Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension

Silvestru Sever Dragomir; Mohamed Jleli; Bessem Samet

doi:10.1515/math-2024-0028

Article Open Access

Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension

Silvestru Sever Dragomir , Mohamed Jleli and Bessem Samet

Published/Copyright: August 2, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we establish Hermite-Hadamard-type inequalities for the two classes of functions X ± λ ( Ω ) = { f ∈ C 2 ( Ω ) : Δ f ± λ f ≥ 0 } , where λ > 0 and Ω is an open subset of R 2 . We also obtain a characterization of the set X − λ ( Ω ) . Notice that in the one-dimensional case, if Ω = I (an open interval of R ) and λ = ρ 2 , ρ > 0 , then X λ ( Ω ) (resp. X − λ ( Ω ) ) reduces to the class of functions f ∈ C 2 ( I ) such that f is trigonometrically ρ -convex (resp. hyperbolic ρ -convex) on I .

Keywords: Hermite-Hadamard-type inequalities; trigonometrically ρ-convex functions; hyperbolic ρ-convex functions; Bessel functions

MSC 2010: 26B25; 26A51; 26D15; 35A23

1 Introduction

Convex functions are widely used in various branches of mathematics, such as functional analysis, optimization theory, and numerical analysis. One of the important inequalities related to convex functions is the (right-side) Hermite-Hadamard inequality [1,2] that can be stated as follows: Let σ 1 < σ 2 and f : [ σ 1 , σ 2 ] → R be a convex function. Then

(1.1) 1 σ 2 − σ 1 ∫ σ 1 σ 2 f ( x ) d x ≤ ∑ j = 1 2 f ( σ j ) 2 .

Inequality (1.1) has been refined and generalized in various directions. For more details, we refer to the monograph [3] collecting many results in that direction.

In [4], among other results, the first author extended inequality (1.1) to the class of convex functions on a disk. Namely, he proved that, if f : B ( C , R ) ¯ → R is convex, then

(1.2) 1 π R 2 ∫ B ( C , R ) f ( z ) d z ≤ 1 2 π R ∫ ∂ B ( C , R ) f ( γ ) d l ( γ ) ,

where B ( C , R ) is the open ball of R 2 with center C ∈ R 2 and radius R > 0 , ∂ B ( C , R ) is the boundary of B ( C , R ) and B ( C , R ) ¯ is the closure of B ( C , R ) . For further results related to Hermite-Hadamard-type inequalities in higher dimensions, see [5–11] and the references therein.

Let ρ > 0 and f : I → R , where I is an interval of R . We say that f is a trigonometrically ρ -convex function on I [12], if for all a , b ∈ I with 0 < b − a < π ρ , we have

f ( x ) ≤ sin [ ρ ( b − x ) ] sin [ ρ ( b − a ) ] f ( a ) + sin [ ρ ( x − a ) ] sin [ ρ ( b − a ) ] f ( b ) , a ≤ x ≤ b .

In [13], the first author proved (among many other results) that, if f : I → R is a trigonometrically ρ -convex function on I , then for all a , b ∈ I with 0 < b − a < π ρ , we have

∫ a b f ( x ) d x ≤ f ( a ) + f ( b ) ρ tan ρ ( b − a ) 2 .

For other results related to trigonometrically ρ -convex functions, see [14–17] and the references therein.

Let ρ > 0 and f : I → R . Assume that for all a , b ∈ I with a < b , we have

f ( x ) ≤ sinh [ ρ ( b − x ) ] sinh [ ρ ( b − a ) ] f ( a ) + sinh [ ρ ( x − a ) ] sinh [ ρ ( b − a ) ] f ( b ) , a ≤ x ≤ b .

In this case, the function f is said to be a hyperbolic ρ -convex function on I [18]. The first author in [19] established that if f : I → R is a hyperbolic ρ -convex function on I , then for all a , b ∈ I with a < b , we have

∫ a b f ( x ) d x ≤ f ( a ) + f ( b ) ρ tanh ρ ( b − a ) 2 .

We also refer to [20,21] for further results related to hyperbolic ρ -convex functions.

On the other hand, it was shown in [12] that f ∈ C 2 ( I ) is trigonometrically ρ -convex on I if and only if f satisfies the second-order differential inequality:

(1.3) f ″ + ρ 2 f ≥ 0 in I .

In [18], it was proven that f ∈ C 2 ( I ) is hyperbolic ρ -convex on I if and only if f satisfies the second-order differential inequality:

(1.4) f ″ − ρ 2 f ≥ 0 in I .

In this article, motivated by the aforementioned cited works, we establish Hermite-Hadamard-type inequalities for the two classes of functions

X ± λ ( Ω ) = { f ∈ C 2 ( Ω ) : Δ f ± λ f ≥ 0 } ,

where λ > 0 , Ω is an open subset of R 2 , and Δ denotes, as usual, the Laplacian operator. Moreover, we prove a characterization of the set X − λ ( Ω ) . Remark that in the one-dimensional case, by (1.3), if Ω = I and λ = ρ 2 , ρ > 0 , then X λ ( Ω ) reduces to the class of functions f ∈ C 2 ( I ) such that f is trigonometrically ρ -convex on I . Similarly, in the one-dimensional case, by (1.4), if Ω = I and λ = ρ 2 , ρ > 0 , then X − λ ( Ω ) reduces to the class of functions f ∈ C 2 ( I ) , such that f is hyperbolic ρ -convex on I .

The structure of the rest of the article is as follows. In Section 2, we study some properties of the sets X ± λ ( Ω ) . In Section 3, we derive Hermite-Hadamard-type inequalities for both classes of functions X ± λ ( Ω ) . Finally, in Section 5, we establish a characterization of the set X − λ ( Ω ) .

2 The classes of functions X ± λ ( Ω )

Throughout this article, by Ω , we mean an open subset of R 2 . For λ > 0 , let

X ± λ ( Ω ) = { f ∈ C 2 ( Ω ) : Δ f ± λ f ≥ 0 } .

We also consider the class of functions

X 0 ( Ω ) = { f ∈ C 2 ( Ω ) : Δ f ≥ 0 } .

In this section, we provide some properties of the above classes of functions.

Proposition 2.1

Let λ > 0 and 0 ≤ θ ≤ 1 . The following properties hold:

⋂ μ > 0 X + μ ( Ω ) ⊂ X 0 ( Ω ) .
f ∈ X 0 ( Ω ) , f ≥ 0 ⇒ f ∈ ⋂ μ > 0 X + μ ( Ω ) .
f ∈ X + λ ( Ω ) , f ≥ 0 ⇒ f ∈ ⋂ μ ≥ λ X + μ ( Ω ) .
f ∈ X + λ ( Ω ) , α ≥ 0 ( α ∈ R ) ⇒ α f ∈ X + λ ( Ω ) .
f , g ∈ X + λ ( Ω ) ⇒ f + g ∈ X + λ ( Ω ) .
f ∈ X θ λ ( Ω ) , g ∈ X ( 1 − θ ) λ ( Ω ) , f ≥ 0 , g ≥ 0 , ∇ f ⋅ ∇ g ≥ 0 ⇒ f g ∈ X + λ ( Ω ) , where ∇ is the gradient operator and ⋅ is the inner product in R 2 .

Proof

We only prove (i) and (vi). The other properties are immediate, so we omit the details.

(i) Let f ∈ ⋂ μ > 0 X + μ ( Ω ) . If z ∈ Ω is fixed, then for all μ > 0 , we have

Δ f ( z ) + μ f ( z ) ≥ 0 .

Passing to the limit as μ → 0 + , we obtain Δ f ( z ) ≥ 0 , which shows that f ∈ X 0 ( Ω ) .

(vi) Let f ∈ X θ λ ( Ω ) and g ∈ X ( 1 − θ ) λ ( Ω ) be such that

f ≥ 0 , g ≥ 0 , ∇ f ⋅ ∇ g ≥ 0 .

By the property

Δ ( f g ) = f Δ g + g Δ f + 2 ∇ f ⋅ ∇ g ,

we obtain

Δ ( f g ) + λ f g = Δ ( f g ) + λ θ f g + ( 1 − θ ) λ f g = f ( Δ g + ( 1 − θ ) λ g ) + g ( Δ f + λ θ f ) + 2 ∇ f ⋅ ∇ g ≥ 0 ,

which shows that f g ∈ X + λ ( Ω ) .□

Similarly, we have the following properties.

Proposition 2.2

Let λ > 0 and 0 ≤ θ ≤ 1 . The following properties hold:

⋂ μ > 0 X − μ ( Ω ) ⊂ X 0 ( Ω ) .
f ∈ X 0 ( Ω ) , f ≤ 0 ⇒ f ∈ ⋂ μ > 0 X − μ ( Ω ) .
f ∈ X − λ ( Ω ) , f ≥ 0 ⇒ f ∈ ⋂ 0 ≤ μ ≤ λ X + μ ( Ω ) .
f ∈ X − λ ( Ω ) , α ≥ 0 ( α ∈ R ) ⇒ α f ∈ X − λ ( Ω ) .
f , g ∈ X − λ ( Ω ) ⇒ f + g ∈ X − λ ( Ω ) .
f ∈ X − θ λ ( Ω ) , g ∈ X − ( 1 − θ ) λ ( Ω ) , f ≥ 0 , g ≥ 0 , ∇ f ⋅ ∇ g ≥ 0 ⇒ f g ∈ X − λ ( Ω ) .

3 Hermite-Hadamard-type inequalities on a disk

In this section, we are concerned with the extension of inequality (1.2) to the classes of functions X ± λ ( Ω ) . We need to recall some properties related to Bessel functions. For more details about such functions, we refer to [22].

3.1 Some preliminaries

For all n ∈ N ( n is a natural number), let J n be the Bessel function of the first kind of order n . This function can be written as an infinite power series as follows:

J n ( x ) = ∑ ι = 0 ∞ ( − 1 ) ι ι ! ( n + ι ) ! x 2 n + 2 ι .

Let j n , 1 be the first positive zero of J n .

Lemma 3.1

Let n ∈ N . We have the following properties:

The function J n satisfies the Bessel differential equation:
x 2 J n ″ ( x ) + x J n ′ ( x ) + ( x 2 − n 2 ) J n ( x ) = 0 .
J n ′ ( x ) = − J n + 1 ( x ) + n x J n ( x ) .
J n ( x ) ∼ 1 n ! x 2 n as x → 0 .
The function J 0 is decreasing in the interval [ 0 , j 0 , 1 ] .

Similarly, for all n ∈ N , let I n be the modified Bessel function of the first kind of order n . This function can be written as an infinite power series as follows:

I n ( x ) = x 2 n ∑ ι = 0 ∞ 1 ι ! ( n + ι ) ! x 2 4 ι .

Lemma 3.2

Let n ∈ N . We have the following properties:

The function I n satisfies the modified Bessel differential equation:
x 2 I n ″ ( x ) + x I n ′ ( x ) − ( x 2 + n 2 ) I n ( x ) = 0 .
I 0 ′ ( x ) = I 1 ( x ) .
I n ( x ) ∼ 1 n ! x 2 n as x → 0 .

3.2 The class of functions X + λ ( Ω )

The following result is an extension of inequality (1.2) to the class of functions X + λ ( Ω ) .

Theorem 3.3

Let λ > 0 and f ∈ X + λ ( Ω ) . Assume that there exist C ∈ Ω and R ∈ 0 , j 0 , 1 λ such that B ( C , R ) ¯ ⊂ Ω . Then, the following inequality holds:

(3.1) ∫ B ( C , R ) f ( z ) d z ≤ J 1 ( λ R ) λ J 0 ( λ R ) ∫ ∂ B ( C , R ) f ( γ ) d l ( γ ) .

Proof

Let C ∈ Ω and R ∈ 0 , j 0 , 1 λ be such that B ( C , R ) ¯ ⊂ Ω . We introduce the function

(3.2) ξ ( z ) = 1 λ J 0 ( λ ‖ z − C ‖ ) J 0 ( λ R ) − 1 , z ∈ B ( C , R ) ¯ .

We first establish some properties of the above function. Since 0 < R ≤ j 0 , 1 λ , then J 0 ( λ R ) ≠ 0 , and the function ξ is well defined. Taking ‖ z − C ‖ = R in (3.2), we obtain

(3.3) ξ ( z ) = 0 , z ∈ ∂ B ( C , R ) .

On the other hand, from the property (iv) in Lemma 3.1, we deduce that for all z ∈ B ( C , R ) ¯ ,

J 0 ( λ ‖ z − C ‖ ) ≤ J 0 ( λ R ) ,

which shows that

(3.4) ξ ( z ) ≥ 0 , z ∈ B ( C , R ) ¯ .

For all 0 ≤ r ≤ R , let

ξ ˜ ( r ) = 1 λ J 0 ( λ r ) J 0 ( λ R ) − 1 .

Since ξ is a radial function, that is,

ξ ( z ) = ξ ˜ ( r ) , r = ‖ z − C ‖ ,

then for all z ∈ B ( C , R ) , we have

(3.5) Δ ξ ( z ) = ζ ˜ ″ ( r ) + 1 r ξ ˜ ′ ( r ) .

Furthermore, we have

(3.6) ξ ˜ ′ ( r ) = 1 λ J 0 ′ ( λ r ) J 0 ( λ R )

and

ξ ˜ ″ ( r ) = J 0 ″ ( λ r ) J 0 ( λ R ) ,

which imply by (3.5) that

Δ ξ ( z ) = J 0 ″ ( λ r ) J 0 ( λ R ) + 1 λ r J 0 ′ ( λ r ) J 0 ( λ R ) = 1 J 0 ( λ R ) ( λ r ) 2 ( ( λ r ) 2 J 0 ″ ( λ r ) + λ r J 0 ′ ( λ r ) ) .

Then, from the property (i) in Lemma 3.1, we deduce that

Δ ξ ( z ) = 1 J 0 ( λ R ) ( λ r ) 2 ( − ( λ r ) 2 J 0 ( λ r ) ) = − J 0 ( λ r ) J 0 ( λ R ) = − 1 − J 0 ( λ r ) J 0 ( λ R ) − 1 = − 1 − λ ξ ˜ ( r ) = − 1 − λ ξ ( z ) ,

that is,

(3.7) Δ ξ ( z ) + λ ξ ( z ) = − 1 , z ∈ B ( C , R ) .

We now use (3.7) to obtain

(3.8) ∫ B ( C , R ) f ( z ) d z = − ∫ B ( C , R ) Δ ξ ( z ) f ( z ) d z − λ ∫ B ( C , R ) f ( z ) ξ ( z ) d z .

Moreover, thanks to Green’s formula, we have

(3.9) − ∫ B ( C , R ) Δ ξ ( z ) f ( z ) d z = ∫ B ( C , R ) ∇ f ( z ) ⋅ ∇ ξ ( z ) d z − ξ ˜ ′ ( R ) ∫ ∂ B ( C , R ) f ( γ ) d l ( γ ) .

By using again Green’s formula together with (3.3), we obtain

(3.10) − ∫ B ( C , R ) Δ f ( z ) ξ ( z ) d z = ∫ B ( C , R ) ∇ f ( z ) ⋅ ∇ ξ ( z ) d z .

Then, it follows from (3.8), (3.9), and (3.10) that

∫ B ( C , R ) f ( z ) d z = − ξ ˜ ′ ( R ) ∫ ∂ B ( C , R ) f ( γ ) d l ( γ ) − ∫ B ( C , R ) ( Δ f ( z ) + λ f ( z ) ) ξ ( z ) d z .

We also have by (3.6) and the property (ii) of Lemma 3.1 that

ξ ˜ ′ ( R ) = − 1 λ J 1 ( λ R ) J 0 ( λ R ) .

Therefore, it holds that

(3.11) ∫ B ( C , R ) f ( z ) d z = 1 λ J 1 ( λ R ) J 0 ( λ R ) ∫ ∂ B ( C , R ) f ( γ ) d l ( γ ) − ∫ B ( C , R ) ( Δ f ( z ) + λ f ( z ) ) ξ ( z ) d z .

On the other hand, due to (3.4) and since f ∈ X + λ ( Ω ) , we have

(3.12) ∫ B ( C , R ) ( Δ f ( z ) + λ f ( z ) ) ξ ( z ) d z ≥ 0 .

Finally, (3.1) follows from (3.11) and (3.12).□

Remark 3.4

By the property (iii) in Lemma 3.1, one has

lim λ → 0 + J 1 ( λ R ) λ J 0 ( λ R ) = R 2 .

Hence, if λ → 0 + , (3.1) reduces to (1.2).

3.3 The class of functions X ‒ λ ( Ω )

The following result is an extension of inequality (1.2) to the class of functions X − λ ( Ω )

Theorem 3.5

Let λ > 0 and f ∈ X − λ ( Ω ) . Then, for all C ∈ Ω and R > 0 with B ( C , R ) ¯ ⊂ Ω , we have

(3.13) ∫ B ( C , R ) f ( z ) d z ≤ I 1 ( λ R ) λ I 0 ( λ R ) ∫ ∂ B ( C , R ) f ( γ ) d l ( γ ) .

Proof

Let C ∈ Ω and R > 0 be such that B ( C , R ) ¯ ⊂ Ω . We consider the function

(3.14) ζ ( z ) = 1 λ 1 − I 0 ( λ ‖ z − C ‖ ) I 0 ( λ R ) , z ∈ B ( C , R ) ¯ .

Taking ‖ z − C ‖ = R in (3.14), we obtain

(3.15) ζ ( z ) = 0 , z ∈ ∂ B ( C , R ) .

On the other hand, from the property (ii) in Lemma 3.2, the function I 0 is nondecreasing in the interval [ 0 , + ∞ ] . Hence, for all z ∈ B ( C , R ) ¯ , we have

I 0 ( λ ‖ z − C ‖ ) ≤ I 0 ( λ R ) ,

which shows that

(3.16) ζ ( z ) ≥ 0 , z ∈ B ( C , R ) ¯ .

For all 0 ≤ r ≤ R , let

ζ ˜ ( r ) = 1 λ 1 − I 0 ( λ r ) I 0 ( λ R ) .

Since ζ is a radial function, that is,

ζ ( z ) = ζ ˜ ( r ) , r = ‖ z − C ‖ ,

then for all z ∈ B ( C , R ) , we have

(3.17) Δ ζ ( z ) = ζ ˜ ″ ( r ) + 1 r ζ ˜ ′ ( r ) .

Furthermore, we have

(3.18) ζ ˜ ′ ( r ) = − 1 λ I 0 ′ ( λ r ) I 0 ( λ R )

and

ζ ˜ ″ ( r ) = − I 0 ″ ( λ r ) I 0 ( λ R ) ,

which imply by (3.17) that

Δ ξ ( z ) = − I 0 ″ ( λ r ) I 0 ( λ R ) − 1 λ r I 0 ′ ( λ r ) I 0 ( λ R ) = − 1 I 0 ( λ R ) ( λ r ) 2 ( ( λ r ) 2 I 0 ″ ( λ r ) + λ r I 0 ′ ( λ r ) ) .

Then, from the property (i) in Lemma 3.2, we deduce that

Δ ζ ( z ) = − 1 J 0 ( λ R ) ( λ r ) 2 ( ( λ r ) 2 I 0 ( λ r ) ) = − I 0 ( λ r ) I 0 ( λ R ) = − 1 + 1 − I 0 ( λ r ) I 0 ( λ R ) = − 1 + λ ζ ˜ ( r ) = − 1 + λ ζ ( z ) ,

that is,

(3.19) Δ ζ ( z ) − λ ζ ( z ) = − 1 , z ∈ B ( C , R ) .

Then, in view of (3.19), we have

(3.20) ∫ B ( C , R ) f ( z ) d z = − ∫ B ( C , R ) Δ ζ ( z ) f ( z ) d z + λ ∫ B ( C , R ) f ( z ) ζ ( z ) d z .

On the other hand, by Green’s formula and (3.15), we have

(3.21) − ∫ B ( C , R ) Δ ζ ( z ) f ( z ) d z = − ∫ B ( C , R ) Δ f ( z ) ζ ( z ) d z − ζ ˜ ′ ( R ) ∫ ∂ B ( C , R ) f ( γ ) d l ( γ ) .

Combining (3.20) with (3.21), we obtain

(3.22) ∫ B ( C , R ) f ( z ) d z = − ζ ˜ ′ ( R ) ∫ ∂ B ( C , R ) f ( γ ) d l ( γ ) − ∫ B ( C , R ) ( Δ f ( z ) − λ f ( z ) ) ζ ( z ) d z .

Making use of (3.18) and the property (ii) in Lemma 3.2, we obtain

ζ ˜ ′ ( R ) = − 1 λ I 1 ( λ R ) I 0 ( λ R ) ,

which implies together with (3.22) that

∫ B ( C , R ) f ( z ) d z = I 1 ( λ R ) λ I 0 ( λ R ) ∫ ∂ B ( C , R ) f ( γ ) d l ( γ ) − ∫ B ( C , R ) ( Δ f ( z ) − λ f ( z ) ) ζ ( z ) d z .

Finally, since f ∈ X − λ ( Ω ) and ζ ≥ 0 by (3.16), the aforementioned inequality yields (3.13).□

Remark 3.6

By the property (iii) in Lemma 3.2, one has

lim λ → 0 + I 1 ( λ R ) λ I 0 ( λ R ) = R 2 .

Hence, if λ → 0 + , (3.13) reduces to (1.2).

Let Ω i , i = 1 , 2 , be two open subsets of R 2 . For λ = ( λ 1 , λ 2 ) ∈ R 2 with λ i > 0 , i = 1 , 2 , we consider the class of functions

X − ( λ 1 , λ 2 ) ( Ω 1 × Ω 2 ) = { f = f ( x , y ) ∈ C 2 ( Ω 1 × Ω 2 ) : Δ x f − λ 1 f ≥ 0 , Δ y f − λ 2 f ≥ 0 } ,

where for all x = ( x 1 , x 2 ) ∈ Ω 1 and y = ( y 1 , y 2 ) ∈ Ω 2 ,

Δ x f ( x , y ) = ∂ 2 f ∂ x 1 2 ( x , y ) + ∂ 2 f ∂ x 2 2 ( x , y )

and

Δ y f ( x , y ) = ∂ 2 f ∂ y 1 2 ( x , y ) + ∂ 2 f ∂ y 2 2 ( x , y ) .

Remark that f = f ( x , y ) ∈ X − ( λ 1 , λ 2 ) ( Ω 1 × Ω 2 ) means that for all ( x , y ) ∈ Ω 1 × Ω 2 ,

f ( ⋅ , y ) ∈ X − λ 1 ( Ω 1 ) and f ( x , ⋅ ) ∈ X − λ 2 ( Ω 2 ) ,

where

f ( ⋅ , y ) : Ω 1 ∋ x ↦ f ( x , y ) ∈ R and f ( x , ⋅ ) : Ω 2 ∋ y ↦ f ( x , y ) ∈ R .

From Theorem 3.5, we deduce the following Hermite-Hadamard-type inequalities for the class of functions X − ( λ 1 , λ 2 ) ( Ω 1 × Ω 2 ) .

Corollary 3.7

Let Ω i , i = 1 , 2 , be two open subsets of R 2 and λ = ( λ 1 , λ 2 ) ∈ R 2 with λ i > 0 , i = 1 , 2 . If f ∈ X − ( λ 1 , λ 2 ) ( Ω 1 × Ω 2 ) , then for all ( C 1 , C 2 ) ∈ Ω 1 × Ω 2 and R 1 , R 2 > 0 with B ( C i , R i ) ¯ ⊂ Ω i , i = 1 , 2 , we have

(3.23) ∫ B ( C 1 , R 1 ) × B ( C 2 , R 2 ) f ( x , y ) d x d y ≤ I 1 ( λ 1 R 1 ) 2 λ 1 I 0 ( λ 1 R 1 ) ∫ ∂ B ( C 1 , R 1 ) ∫ B ( C 2 , R 2 ) f ( γ , y ) d y d l ( γ ) + I 1 ( λ 2 R 2 ) 2 λ 2 I 0 ( λ 2 R 2 ) ∫ ∂ B ( C 2 , R 2 ) ∫ B ( C 1 , R 1 ) f ( x , σ ) d x d l ( σ ) ≤ I 1 ( λ 1 R 1 ) I 1 ( λ 2 R 2 ) λ 1 λ 2 I 0 ( λ 1 R 1 ) I 0 ( λ 2 R 2 ) ∫ ∂ B ( C 1 , R 1 ) × ∂ B ( C 2 , R 2 ) f ( γ , σ ) d l ( γ ) d l ( σ ) .

Proof

Let R i > 0 be such that B ( C i , R i ) ¯ ⊂ Ω i , i = 1 , 2 . Let y ∈ Ω 2 be fixed. Since f ( ⋅ , y ) ∈ X − λ 1 ( Ω 1 ) , then by Theorem 3.5 we have

∫ B ( C 1 , R 1 ) f ( x , y ) d x ≤ I 1 ( λ 1 R 1 ) λ 1 I 0 ( λ 1 R 1 ) ∫ ∂ B ( C 1 , R 1 ) f ( γ , y ) d l ( γ ) .

Integrating the aforementioned inequality over y ∈ B ( C 2 , R 2 ) , we obtain

(3.24) ∫ B ( C 1 , R 1 ) × B ( C 2 , R 2 ) f ( x , y ) d x d y ≤ I 1 ( λ 1 R 1 ) λ 1 I 0 ( λ 1 R 1 ) ∫ ∂ B ( C 1 , R 1 ) ∫ B ( C 2 , R 2 ) f ( γ , y ) d y d l ( γ ) .

Similarly, for a fixed x ∈ Ω 1 , since f ( x , ⋅ ) ∈ X − λ 2 ( Ω 2 ) , then by Theorem 3.5, we have

∫ B ( C 2 , R 2 ) f ( x , y ) d y ≤ I 1 ( λ 2 R 2 ) λ 2 I 0 ( λ 2 R 2 ) ∫ ∂ B ( C 2 , R 2 ) f ( x , σ ) d l ( σ ) .

Integrating the aforementioned inequality over x ∈ B ( C 1 , R 1 ) , we obtain

(3.25) ∫ B ( C 1 , R 1 ) × B ( C 2 , R 2 ) f ( x , y ) d x d y ≤ I 1 ( λ 2 R 2 ) λ 2 I 0 ( λ 2 R 2 ) ∫ ∂ B ( C 2 , R 2 ) ∫ B ( C 1 , R 1 ) f ( x , σ ) d x d l ( σ ) .

By summing (3.24) and (3.25), we obtain

∫ B ( C 1 , R 1 ) × B ( C 2 , R 2 ) f ( x , y ) d x d y ≤ I 1 ( λ 1 R 1 ) 2 λ 1 I 0 ( λ 1 R 1 ) ∫ ∂ B ( C 1 , R 1 ) ∫ B ( C 2 , R 2 ) f ( γ , y ) d y d l ( γ ) + I 1 ( λ 2 R 2 ) 2 λ 2 I 0 ( λ 2 R 2 ) ∫ ∂ B ( C 2 , R 2 ) ∫ B ( C 1 , R 1 ) f ( x , σ ) d x d l ( σ ) ,

which proves the first inequality in (3.23). On the other hand, since for all γ ∈ ∂ B ( C 1 , R 1 ) , the function f ( γ , ⋅ ) ∈ X − λ 2 ( Ω 2 ) , then by Theorem 3.5, we have

∫ B ( C 2 , R 2 ) f ( γ , y ) d y ≤ I 1 ( λ 2 R 2 ) λ 2 I 0 ( λ 2 R 2 ) ∫ ∂ B ( C 2 , R 2 ) f ( γ , σ ) d l ( σ ) .

Integrating the aforementioned inequality over γ ∈ ∂ B ( C 1 , R 1 ) , we obtain

(3.26) ∫ ∂ B ( C 1 , R 1 ) ∫ B ( C 2 , R 2 ) f ( γ , y ) d y d l ( γ ) ≤ I 1 ( λ 2 R 2 ) λ 2 I 0 ( λ 2 R 2 ) ∫ ∂ B ( C 1 , R 1 ) × ∂ B ( C 2 , R 2 ) f ( γ , σ ) d l ( γ ) d l ( σ ) .

Similarly, since for all σ ∈ ∂ B ( C 2 , R 2 ) , the function f ( ⋅ , σ ) ∈ X − λ 1 ( Ω 1 ) , then by Theorem 3.5, we have

∫ B ( C 1 , R 1 ) f ( x , σ ) d x ≤ I 1 ( λ 1 R 1 ) λ 1 I 0 ( λ 1 R 1 ) ∫ ∂ B ( C 1 , R 1 ) f ( γ , σ ) d l ( γ ) ,

which implies after integration over σ ∈ ∂ B ( C 2 , R 2 ) that

(3.27) ∫ ∂ B ( C 2 , R 2 ) ∫ B ( C 1 , R 1 ) f ( x , σ ) d x d l ( σ ) ≤ I 1 ( λ 1 R 1 ) λ 1 I 0 ( λ 1 R 1 ) ∫ ∂ B ( C 1 , R 1 ) × ∂ B ( C 2 , R 2 ) f ( γ , σ ) d l ( γ ) d l ( σ ) .

Then, it follows from (3.26) and (3.27) that

I 1 ( λ 1 R 1 ) 2 λ 1 I 0 ( λ 1 R 1 ) ∫ ∂ B ( C 1 , R 1 ) ∫ B ( C 2 , R 2 ) f ( γ , y ) d y d l ( γ ) + I 1 ( λ 2 R 2 ) 2 λ 2 I 0 ( λ 2 R 2 ) ∫ ∂ B ( C 2 , R 2 ) ∫ B ( C 1 , R 1 ) f ( x , σ ) d x d l ( σ ) ≤ I 1 ( λ 1 R 1 ) I 1 ( λ 2 R 2 ) λ 1 λ 2 I 0 ( λ 1 R 1 ) I 0 ( λ 2 R 2 ) ∫ ∂ B ( C 1 , R 1 ) × ∂ B ( C 2 , R 2 ) f ( γ , σ ) d l ( γ ) d l ( σ ) ,

which proves the second inequality in (3.23).□

4 Characterization of the class of functions X ‒ λ ( Ω )

In this section, we show that inequality (3.13) provides a characterization of the class of functions X − λ ( Ω ) . Namely, we have the following result.

Theorem 4.1

Let f ∈ C 2 ( Ω ) and λ > 0 . Assume that for all C ∈ Ω and R > 0 with B ( C , R ) ¯ ⊂ Ω , (3.13) holds. Then f ∈ X − λ ( Ω ) .

Proof

Let C ∈ Ω and ε > 0 small enough so that B ( C , ε ) ¯ ⊂ Ω . Let

ζ ε ( z ) = ζ ε ˜ ( r ) = 1 λ 1 − I 0 ( λ r ) I 0 ( λ ε ) , z ∈ B ( C , ε ) ¯ , r = ‖ z − C ‖ .

From the proof of Theorem 3.5, we have

(4.1) ζ ε ( z ) = 0 , z ∈ ∂ B ( C , ε ) ,

(4.2) ζ ε ( z ) ≥ 0 , z ∈ B ( C , ε ) ¯ ,

(4.3) Δ ζ ε ( z ) − λ ζ ε ( z ) = − 1 , z ∈ B ( C , ε ) ,

(4.4) ζ ε ˜ ′ ( ε ) = − 1 λ I 1 ( λ ε ) I 0 ( λ ε ) .

Making use of (5.1), (5.3), (5.4), and using two times Green’s formula, we obtain

∫ B ( C , ε ) f ( z ) d z = − ∫ B ( C , ε ) Δ ζ ε ( z ) f ( z ) d z + λ ∫ B ( C , ε ) f ( z ) ζ ε ( z ) d z = − ∫ B ( C , ε ) ( Δ f ( z ) − λ f ( z ) ) ζ ε ( z ) d z − ζ ε ˜ ′ ( ε ) ∫ ∂ B ( C , ε ) f ( γ ) d l ( γ ) = − ∫ B ( C , ε ) ( Δ f ( z ) − λ f ( z ) ) ζ ε ( z ) d z + I 1 ( λ ε ) λ I 0 ( λ ε ) ∫ ∂ B ( C , ε ) f ( γ ) d l ( γ ) ,

that is,

∫ B ( C , ε ) f ( z ) d z − I 1 ( λ ε ) λ I 0 ( λ ε ) ∫ ∂ B ( C , ε ) f ( γ ) d l ( γ ) = − ∫ B ( C , ε ) ( Δ f ( z ) − λ f ( z ) ) ζ ε ( z ) d z .

Then, from (3.13) with R = ε , we deduce that

− ∫ B ( C , ε ) ( Δ f ( z ) − λ f ( z ) ) ζ ε ( z ) d z ≤ 0 .

Since ζ ε ≥ 0 by (5.2), we deduce the existence of z ε ∈ B ( C , ε ) so that

Δ f ( z ε ) − λ f ( z ε ) ≥ 0 .

Furthermore, since f ∈ C 2 ( Ω ) , passing to the limit as ε → 0 + in the aforementioned inequality, we obtain

Δ f ( C ) − λ f ( C ) ≥ 0 .

Since C is an arbitrary point of Ω , it holds that Δ f − λ f ≥ 0 in Ω , which shows that f ∈ X − λ ( Ω ) .□

5 Characterization of the class of functions X ‒ λ ( Ω )

In this section, we show that inequality (3.13) provides a characterization of the class of functions X − λ ( Ω ) . Namely, we have the following result.

Theorem 5.1

Let f ∈ C 2 ( Ω ) and λ > 0 . Assume that for all C ∈ Ω and R > 0 with B ( C , R ) ¯ ⊂ Ω , (3.13) holds. Then f ∈ X − λ ( Ω ) .

Proof

Let C ∈ Ω and ε > 0 small enough so that B ( C , ε ) ¯ ⊂ Ω . Let

ζ ε ( z ) = ζ ε ˜ ( r ) = 1 λ 1 − I 0 ( λ r ) I 0 ( λ ε ) , z ∈ B ( C , ε ) ¯ , r = ‖ z − C ‖ .

From the proof of Theorem 3.5, we have

(5.1) ζ ε ( z ) = 0 , z ∈ ∂ B ( C , ε ) ,

(5.2) ζ ε ( z ) ≥ 0 , z ∈ B ( C , ε ) ¯ ,

(5.3) Δ ζ ε ( z ) − λ ζ ε ( z ) = − 1 , z ∈ B ( C , ε ) ,

(5.4) ζ ε ˜ ′ ( ε ) = − 1 λ I 1 ( λ ε ) I 0 ( λ ε ) .

Making use of (5.1), (5.3), (5.4), and using two times Green’s formula, we obtain

that is,

∫ B ( C , ε ) f ( z ) d z − I 1 ( λ ε ) λ I 0 ( λ ε ) ∫ ∂ B ( C , ε ) f ( γ ) d l ( γ ) = − ∫ B ( C , ε ) ( Δ f ( z ) − λ f ( z ) ) ζ ε ( z ) d z .

Then, from (3.13) with R = ε , we deduce that

− ∫ B ( C , ε ) ( Δ f ( z ) − λ f ( z ) ) ζ ε ( z ) d z ≤ 0 .

Since ζ ε ≥ 0 by (5.2), we deduce the existence of z ε ∈ B ( C , ε ) so that

Δ f ( z ε ) − λ f ( z ε ) ≥ 0 .

Furthermore, since f ∈ C 2 ( Ω ) , passing to the limit as ε → 0 + in the above inequality, we obtain

Δ f ( C ) − λ f ( C ) ≥ 0 .

Since C is an arbitrary point of Ω , it holds that Δ f − λ f ≥ 0 in Ω , which shows that f ∈ X − λ ( Ω ) .□

From Theorems 3.5 and 5.1, we deduce the following characterization.

Corollary 5.2

Let f ∈ C 2 ( Ω ) and λ > 0 . The following statements are equivalent:

f ∈ X − λ ( Ω ) .
For all C ∈ Ω and R > 0 with B ( C , R ) ¯ ⊂ Ω , (3.13) holds.

6 Conclusion

We considered the two classes of functions X ± λ ( Ω ) , where Ω is an open subset of R 2 and λ > 0 . We extended inequality (1.2) to functions f ∈ X ± λ ( Ω ) (Theorems 3.3 and 3.5). We also obtained a characterization of the class of functions X − λ ( Ω ) (Corollary 5.2). In this article, only the two-dimensional case is considered. It will be interesting to extend the obtained results to the N -dimensional case (i.e., Ω is an open subset of R N ). It will be also interesting to extend the obtained results to the more general classes of functions

X ± λ , α ( Ω ) = { f ∈ C 2 ( Ω ) : div ( α ∇ f ) ± λ f ≥ 0 } ,

where α ∈ C 1 ( Ω ) . Here, div is the divergence operator. Notice that, if α ≡ 1 , then X ± λ , α ( Ω ) = X ± λ ( Ω ) .

Acknowledgement

The third author was supported by Researchers Supporting Project number (RSP2024R4), King Saud University, Riyadh, Saudi Arabia.

Author contributions: All authors contributed equally to this work and have read and agreed to the published version of the manuscript.
Conflict of interest: The authors state no conflicts of interest. Bessem Samet was a Guest Editor of the Open Mathematics journal and was not involved in the review and decision-making process of this article.
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: 2024-02-10

Revised: 2024-05-11

Accepted: 2024-05-16

Published Online: 2024-08-02

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

Articles in the same Issue

https://doi.org/10.1515/math-2024-0028

Keywords for this article

Hermite-Hadamard-type inequalities; trigonometrically ρ-convex functions; hyperbolic ρ-convex functions; Bessel functions

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