3] is a special case of (1.1) with υ ≡ 1 . Inequalities of type (1.1) are very useful in applied mathematics, in particular, in convex analysis and numerical integration. As references related to the applications of such inequalities, we suggest the studies [4–6] and the monograph [7].

Fejér’s inequality as well as the Hermite-Hadamard inequality ((1.1) with υ ≡ 1 ) have been extended and generalized in various directions. For more details, we refer to the survey paper [8]. We also refer to [9–12], where the Hermite-Hadamard inequality was studied for different kinds of convex functions. Some results related to multidimensional versions of the Hermite-Hadamard inequality can be found in [13,14]. We also refer to [15,16] for some fractional versions of this inequality.

In Section 3, we are concerned with an extension of (1.1) to the class of sub-biharmonic functions. This class of functions was first introduced by Bradford [17]. Namely, we are concerned with an extension of (1.1) to the set of functions f ∈ C 4 ( I ) satisfying f ″ ″ ≤ 0 is derived. The obtained inequality holds for any nonnegative weight function υ ∈ C ( I ) without requiring any symmetry condition. Next, several interesting results are deduced. For instance, if we assume in addition that f is convex and υ is symmetric w.r.t. α + β 2 , a refinement of (1.1) is obtained. Our approach makes use of some tools from ordinary differential equations.

In Section 4, we introduce the class of sub-biharmonic functions on the coordinates and establish a Fejér-type inequality for such functions. Namely, we consider the set of functions f ∈ C 4 ( I × I ) satisfying ∂ j 4 f ≤ 0 , j = 1 , 2 , where ∂ j 4 f denotes the fourth-order partial derivative of f w.r.t. the j th variable. This study is motivated by Dragomir [14], where he extended the Hermite-Hadamard inequality to the class of convex functions on the coordinates. Let us recall that f = f ( t , s ) : I × I → R is convex on the coordinates, if for all ( t , s ) ∈ I × I , the functions f ( t , ⋅ ) : s ∋ I → f ( t , s ) ∈ R and f ( ⋅ , s ) : t ∋ I → f ( t , s ) ∈ R are convex. For this class of functions, Dragomir [14] proved that for all j ∈ { 1 , 2 } and α j , β j ∈ I with α j < β j , it holds that

(1.2) ∫ α 1 β 1 ∫ α 2 β 2 f ( t , s ) d s d t ≤ β 2 − α 2 4 ∫ α 1 β 1 ( f ( t , α 2 ) + f ( t , β 2 ) ) d t + β 1 − α 1 4 ∫ α 2 β 2 ( f ( α 1 , s ) + f ( β 1 , s ) ) d s .

We mention that in [18], the authors introduced the notion of the convexity in a direction, which is more general than the notion of convexity on the coordinates. Namely, the convexity in the t -coordinate matches with the convexity in the direction ( 1 , 0 ) , and the convexity in the s -coordinate matches with the convexity in the direction ( 0 , 1 ) . In Section 4, we establish a weighted version of (1.2) for the class of sub-biharmonic functions on the coordinates, always without any symmetry condition on the weight function. Next, we study some special cases of weight functions. In particular, if in addition, f is convex on the coordinates and υ ≡ 1 , an interesting refinement of (1.2) is obtained.

The next section of this work is devoted to some preliminaries that will be used in the proofs of the main results.

2 Preliminaries

Let α , β ∈ R with α < β . Given υ ∈ C ( [ α , β ] ) , we consider the fourth-order boundary value problem

(2.1) φ ″ ″ ( t ) = υ ( t ) , α < t < β , φ ( α ) = φ ( β ) = 0 , φ ′ ( α ) = φ ′ ( β ) = 0 .

For all α ≤ t , s ≤ β , let

G ( t , s ) = 1 6 ( β − α ) 3 ( s − α ) 2 ( β − t ) 2 [ ( t − s ) ( β − α ) + 2 ( β − s ) ( t − α ) ] if α ≤ s ≤ t ≤ β , ( t − α ) 2 ( β − s ) 2 [ ( s − t ) ( β − α ) + 2 ( β − t ) ( s − α ) ] if α ≤ t ≤ s ≤ β .

We can easily observe that G ≥ 0 and G is symmetric.

The proof of the following result is based on the Green-function method. For completeness, we give the details of the proof.

Lemma 2.1

Problem (2.1) admits a unique solution φ ∈ C 4 ( [ α , β ] ) , which is given by

(2.2) φ ( t ) = ∫ α β G ( t , s ) υ ( s ) d s , α ≤ t ≤ β .

Proof

The uniqueness of solutions to (2.1) is obvious. We have to just show that φ satisfies (2.1). The calculations of the successive derivatives of G give us that

∂ G ∂ t ( t , s ) = 1 2 ( β − α ) 3 ( s − α ) 2 ( β − t ) ( α β − 2 α s + α t + β 2 − 3 β t + 2 s t ) if α ≤ s ≤ t ≤ β , ( β − s ) 2 ( α − t ) ( α 2 + α β − 3 α t − 2 β s + β t + 2 s t ) if α ≤ t ≤ s ≤ β , ∂ 2 G ∂ t 2 ( t , s ) = 1 ( β − α ) 3 ( s − α ) 2 ( α s − α t − 2 β 2 + β s + 3 β t − 2 s t ) if α ≤ s ≤ t ≤ β , ( β − s ) 2 ( − 2 α 2 + α s + 3 α t + β s − β t − 2 s t ) if α ≤ t ≤ s ≤ β

and

∂ 3 G ∂ t 3 ( t , s ) = 1 ( β − α ) 3 ( s − α ) 2 ( − α + 3 β − 2 s ) if α ≤ s ≤ t ≤ β , ( β − s ) 2 ( 3 α − β − 2 s ) if α ≤ t ≤ s ≤ β .

From the definition of G and the above calculations, we have

G ( α , ⋅ ) = G ( β , ⋅ ) = ∂ G ∂ t ( α , ⋅ ) = ∂ G ∂ t ( β , ⋅ ) = 0 ,

which shows that

φ ( α ) = φ ( β ) = φ ′ ( α ) = φ ′ ( β ) = 0 .

We also have

φ ‴ ( t ) = ∫ α β ∂ 3 G ∂ t 3 ( t , s ) υ ( s ) d s = 1 ( β − α ) 3 ∫ α t ( s − α ) 2 ( − α + 3 β − 2 s ) υ ( s ) d s + ∫ t β ( β − s ) 2 ( 3 α − β − 2 s ) υ ( s ) d s ,

which implies (after differentiation in variable t and simplifications) that

φ ″ ″ ( t ) = 1 ( β − α ) 3 ( ( t − α ) 2 ( − α + 3 β − 2 t ) − ( β − t ) 2 ( 3 α − β − 2 t ) ) υ ( t ) = υ ( t ) .

Then, the function φ given by (2.2) solves (2.1).□

From the calculations made in the proof of the above lemma, we obtain the following identities.

Lemma 2.2

We have

φ ″ ( α ) = 1 ( β − α ) 2 ∫ α β ( β − s ) 2 ( s − α ) υ ( s ) d s , φ ″ ( β ) = 1 ( β − α ) 2 ∫ α β ( s − α ) 2 ( β − s ) υ ( s ) d s , φ ‴ ( α ) = 1 ( β − α ) 3 ∫ α β ( β − s ) 2 ( 3 α − β − 2 s ) υ ( s ) d s , φ ‴ ( β ) = 1 ( β − α ) 3 ∫ α β ( s − α ) 2 ( 3 β − α − 2 s ) υ ( s ) d s .

3 Fejér-type inequalities for sub-biharmonic functions

We now consider the set of sub-biharmonic functions

C 4 , − ( I ) = { f ∈ C 4 ( I ) : f ″ ″ ≤ 0 } .

This class of functions was first introduced by Bradford [17]. Our main result is stated below.

Theorem 3.1

Let f ∈ C 4 , − ( I ) and υ ∈ C ( I ) be a nonnegative function. For every α , β ∈ I with α < β , it holds that

(3.1) ∫ α β f ( s ) υ ( s ) d s ≤ 1 ( β − α ) 3 f ( β ) ∫ α β ( s − α ) 2 ( 3 β − α − 2 s ) υ ( s ) d s − f ( α ) ∫ α β ( β − s ) 2 ( 3 α − β − 2 s ) υ ( s ) d s − 1 ( β − α ) 2 f ′ ( β ) ∫ α β ( s − α ) 2 ( β − s ) υ ( s ) d s − f ′ ( α ) ∫ α β ( β − s ) 2 ( s − α ) υ ( s ) d s .

Proof

Let α , β ∈ I with α < β . Let φ ∈ C 4 ( [ α , β ] ) be the function given by (2.2). By Lemma 2.1, we have

(3.2) ∫ α β f ( t ) υ ( t ) d t = ∫ α β f ( t ) φ ″ ″ ( t ) d t .

Using integrations by parts, we obtain

(3.3) ∫ α β f ( t ) φ ″ ″ ( t ) d t = [ φ ‴ ( t ) f ( t ) ] t = α β − ∫ α β f ′ ( t ) φ ‴ ( t ) d t = [ φ ‴ ( t ) f ( t ) ] t = α β − [ φ ″ ( t ) f ′ ( t ) ] t = α β + ∫ α β f ″ ( t ) φ ″ ( t ) d t = [ φ ‴ ( t ) f ( t ) ] t = α β − [ φ ″ ( t ) f ′ ( t ) ] t = α β + [ φ ′ ( t ) f ″ ( t ) ] t = α β − ∫ α β f ‴ ( t ) φ ′ ( t ) d t = [ φ ‴ ( t ) f ( t ) ] t = α β − [ φ ″ ( t ) f ′ ( t ) ] t = α β + [ φ ′ ( t ) f ″ ( t ) ] t = α β − [ φ ( t ) f ‴ ( t ) ] t = α β + ∫ α β f ″ ″ ( t ) φ ( t ) d t .

Since f ″ ″ ≤ 0 and φ ≥ 0 (because G ≥ 0 and υ ≥ 0 ), it holds that

(3.4) ∫ α β f ( t ) φ ″ ″ ( t ) d t ≤ [ φ ‴ ( t ) f ( t ) ] t = α β − [ φ ″ ( t ) f ′ ( t ) ] t = α β + [ φ ′ ( t ) f ″ ( t ) ] t = α β − [ φ ( t ) f ‴ ( t ) ] t = α β .

On the other hand, by Lemma 2.1, we know that

φ ( α ) = φ ( β ) = φ ′ ( α ) = φ ′ ( β ) = 0 .

Then, we have

[ φ ‴ ( t ) f ( t ) ] t = α β − [ φ ″ ( t ) f ′ ( t ) ] t = α β + [ φ ′ ( t ) f ″ ( t ) ] t = α β − [ φ ( t ) f ‴ ( t ) ] t = α β = [ φ ‴ ( t ) f ( t ) ] t = α β − [ φ ″ ( t ) f ′ ( t ) ] t = α β = φ ‴ ( β ) f ( β ) − φ ‴ ( α ) f ( α ) − φ ″ ( β ) f ′ ( β ) + φ ″ ( α ) f ′ ( α ) .

We now make use of Lemma 2.2 to obtain

(3.5) [ φ ‴ ( t ) f ( t ) ] t = α β − [ φ ″ ( t ) f ′ ( t ) ] t = α β + [ φ ′ ( t ) f ″ ( t ) ] t = α β − [ φ ( t ) f ‴ ( t ) ] t = α β = f ( β ) ( β − α ) 3 ∫ α β ( s − α ) 2 ( 3 β − α − 2 s ) υ ( s ) d s − f ( α ) ( β − α ) 3 ∫ α β ( β − s ) 2 ( 3 α − β − 2 s ) υ ( s ) d s − f ′ ( β ) ( β − α ) 2 ∫ α β ( s − α ) 2 ( β − s ) υ ( s ) d s + f ′ ( α ) ( β − α ) 2 ∫ α β ( β − s ) 2 ( s − α ) υ ( s ) d s .

Hence, in view of (3.2), (3.4), and (3.5), we obtain (3.1).□

Remark 3.2

We point out that (3.1) is sharp. Namely, for f ∈ C 4 , − ( I ) , the equality

∫ α β f ( s ) υ ( s ) d s = 1 ( β − α ) 3 f ( β ) ∫ α β ( s − α ) 2 ( 3 β − α − 2 s ) υ ( s ) d s − f ( α ) ∫ α β ( β − s ) 2 ( 3 α − β − 2 s ) υ ( s ) d s − 1 ( β − α ) 2 f ′ ( β ) ∫ α β ( s − α ) 2 ( β − s ) υ ( s ) d s − f ′ ( α ) ∫ α β ( β − s ) 2 ( s − α ) υ ( s ) d s

holds if and only if f ″ ″ ≡ 0 (i.e., f is a polynomial function of degree 3). Indeed, from (3.3), the above equality holds if and only if

∫ α β f ″ ″ ( t ) φ ( t ) d t = 0 .

Since f ″ ″ φ ≤ 0 and f ″ ″ φ is continuous, the above condition is equivalent to f ″ ″ = 0 .

We next consider some examples of weight functions υ ∈ C ( I ) .

Let us assume that υ is symmetric w.r.t. α + β 2 , i.e.,

υ ( α + β − t ) = υ ( t ) , α ≤ t ≤ β .

In this case, by the change in variable x = α + β − s , we obtain

(3.6) ∫ α β ( s − α ) 2 ( 3 β − α − 2 s ) υ ( s ) d s = ∫ α β ( β − x ) 2 ( 3 β − α − 2 α − 2 β + 2 x ) υ ( α + β − x ) d x = ∫ α β ( β − x ) 2 ( β − 3 α + 2 x ) υ ( x ) d x = ∫ α β ( β − s ) 2 ( β − 3 α + 2 s ) υ ( s ) d s ,

which implies that

f ( β ) ∫ α β ( s − α ) 2 ( 3 β − α − 2 s ) υ ( s ) d s − f ( α ) ∫ α β ( β − s ) 2 ( 3 α − β − 2 s ) υ ( s ) d s = ( f ( α ) + f ( β ) ) ∫ α β ( s − α ) 2 ( 3 β − α − 2 s ) υ ( s ) d s = f ( α ) + f ( β ) 2 ∫ α β [ ( s − α ) 2 ( 3 β − α − 2 s ) + ( β − s ) 2 ( β − 3 α + 2 s ) ] υ ( s ) d s = f ( α ) + f ( β ) 2 ( β − α ) 3 ∫ α β υ ( s ) d s .

Then, it holds that

(3.7) 1 ( β − α ) 3 f ( β ) ∫ α β ( s − α ) 2 ( 3 β − α − 2 s ) υ ( s ) d s − f ( α ) ∫ α β ( β − s ) 2 ( 3 α − β − 2 s ) υ ( s ) d s = f ( α ) + f ( β ) 2 ∫ α β υ ( s ) d s .

Similarly, we have

(3.8) ∫ α β ( s − α ) 2 ( β − s ) υ ( s ) d s = ∫ α β ( β − x ) 2 ( x − α ) υ ( α + β − x ) d x = ∫ α β ( β − x ) 2 ( x − α ) υ ( x ) d x = ∫ α β ( β − s ) 2 ( s − α ) υ ( s ) d s ,

which implies that

f ′ ( β ) ∫ α β ( s − α ) 2 ( β − s ) υ ( s ) d s − f ′ ( α ) ∫ α β ( β − s ) 2 ( s − α ) υ ( s ) d s = ( f ′ ( β ) − f ′ ( α ) ) ∫ α β ( s − α ) 2 ( β − s ) υ ( s ) d s = f ′ ( β ) − f ′ ( α ) 2 ∫ α β [ ( s − α ) 2 ( β − s ) + ( β − s ) 2 ( s − α ) ] υ ( s ) d s = f ′ ( β ) − f ′ ( α ) 2 ( β − α ) ∫ α β ( β − s ) ( s − α ) υ ( s ) d s .

Then, it holds that

(3.9) − 1 ( β − α ) 2 f ′ ( β ) ∫ α β ( s − α ) 2 ( β − s ) υ ( s ) d s − f ′ ( α ) ∫ α β ( β − s ) 2 ( s − α ) υ ( s ) d s = − f ′ ( β ) − f ′ ( α ) 2 ( β − α ) ∫ α β ( β − s ) ( s − α ) υ ( s ) d s .

Therefore, from Theorem 3.1, (3.7) and (3.9), we deduce the following result.

Corollary 3.3

Let f ∈ C 4 , − ( I ) and υ ∈ C ( I ) be a nonnegative function. Assume that there exist α , β ∈ I with α < β such that υ is symmetric w.r.t. α + β 2 . Then, it holds that

(3.10) ∫ α β f ( s ) υ ( s ) d s ≤ f ( α ) + f ( β ) 2 ∫ α β υ ( s ) d s − f ′ ( β ) − f ′ ( α ) 2 ( β − α ) ∫ α β ( β − s ) ( s − α ) υ ( s ) d s .

It is interesting to observe that if f ∈ C 4 , − ( I ) is convex (so f ′ is nondecreasing), then from Corollary 3.3, we deduce the following refinement of Fejér’s inequality.

Corollary 3.4

Let f ∈ C 4 , − ( I ) be a convex function and υ ∈ C ( I ) be a nonnegative function. Assume that there exist α , β ∈ I with α < β such that υ is symmetric w.r.t. α + β 2 . Then, it holds that

∫ α β f ( s ) υ ( s ) d s ≤ f ( α ) + f ( β ) 2 ∫ α β υ ( s ) d s − f ′ ( β ) − f ′ ( α ) 2 ( β − α ) ∫ α β ( β − s ) ( s − α ) υ ( s ) d s ≤ f ( α ) + f ( β ) 2 ∫ α β υ ( s ) d s .

The following numerical example illustrates the above result.

Example 3.5

Let I = ] 0 , + ∞ [ . Consider the functions f , υ : I → R defined by

f ( t ) = t 5 2 , υ ( t ) = 3 2 − t 2 , t ∈ I .

Then, f is convex and f ‴ ′ ( t ) = − 15 16 t − 3 2 < 0 . This shows that f ∈ C 4 , − ( I ) . Let α = 1 and β = 2 . For all α ≤ t ≤ β , we have

υ ( α + β − t ) = υ ( 3 − t ) = 3 2 − ( 3 − t ) 2 = t − 3 2 2 = υ ( t ) .

This shows that υ is symmetric w.r.t. α + β 2 . On the other hand, elementary calculations give us that

∫ α β f ( s ) υ ( s ) d s = ∫ 1 2 s 5 2 3 2 − s 2 d s = 136 2 − 73 462 , ∫ α β υ ( s ) d s = ∫ 1 2 3 2 − s 2 d s = 1 12 , ∫ α β ( β − s ) ( s − α ) υ ( s ) d s = ∫ 1 2 ( 2 − s ) ( s − 1 ) 3 2 − s 2 d s = 1 120 , f ( α ) + f ( β ) 2 = f ( 1 ) + f ( 2 ) 2 = 1 + 2 5 2 2 , f ′ ( β ) − f ′ ( α ) 2 ( β − α ) = f ′ ( 2 ) − f ′ ( 1 ) 2 = 5 4 2 3 2 − 1 .

Using the above numerical values, we obtain

∫ α β f ( s ) υ ( s ) d s ≈ 0.25829663308 , f ( α ) + f ( β ) 2 ∫ α β υ ( s ) d s − f ′ ( β ) − f ′ ( α ) 2 ( β − α ) ∫ α β ( β − s ) ( s − α ) υ ( s ) d s ≈ 0.25832281117 , f ( α ) + f ( β ) 2 ∫ α β υ ( s ) d s ≈ 0.27736892706 ,

which confirms the estimates provided by Corollary 3.4.

We now take υ ≡ 1 . In this case, from Corollary 3.3, we deduce the following result.

Corollary 3.6

Let f ∈ C 4 , − ( I ) . Then, for all α , β ∈ I with α < β , it holds that

(3.11) 1 β − α ∫ α β f ( s ) d s ≤ f ( α ) + f ( β ) 2 − f ′ ( β ) − f ′ ( α ) 12 ( β − α ) .

Note that if f ∈ C 4 , − ( I ) is convex, then (3.11) is a refinement of Hermite-Hadamard inequality. Namely, in this case, we obtain the following result.

Corollary 3.7

Let f ∈ C 4 , − ( I ) be a convex function. Then, for all α , β ∈ I with α < β , it holds that

1 β − α ∫ α β f ( s ) d s ≤ f ( α ) + f ( β ) 2 − f ′ ( β ) − f ′ ( α ) 12 ( β − α ) ≤ f ( α ) + f ( β ) 2 .

Consider now the case υ ( t ) = t ξ , t ∈ I = [ 0 , + ∞ [ , where ξ ≥ 0 . In this case, for all β > 0 = α , elementary calculations give us that

∫ α β ( s − α ) 2 ( 3 β − α − 2 s ) υ ( s ) d s = ∫ 0 β ( 3 β − 2 s ) s ξ + 2 d s = ( ξ + 6 ) β ξ + 4 ( ξ + 3 ) ( ξ + 4 ) ∫ α β ( β − s ) 2 ( 3 α − β − 2 s ) υ ( s ) d s = − ∫ 0 β ( β − s ) 2 ( β + 2 s ) s ξ d s = − 6 β ξ + 4 ( ξ + 1 ) ( ξ + 3 ) ( ξ + 4 ) , ∫ α β ( s − α ) 2 ( β − s ) υ ( s ) d s = ∫ 0 β ( β − s ) s ξ + 2 d s = β ξ + 4 ( ξ + 3 ) ( ξ + 4 ) , ∫ α β ( β − s ) 2 ( s − α ) υ ( s ) d s = ∫ 0 β ( β − s ) 2 s ξ + 1 d s = 2 β ξ + 4 ( ξ + 2 ) ( ξ + 3 ) ( ξ + 4 ) .

Then, from Theorem 3.1, we deduce the following result.

Corollary 3.8

Let f ∈ C 4 , − ( [ 0 , + ∞ [ ) and ξ ≥ 0 . Then, for all β > 0 , it holds that

(3.12) 1 β ξ + 1 ∫ 0 β f ( s ) s ξ d s ≤ 6 ( ξ + 1 ) ( ξ + 3 ) ( ξ + 4 ) f ( 0 ) + ( ξ + 6 ) ( ξ + 3 ) ( ξ + 4 ) f ( β ) + 2 β ( ξ + 2 ) ( ξ + 3 ) ( ξ + 4 ) f ′ ( 0 ) − β ( ξ + 3 ) ( ξ + 4 ) f ′ ( β ) .

4 Sub-biharmonic functions on the coordinates

In this section, we introduce the sub-biharmonic functions on the coordinates and obtain new Fejér-type inequalities for this class of functions.

Definition 4.1

Let f = f ( t , s ) : I × I → R be a fourth continuously differentiable function. We say that f is sub-biharmonic on the coordinates, if for all t , s ∈ I , we have

∂ 1 4 f ( t , s ) ≤ 0 , ∂ 2 4 f ( t , s ) ≤ 0 .

We denote by C 4 , − ( I 2 ) the set of fourth continuously differentiable functions that are sub-biharmonic on the coordinates. Namely,

C 4 , − ( I 2 ) = { f ∈ C 4 ( I × I ) : ∂ 1 4 f ≤ 0 , ∂ 2 4 f ≤ 0 } .

We give below some examples of functions f ∈ C 4 , − ( I 2 ) .

Example 4.2

Let f 1 , f 2 ∈ C 4 , − ( I ) . The function

f ( t , s ) = f 1 ( t ) + f 2 ( s ) , ( t , s ) ∈ I × I

belongs to C 4 , − ( I 2 ) .

Example 4.3

Let f 1 , f 2 ∈ C 4 , − ( I ) . If f 1 , f 2 ≥ 0 , then the function

f ( t , s ) = f 1 ( t ) f 2 ( s ) , ( t , s ) ∈ I × I

belongs to C 4 , − ( I 2 ) .

Example 4.4

Let g ∈ C 4 , − ( I ) , where I = R . The function

f ( t , s ) = g ( t s ) , ( t , s ) ∈ I × I

belongs to C 4 , − ( I 2 ) .

We have the following result.

Theorem 4.5

Let f ∈ C 4 , − ( I 2 ) and υ ∈ C ( I × I ) be a nonnegative function. Then, for all j ∈ { 1 , 2 } and α j , β j ∈ I with α j < β j , it holds that

(4.1) ∫ α 1 β 1 ∫ α 2 β 2 f ( t , s ) υ ( t , s ) d s d t ≤ ∫ α 1 β 1 ∫ α 2 β 2 f ( t , β 2 ) ( s − α 2 ) 2 ( 3 β 2 − α 2 − 2 s ) − f ( t , α 2 ) ( β 2 − s ) 2 ( 3 α 2 − β 2 − 2 s ) 2 ( β 2 − α 2 ) 3 υ ( t , s ) d s d t + ∫ α 1 β 1 ∫ α 2 β 2 f ( β 1 , s ) ( t − α 1 ) 2 ( 3 β 1 − α 1 − 2 t ) − f ( α 1 , s ) ( β 1 − t ) 2 ( 3 α 1 − β 1 − 2 t ) 2 ( β 1 − α 1 ) 3 υ ( t , s ) d s d t − ∫ α 1 β 1 ∫ α 2 β 2 ∂ 2 f ( t , β 2 ) ( s − α 2 ) 2 ( β 2 − s ) − ∂ 2 f ( t , α 2 ) ( β 2 − s ) 2 ( s − α 2 ) 2 ( β 2 − α 2 ) 2 υ ( t , s ) d s d t − ∫ α 1 β 1 ∫ α 2 β 2 ∂ 1 f ( β 1 , s ) ( t − α 1 ) 2 ( β 1 − t ) − ∂ 1 f ( α 1 , s ) ( β 1 − t ) 2 ( t − α 1 ) 2 ( β 1 − α 1 ) 2 υ ( t , s ) d s d t .

Proof

For a fixed t ∈ I , let f ( t , ⋅ ) : I ∋ s ↦ f ( t , s ) . Similarly, for a fixed s ∈ I , let f ( ⋅ , s ) : I ∋ t ↦ f ( t , s ) . In a similar way, we define υ ( t , ⋅ ) and υ ( ⋅ , s ) . From the definition of C 4 , − ( I 2 ) , for all t , s ∈ I , one has f ( t , ⋅ ) , f ( ⋅ , s ) ∈ C 4 , − ( I ) . Let α j , β j ∈ I with α j < β j and j ∈ { 1 , 2 } . By Theorem 3.1, for all α 1 ≤ t ≤ β 1 , we have

(4.2) ∫ α 2 β 2 f ( t , s ) υ ( t , s ) d s = ∫ α 2 β 2 f ( t , ⋅ ) ( s ) υ ( t , ⋅ ) ( s ) d s ≤ f ( t , β 2 ) ( β 2 − α 2 ) 3 ∫ α 2 β 2 ( s − α 2 ) 2 ( 3 β 2 − α 2 − 2 s ) υ ( t , s ) d s − f ( t , α 2 ) ( β 2 − α 2 ) 3 ∫ α 2 β 2 ( β 2 − s ) 2 ( 3 α 2 − β 2 − 2 s ) υ ( t , s ) d s − ∂ 2 f ( t , β 2 ) ( β 2 − α 2 ) 2 ∫ α 2 β 2 ( s − α 2 ) 2 ( β 2 − s ) υ ( t , s ) d s − ∂ 2 f ( t , α 2 ) ( β 2 − α 2 ) 2 ∫ α 2 β 2 ( β 2 − s ) 2 ( s − α 2 ) υ ( t , s ) d s .

Integrating w.r.t. t ∈ ] α 1 , β 1 [ , we obtain

(4.3) ∫ α 1 β 1 ∫ α 2 β 2 f ( t , s ) υ ( t , s ) d s d t ≤ ∫ α 1 β 1 f ( t , β 2 ) ( β 2 − α 2 ) 3 ∫ α 2 β 2 ( s − α 2 ) 2 ( 3 β 2 − α 2 − 2 s ) υ ( t , s ) d s d t − ∫ α 1 β 1 f ( t , α 2 ) ( β 2 − α 2 ) 3 ∫ α 2 β 2 ( β 2 − s ) 2 ( 3 α 2 − β 2 − 2 s ) υ ( t , s ) d s d t − ∫ α 1 β 1 ∂ 2 f ( t , β 2 ) ( β 2 − α 2 ) 2 ∫ α 2 β 2 ( s − α 2 ) 2 ( β 2 − s ) υ ( t , s ) d s d t − ∫ α 1 β 1 ∂ 2 f ( t , α 2 ) ( β 2 − α 2 ) 2 ∫ α 2 β 2 ( β 2 − s ) 2 ( s − α 2 ) υ ( t , s ) d s d t .

Similarly, by Theorem 3.1, for all α 2 ≤ s ≤ β 2 , we have

(4.4) ∫ α 1 β 1 f ( t , s ) υ ( t , s ) d t = ∫ α 1 β 1 f ( ⋅ , s ) ( t ) υ ( ⋅ , s ) ( t ) d t ≤ f ( β 1 , s ) ( β 1 − α 1 ) 3 ∫ α 1 β 1 ( τ − α 1 ) 2 ( 3 β 1 − α 1 − 2 τ ) υ ( τ , s ) d τ − f ( α 1 , s ) ( β 1 − α 1 ) 3 ∫ α 1 β 1 ( β 1 − τ ) 2 ( 3 α 1 − β 1 − 2 τ ) υ ( τ , s ) d τ − ∂ 1 f ( β 1 , s ) ( β 1 − α 1 ) 2 ∫ α 1 β 1 ( τ − α 1 ) 2 ( β 1 − τ ) υ ( τ , s ) d τ − ∂ 1 f ( α 1 , s ) ( β 1 − α 1 ) 2 ∫ α 1 β 1 ( β 1 − τ ) 2 ( τ − α 1 ) υ ( τ , s ) d τ .

Integrating w.r.t. s ∈ ] α 2 , β 2 [ , we obtain

(4.5) ∫ α 1 β 1 ∫ α 2 β 2 f ( t , s ) υ ( t , s ) d s d t ≤ ∫ α 2 β 2 f ( β 1 , s ) ( β 1 − α 1 ) 3 ∫ α 1 β 1 ( τ − α 1 ) 2 ( 3 β 1 − α 1 − 2 τ ) υ ( τ , s ) d τ d s − ∫ α 2 β 2 f ( α 1 , s ) ( β 1 − α 1 ) 3 ∫ α 1 β 1 ( β 1 − τ ) 2 ( 3 α 1 − β 1 − 2 τ ) υ ( τ , s ) d τ d s − ∫ α 2 β 2 ∂ 1 f ( β 1 , s ) ( β 1 − α 1 ) 2 ∫ α 1 β 1 ( τ − α 1 ) 2 ( β 1 − τ ) υ ( τ , s ) d τ d s + ∫ α 2 β 2 ∂ 1 f ( α 1 , s ) ( β 1 − α 1 ) 2 ∫ α 1 β 1 ( β 1 − τ ) 2 ( τ − α 1 ) υ ( τ , s ) d τ d s .

Finally, adding (4.3) to (4.5), we obtain (4.1).

Remark 4.6

Note that (4.1) is sharp. Namely, from Remark 3.3, the equality holds in (4.2) if and only if ∂ 2 4 f ≡ 0 . Similarly, the equality holds in (4.4) if and only if ∂ 1 4 f ≡ 0 . Consequently, for all f ∈ C 4 ( I × I ) satisfying

∂ 1 4 f ≡ 0 , ∂ 2 4 f ≡ 0 ,

the equality holds in (4.1), i.e., for any function f of the form

f ( t , s ) = ∑ 0 ≤ i , j ≤ 3 a i j s i t j , t , s ∈ I ,

where the coefficients a i j are constants, the equality holds in (4.1).

Consider now the case when

(4.6) υ ( t , α 2 + β 2 − s ) = υ ( t , s ) , υ ( α 1 + β 1 − t , s ) = υ ( t , s )

for all ( t , s ) ∈ [ α 1 , β 1 ] × [ α 2 , β 2 ] . In this case, by (3.6), one has

∫ α 2 β 2 ( s − α 2 ) 2 ( 3 β 2 − α 2 − 2 s ) υ ( t , s ) d s = ∫ α 2 β 2 ( β 2 − s ) 2 ( β 2 − 3 α 2 + 2 s ) υ ( t , s ) d s .

Hence,

f ( t , β 2 ) ∫ α 2 β 2 ( s − α 2 ) 2 ( 3 β 2 − α 2 − 2 s ) υ ( t , s ) d s − f ( t , α 2 ) ∫ α 2 β 2 ( β 2 − s ) 2 ( 3 α 2 − β 2 − 2 s ) υ ( t , s ) d s = ( f ( t , α 2 ) + f ( t , β 2 ) ) ∫ α 2 β 2 ( s − α 2 ) 2 ( 3 β 2 − α 2 − 2 s ) υ ( t , s ) d s = f ( t , α 2 ) + f ( t , β 2 ) 2 ∫ α 2 β 2 [ ( s − α 2 ) 2 ( 3 β 2 − α 2 − 2 s ) + ( β 2 − s ) 2 ( β 2 − 3 α 2 + 2 s ) ] υ ( t , s ) d s = f ( t , α 2 ) + f ( t , β 2 ) 2 ( β 2 − α 2 ) 3 ∫ α 2 β 2 υ ( t , s ) d s ,

which yields

(4.7) ∫ α 1 β 1 ∫ α 2 β 2 f ( t , β 2 ) ( s − α 2 ) 2 ( 3 β 2 − α 2 − 2 s ) − f ( t , α 2 ) ( β 2 − s ) 2 ( 3 α 2 − β 2 − 2 s ) 2 ( β 2 − α 2 ) 3 υ ( t , s ) d s d t = ∫ α 1 β 1 ∫ α 2 β 2 f ( t , α 2 ) + f ( t , β 2 ) 4 υ ( t , s ) d s d t .

Similarly, we have

(4.8) ∫ α 1 β 1 ∫ α 2 β 2 f ( β 1 , s ) ( t − α 1 ) 2 ( 3 β 1 − α 1 − 2 t ) − f ( α 1 , s ) ( β 1 − t ) 2 ( 3 α 1 − β 1 − 2 t ) 2 ( β 1 − α 1 ) 3 υ ( t , s ) d s d t = ∫ α 1 β 1 ∫ α 2 β 2 f ( α 1 , s ) + f ( β 1 , s ) 4 υ ( t , s ) d s d t .

We now use (3.8) to obtain

∫ α 2 β 2 ( s − α 2 ) 2 ( β 2 − s ) υ ( t , s ) d s = ∫ α 2 β 2 ( β 2 − s ) 2 ( s − α 2 ) υ ( t , s ) d s .

Hence,

∂ 2 f ( t , β 2 ) ∫ α 2 β 2 ( s − α 2 ) 2 ( β 2 − s ) υ ( t , s ) d s − ∂ 2 f ( t , α 2 ) ∫ α 2 β 2 ( β 2 − s ) 2 ( s − α 2 ) υ ( t , s ) d s = 1 2 ( ∂ 2 f ( t , β 2 ) − ∂ 2 f ( t , α 2 ) ) ∫ α 2 β 2 [ ( s − α 2 ) 2 ( β 2 − s ) + ( β 2 − s ) 2 ( s − α 2 ) ] υ ( t , s ) d s = β 2 − α 2 2 ( ∂ 2 f ( t , β 2 ) − ∂ 2 f ( t , α 2 ) ) ∫ α 2 β 2 ( β 2 − s ) ( s − α 2 ) υ ( t , s ) d s ,

which yields

(4.9) − ∫ α 1 β 1 ∫ α 2 β 2 ∂ 2 f ( t , β 2 ) ( s − α 2 ) 2 ( β 2 − s ) − ∂ 2 f ( t , α 2 ) ( β 2 − s ) 2 ( s − α 2 ) 2 ( β 2 − α 2 ) 2 υ ( t , s ) d s d t = − 1 4 ( β 2 − α 2 ) ∫ α 1 β 1 ∫ α 2 β 2 ( ∂ 2 f ( t , β 2 ) − ∂ 2 f ( t , α 2 ) ) ( β 2 − s ) ( s − α 2 ) υ ( t , s ) d s d t .

Similarly, we have

(4.10) − ∫ α 1 β 1 ∫ α 2 β 2 ∂ 1 f ( β 1 , s ) ( t − α 1 ) 2 ( β 1 − t ) − ∂ 1 f ( α 1 , s ) ( β 1 − t ) 2 ( t − α 1 ) 2 ( β 1 − α 1 ) 2 υ ( t , s ) d s d t = − 1 4 ( β 1 − α 1 ) ∫ α 1 β 1 ∫ α 2 β 2 ( ∂ 1 f ( β 1 , s ) − ∂ 1 f ( α 1 , s ) ) ( β 1 − t ) ( t − α 1 ) υ ( t , s ) d s d t .

Thus, from Theorem 4.5, (4.7), (4.8), (4.9), and (4.10), we deduce the following result.

Corollary 4.7

Let f ∈ C 4 , − ( I 2 ) and υ ∈ C ( I × I ) be a nonnegative function. Assume that υ satisfies (4.6) for some α j , β j ∈ I with α j < β j and j ∈ { 1 , 2 } . Then, it holds that

(4.11) ∫ α 1 β 1 ∫ α 2 β 2 f ( t , s ) υ ( t , s ) d s d t ≤ ∫ α 1 β 1 ∫ α 2 β 2 f ( t , α 2 ) + f ( t , β 2 ) + f ( α 1 , s ) + f ( β 1 , s ) 4 υ ( t , s ) d s d t − 1 4 ( β 2 − α 2 ) ∫ α 1 β 1 ∫ α 2 β 2 ( ∂ 2 f ( t , β 2 ) − ∂ 2 f ( t , α 2 ) ) ( β 2 − s ) ( s − α 2 ) υ ( t , s ) d s d t − 1 4 ( β 1 − α 1 ) ∫ α 1 β 1 ∫ α 2 β 2 ( ∂ 1 f ( β 1 , s ) − ∂ 1 f ( α 1 , s ) ) ( β 1 − t ) ( t − α 1 ) υ ( t , s ) d s d t .

Taking υ ≡ 1 in (4.11), we obtain the following result.

Corollary 4.8

Let f ∈ C 4 , − ( I 2 ) . Then, for all j ∈ { 1 , 2 } and α j , β j ∈ I with α j < β j , it holds that

∫ α 1 β 1 ∫ α 2 β 2 f ( t , s ) d s d t ≤ β 2 − α 2 4 ∫ α 1 β 1 ( f ( t , α 2 ) + f ( t , β 2 ) ) d t + β 1 − α 1 4 ∫ α 2 β 2 ( f ( α 1 , s ) + f ( β 1 , s ) ) d s − ( β 2 − α 2 ) 2 24 ∫ α 1 β 1 ( ∂ 2 f ( t , β 2 ) − ∂ 2 f ( t , α 2 ) ) d t + ( β 1 − α 1 ) 2 24 ∫ α 2 β 2 ( ∂ 1 f ( β 1 , s ) − ∂ 1 f ( α 1 , s ) ) d s .

Assume now that f = f ( t , s ) ∈ C 4 , − ( I 2 ) is convex on the coordinates. In this case, for all t , s ∈ I , ∂ 1 f ( ⋅ , s ) and ∂ 2 f ( t , ⋅ ) are nondecreasing functions. In this case, from Corollary 4.8, we deduce the following interesting refinement of Dragomir inequality (1.2).

Corollary 4.9

Let f ∈ C 4 , − ( I 2 ) be convex on the coordinates. Then, for all j ∈ { 1 , 2 } and α j , β j ∈ I with α j < β j , it holds that

∫ α 1 β 1 ∫ α 2 β 2 f ( t , s ) d s d t ≤ β 2 − α 2 4 ∫ α 1 β 1 ( f ( t , α 2 ) + f ( t , β 2 ) ) d t + β 1 − α 1 4 ∫ α 2 β 2 ( f ( α 1 , s ) + f ( β 1 , s ) ) d s − ( β 2 − α 2 ) 2 24 ∫ α 1 β 1 ( ∂ 2 f ( t , β 2 ) − ∂ 2 f ( t , α 2 ) ) d t + ( β 1 − α 1 ) 2 24 ∫ α 2 β 2 ( ∂ 1 f ( β 1 , s ) − ∂ 1 f ( α 1 , s ) ) d s ≤ β 2 − α 2 4 ∫ α 1 β 1 ( f ( t , α 2 ) + f ( t , β 2 ) ) d t + β 1 − α 1 4 ∫ α 2 β 2 ( f ( α 1 , s ) + f ( β 1 , s ) ) d s .

5 Conclusion

Using some tools from ordinary differential equations, new Fejér-type inequalities are derived for the class of sub-biharmonic functions. Namely, we first extended inequality (1.1) to the set of functions f ∈ C 4 , − ( I ) for any nonnegative and continuous weight function υ , without any symmetry condition imposed on υ (see Theorem 3.1). Next we studied some special cases of υ . In particular, if f ∈ C 4 , − ( I ) is convex and υ is symmetric w.r.t. α + β 2 , we obtained a refinement of Fejér’s inequality (1.1) (see Corollary 3.4). Finally, after introducing sub-biharmonic functions on the coordinates, a Fejér-type inequality for this class of functions is obtained for any nonnegative and continuous weight function υ , always without assuming any symmetry condition on υ (see Theorem 4.5). In the special case υ ≡ 1 , a refinement of Dragomir’s inequality (1.2) is obtained for the class of sub-biharmonic and convex functions on the coordinates (see Corollary 4.9).

An interesting question is to try to generalize the obtained results in this study to the class of functions f ∈ C 2 k ( I ) satisfying ( − 1 ) k f ( 2 k ) ≤ 0 , where k is a positive integer and f ( 2 k ) is the derivative of order k of f .

  1. Funding information: The author was supported by Researchers Supporting Project number (RSP2024R57), King Saud University, Riyadh, Saudi Arabia.

  2. Author contributions: The author confirms sole responsibility for the conception of the study, presented results, and manuscript preparation.

  3. Conflict of interest: The author states no conflicts of interest.

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

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Received: 2023-12-24
Revised: 2024-05-27
Accepted: 2024-07-01
Published Online: 2024-10-14

© 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-2024-0035
Keywords for this article
Fejér’s inequality; the Hermite-Hadamard inequality; convex function; sub-biharmonic function
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  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  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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