Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Next Article in Journal
A Relational Semantics for Ockham’s Modalities
Next Article in Special Issue
Initial Coefficients Upper Bounds for Certain Subclasses of Bi-Prestarlike Functions
Previous Article in Journal
Numerical Investigation by Cut-Cell Approach for Turbulent Flow through an Expanded Wall Channel
Previous Article in Special Issue
Measure-Based Extension of Continuous Functions and p-Average-Slope-Minimizing Regression
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

On Hermite–Hadamard-Type Inequalities for Functions Satisfying Second-Order Differential Inequalities

1
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11566, Saudi Arabia
2
Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia
*
Author to whom correspondence should be addressed.
Axioms 2023, 12(5), 443; https://doi.org/10.3390/axioms12050443
Submission received: 20 March 2023 / Revised: 13 April 2023 / Accepted: 26 April 2023 / Published: 29 April 2023
(This article belongs to the Special Issue Theory of Functions and Applications)

Abstract

:
Hermite–Hadamard inequality is a double inequality that provides an upper and lower bounds of the mean (integral) of a convex function over a certain interval. Moreover, the convexity of a function can be characterized by each of the two sides of this inequality. On the other hand, it is well known that a twice differentiable function is convex, if and only if it admits a nonnegative second-order derivative. In this paper, we obtain a characterization of a class of twice differentiable functions (including the class of convex functions) satisfying second-order differential inequalities. Some special cases are also discussed.

1. Introduction

Inequalities involving convex functions are very useful in many branches of mathematics. The Hermite–Hadamard inequality is the one of the most important inequality for convex functions. This inequality provides an upper and lower bounds of the mean of a convex function over a certain interval. It is mostly used in mathematics to study the properties of convex functions and their applications in optimization and approximation theory, see, e.g., [1,2,3].
A real-valued function f defined in an interval I is convex if:
f ( ι y + ( 1 ι ) z ) ι f ( y ) + ( 1 ι ) f ( z )
for every ι [ 0 , 1 ] and y , z I . If f is twice differentiable, then f is convex, if and only if its second derivative is nonnegative. The Hermite–Hadamard inequality can be stated as follows: Let f be a real-valued convex function in an interval I. Then, for all x , y I with x < y , we have:
f x + y 2 1 y x x y f ( τ ) d τ f ( x ) + f ( y ) 2 .
Many generalizations and extensions of (1) can be found in the literature. For instance, Dragomir and Agarwal [2] studied the following class of functions:
F = f : [ a , b ] R : f   is   differentiable , | f |   is   convex .
They proved that, if f F , then:
1 b a a b f ( x ) d x f ( a ) + f ( b ) 2 b a 8 | f ( a ) | + | f ( b ) | .
Some improvements and extensions of the above result have been obtained by some authors, see, e.g., [4,5,6,7]. Other extensions of (1) to various classes of functions have been obtained: s-convex functions [8,9,10,11], log-convex functions [12,13,14], h-convex functions [15,16], and m-convex functions [17,18,19,20]. For other classes of functions, we refer to [21,22,23,24] and the references therein. Some extensions of Hermite–Hadamard inequality to a higher dimension can be found in [25,26,27,28,29].
It is interesting to notice that each of the two sides of (1) provides a characterization of convex functions. Namely, if f is a real valued continuous function in an interval I, then the following statements are equivalent:
(i)
f is convex;
(ii)
For all x , y I with x < y :
1 y x x y f ( τ ) d τ f x + y 2 ;
(iii)
For all x , y I with x < y :
1 y x x y f ( τ ) d τ f ( x ) + f ( y ) 2 .
The proof of the implication (ii)⇒(i) can be found in ([30], p. 98). For the proof of the implication (iii)⇒(i), we refer to (Problem Q, [31], p. 15). On the other hand, one can check easily that (iii) is equivalent to:
x y f ( τ ) d τ H ( x ) f ( x ) H ( y ) f ( y )
for all x , y I with x < y , where:
H ( z ) = 1 2 ( z x ) ( y z ) , x z y .
Observe that H is the unique (nonnegative) solution to the boundary value problem:
H ( z ) = 1 , x < z < y , H ( x ) = H ( y ) = 0 .
From the above remarks, we deduce that, if f is twice differentiable in I, then f 0 (i.e., f is convex), if and only if (3) holds for all x , y I with x < y . Thus, (3) provides a characterization of twice differentiable functions in I, having a nonnegative second derivative.
Motivated by the above discussion, our aim in this paper is to obtain a characterization of the class of twice continuously differentiable functions f in I, satisfying second-order differential inequalities of the form:
( α f ) + β f + γ 0 ,
where α is twice continuously differentiable in I and β , γ are continuous in I. We shall assume that for all x , y I with x < y , there exists a unique nonnegative solution H to the boundary value problem:
( α ( z ) H ( z ) ) + β ( z ) H ( z ) = 1 , x < z < y , H ( x ) = H ( y ) = 0 .
The rest of the paper is organized as follows. Section 2 is devoted to the main results and their proofs. Namely, we establish a characterization of the class of functions f satisfying differential inequalities of the form (4). In Section 3, we discuss some special cases of (4).

2. Main Results

For any interval J of R , by C n ( J ) , where n 0 is a natural number, we mean the space of n-continuously differentiable functions in J.
Let I be an open interval of R . Let α C 1 ( I ) and β , γ C ( I ) . Throughout this section, it is assumed that for all x , y I with x < y , there exists a unique nonnegative solution H C 2 ( [ x , y ] ) C ( [ x , y ] ) to the Dirichlet boundary value problem:
( α ( z ) H ( z ) ) + β ( z ) H ( z ) = 1 , x < z < y , H ( x ) = H ( y ) = 0 .
We are concerned with the class of functions f C 2 ( I ) satisfying the second-order differential inequality:
( α ( z ) f ( z ) ) + β ( z ) f ( z ) + γ ( z ) 0 , z I .
Our main result, which is stated below, provides a characterization of this class of functions.
Theorem 1. 
Let α C 1 ( I ) , β , γ C ( I ) and f C 2 ( I ) . The following statements are equivalent:
(i)
(6) holds;
(ii)
For all x , y I with x < y , it holds that:
x y f ( τ ) d τ H ( x ) α ( x ) f ( x ) H ( y ) α ( y ) f ( y ) + x y γ ( τ ) H ( τ ) d τ .
Proof. 
Assume that (6) holds. Let x , y I with x < y . Multiplying (6) by H (notice that H 0 ) and integrating over [ x , y ] , we obtain:
x y ( α ( τ ) f ( τ ) ) H ( τ ) d τ + x y β ( τ ) f ( τ ) H ( τ ) d τ x y γ ( τ ) H ( τ ) d τ .
An integration by parts gives us that:
x y ( α ( τ ) f ( τ ) ) H ( τ ) d τ = α ( τ ) f ( τ ) H ( τ ) τ = x y x y f ( τ ) ( α ( τ ) H ( τ ) ) d τ .
On the other hand, by (5), we have H ( x ) = H ( y ) = 0 , which yields:
α ( τ ) f ( τ ) H ( τ ) τ = x y = 0 .
Then, it holds that:
x y ( α ( τ ) f ( τ ) ) H ( τ ) d τ = x y f ( τ ) ( α ( τ ) H ( τ ) ) d τ .
Integrating again by parts, we obtain:
x y ( α ( τ ) f ( τ ) ) H ( τ ) d τ = f ( τ ) α ( τ ) H ( τ ) τ = x y + x y f ( τ ) ( α ( τ ) H ( τ ) ) d τ = H ( y ) α ( y ) f ( y ) + H ( x ) α ( x ) f ( x ) + x y f ( τ ) ( α ( τ ) H ( τ ) ) d τ .
However, due to (5), we have ( α ( τ ) H ( τ ) ) = 1 β ( τ ) H ( τ ) , which yields:
x y ( α ( τ ) f ( τ ) ) H ( τ ) d τ = H ( y ) α ( y ) f ( y ) + H ( x ) α ( x ) f ( x ) x y f ( τ ) d τ x y β ( τ ) f ( τ ) H ( τ ) d τ .
Thus, (7) follows from (8) and (9). This shows that (i)⇒(ii). Assume now that (ii) holds. Let x I be fixed. Then, for all ε > 0 (sufficiently small), we have:
x ε x + ε f ( τ ) d τ H ( x ε ) α ( x ε ) f ( x ε ) H ( x + ε ) α ( x + ε ) f ( x + ε ) + x ε x + ε γ ( τ ) H ( τ ) d τ ,
where H is the unique positive solution to the boundary value problem:
( α ( z ) H ( z ) ) + β ( z ) H ( z ) = 1 , x ε < z < x + ε , H ( x ε ) = H ( x + ε ) = 0 .
Moreover, by (11), we have:
x ε x + ε f ( τ ) d τ = x ε x + ε ( α ( τ ) H ( τ ) ) + β ( τ ) H ( τ ) f ( τ ) d τ .
Integrating by parts, we obtain:
x ε x + ε f ( τ ) d τ = x ε x + ε ( α ( τ ) H ( τ ) ) f ( τ ) d τ x ε x + ε β ( τ ) H ( τ ) f ( τ ) d τ = α ( τ ) H ( τ ) f ( τ ) τ = x ε x + ε + x ε x + ε H ( τ ) α ( τ ) f ( τ ) d τ x ε x + ε β ( τ ) H ( τ ) f ( τ ) d τ = α ( x ε ) H ( x ε ) f ( x ε ) α ( x + ε ) H ( x + ε ) f ( x + ε ) + H ( τ ) α ( τ ) f ( τ ) τ = x ε x + ε x ε x + ε H ( τ ) ( α ( τ ) f ( τ ) ) d τ x ε x + ε β ( τ ) H ( τ ) f ( τ ) d τ .
Since H ( x ε ) = H ( x + ε ) = 0 , we obtain:
x ε x + ε f ( τ ) d τ = α ( x ε ) H ( x ε ) f ( x ε ) α ( x + ε ) H ( x + ε ) f ( x + ε ) x ε x + ε H ( τ ) ( α ( τ ) f ( τ ) ) + β ( τ ) f ( τ ) d τ .
Hence, by (10), it holds that:
x ε x + ε H ( τ ) ( α ( τ ) f ( τ ) ) + β ( τ ) f ( τ ) + γ ( τ ) d τ 0 .
Since H 0 and ( α f ) + β f + γ C ( [ x ε , x + ε ] ) , then there exists z ε [ x ε , x + ε ] such that:
( α ( z ε ) f ( z ε ) ) + β ( z ε ) f ( z ε ) + γ ( z ε ) 0 .
Passing to the limit as ε 0 + in the above inequality, we obtain:
( α ( x ) f ( x ) ) + β ( x ) f ( x ) + γ ( x ) 0 ,
which proves that (6) holds. This shows that (ii)⇒(i). □
Replacing f by f and γ by γ in Theorem 1, we obtain the following result.
Theorem 2. 
Let α C 1 ( I ) , β , γ C ( I ) and f C 2 ( I ) . The following statements are equivalent:
(i)
( α ( z ) f ( z ) ) + β ( z ) f ( z ) + γ ( z ) 0 , z I ;
(ii)
For all x , y I with x < y , it holds that:
x y f ( τ ) d τ H ( x ) α ( x ) f ( x ) H ( y ) α ( y ) f ( y ) + x y γ ( τ ) H ( τ ) d τ .
From Theorem 1, we deduce the following result.
Corollary 1. 
Let α C 1 ( I ) , β , γ C ( I ) and f C 2 ( I ) . If (6) holds, then for all x , y I with x < y , we have:
x y f ( τ ) d τ H 1 ( x ) α ( x ) f ( x ) H 2 ( y ) α ( y ) f ( y ) + x y γ ( τ ) H ( τ ) d τ H 1 x + y 2 H 2 x + y 2 α x + y 2 f x + y 2 ,
where H 1 and H 2 are the unique nonnegative solutions to the boundary value problems:
( α ( z ) H 1 ( z ) ) + β ( z ) H 1 ( z ) = 1 , x < z < x + y 2 H 1 ( x ) = H 1 x + y 2 = 0 ,
( α ( z ) H 2 ( z ) ) + β ( z ) H 2 ( z ) = 1 , x + y 2 < z < y H 2 x + y 2 = H 2 ( y ) = 0 ,
and
H ( z ) = H 1 ( z ) i f x z x + y 2 , H 2 ( z ) i f x + y 2 < z y .
Proof. 
Writing (7) with x + y 2 instead of y, we obtain:
x x + y 2 f ( τ ) d τ H 1 ( x ) α ( x ) f ( x ) H 1 x + y 2 α x + y 2 f x + y 2 + x x + y 2 γ ( τ ) H 1 ( τ ) d τ .
Similarly, writing (7) with x + y 2 instead of x, we obtain:
x + y 2 y f ( τ ) d τ H 2 x + y 2 α x + y 2 f x + y 2 H 2 ( y ) α ( y ) f ( y ) + x + y 2 y γ ( τ ) H 2 ( τ ) d τ .
Adding (16) to (17), we obtain (12). □
Similarly, from Theorem 2, we deduce the following result.
Corollary 2. 
Let α C 1 ( I ) , β , γ C ( I ) and f C 2 ( I ) . If:
( α ( z ) f ( z ) ) + β ( z ) f ( z ) + γ ( z ) 0 , z I ,
then for all x , y I with x < y , we have:
x y f ( τ ) d τ H 1 ( x ) α ( x ) f ( x ) H 2 ( y ) α ( y ) f ( y ) + x y γ ( τ ) ) H ( τ ) d τ H 1 x + y 2 H 2 x + y 2 α x + y 2 f x + y 2 ,
where H 1 (resp. H 2 ) is the unique nonnegative solution to (13) (resp. (14)) and H is defined by (15).
From Corollary 1, we deduce the following refinement of Hermite–Hadamard inequality (see [29]).
Corollary 3. 
Let f C 2 ( I ) be a convex function. Then, for all x , y I with x < y , we have:
1 y x x y f ( τ ) d τ 1 2 f ( x ) + f ( y ) 2 + f x + y 2 .
Proof. 
Taking:
α = 1 , β = γ = 0
in Corollary 1, we obtain:
H 1 ( z ) = 1 4 ( z x ) ( x + y 2 z ) , x z x + y 2
and
H 2 ( z ) = 1 4 ( y z ) ( 2 z x y ) , x + y 2 z y .
Then, by (12), we obtain (18). □
Similarly, from Corollary 2, we deduce the following result.
Corollary 4. 
Let f C 2 ( I ) be a concave function. Then, for all x , y I with x < y , we have
1 y x x y f ( τ ) d τ 1 2 f ( x ) + f ( y ) 2 + f x + y 2 .

3. Applications

In this section, some special cases of Theorems 1 and 2 are discussed. Namely, we provide characterizations of various classes of functions satisfying differential inequalities of type (6). We first consider the classes of functions:
F + = f C 2 ( I ) : f ( z ) , z I
and
F = f C 2 ( I ) : f ( z ) , z I ,
where R is a constant. Observe that for = 0 , F 0 + reduces to the class of twice continuously differentiable convex functions, while F 0 reduces to the class of twice continuously differentiable concave functions. We recall that in [29], Niculescu and Persson proved that, if f F + , then for all x , y I with x < y , it holds that:
f ( x ) + f ( y ) 2 1 y x x y f ( τ ) d τ ( y x ) 2 12 .
Furthermore, if f F , then for all x , y I with x < y , it holds that:
f ( x ) + f ( y ) 2 1 y x x y f ( τ ) d τ ( y x ) 2 12 .
In this section, we show that (21) (resp. (22)) provides a characterization of the class of functions F + (resp. F ). We next consider the classes of functions
G λ + = f C 2 ( I ) : f ( z ) λ f ( z ) 0 , z I
and
G λ = f C 2 ( I ) : f ( z ) λ f ( z ) 0 , z I ,
where λ > 0 . Observe that when λ = 0 , G 0 + reduces to the class of twice continuously differentiable convex functions, while G 0 reduces to the class of twice continuously differentiable concave functions.

3.1. Characterizations of the Classes of Functions F ±

Let I be an open interval of R . Let R . The following result provides a characterization of the class of functions F + defined by (19).
Corollary 5. 
Let f C 2 ( I ) . The following statements are equivalent:
(i)
f F + ;
(ii)
For all x , y I with x < y , (21) holds.
Proof. 
Observe that:
F + = f C 2 ( I ) : ( α ( z ) f ( z ) ) + β ( z ) f ( z ) + γ ( z ) 0 , z I ,
where
α = 1 , β = 0 , γ = .
Hence, by Theorem 1, f F + , if and only if, for all x , y I with x < y , it holds that:
x y f ( τ ) d τ H ( x ) f ( x ) H ( y ) f ( y ) x y H ( τ ) d τ ,
where
H ( z ) = 1 2 ( z x ) ( y z ) , x z y
is the unique (nonnegative) solution to the boundary value problem:
H ( z ) = 1 , x < z < y , H ( x ) = H ( y ) = 0 .
On the other hand, for all x , y I with x < y , we have:
H ( x ) f ( x ) H ( y ) f ( y ) x y H ( τ ) d τ = y x 2 f ( x ) 1 2 ( x y ) f ( y ) 2 x y ( τ 2 + ( x + y ) τ x y ) d τ = ( y x ) f ( x ) + f ( y ) 2 2 τ 3 3 + x + y 2 τ 2 x y τ τ = x y = ( y x ) f ( x ) + f ( y ) 2 12 y 3 x 3 3 x y 2 + 3 x 2 y = ( y x ) f ( x ) + f ( y ) 2 12 ( y x ) 3 ,
which shows that (25) is equivalent to (21). □
Similarly, using Theorem 2 (or replacing f by f and by in Corollary 5), we obtain the following characterization of the class of functions F defined by (20).
Corollary 6. 
Let f C 2 ( I ) . The following statements are equivalent:
(i)
f F ;
(ii)
For all x , y I with x < y , (22) holds.

3.2. Characterizations of the Classes of Functions G λ ±

Let I be an open interval of R and λ > 0 . We first need the following lemma. Its proof is elementary; we omit the details.
Lemma 1. 
For all x , y I with x < y , the following boundary value problem:
H ( z ) λ H ( z ) = 1 , x < z < y , H ( x ) = H ( y ) = 0
admits a unique nonnegative solution given by:
H ( z ) = e λ z e λ x e λ z e λ z e λ y λ e λ x + e λ y , x z y .
The following result provides a characterization of the class of functions G λ + defined by (23).
Corollary 7. 
Let f C 2 ( I ) . The following statements are equivalent:
(i)
f G λ + ;
(ii)
For all x , y I with x < y , it holds that:
x y f ( τ ) d τ e λ y e λ x λ e λ x + e λ y f ( x ) + f ( y ) .
Proof. 
Observe that:
G λ + = f C 2 ( I ) : ( α ( z ) f ( z ) ) + β ( z ) f ( z ) + γ ( z ) 0 , z I ,
where
α = 1 , β = λ , γ = 0 .
Hence, by Theorem 1, f G λ + , if and only if, for all x , y I with x < y , it holds that:
x y f ( τ ) d τ H ( x ) f ( x ) H ( y ) f ( y ) ,
where H is given by (26). On the other hand, for all x , y I with x < y , we have:
H ( z ) = e λ ( x z + y ) e λ z λ e λ x + e λ y , x z y ,
which yields:
H ( x ) = e λ y e λ x λ e λ x + e λ y
and
H ( y ) = e λ x e λ y λ e λ x + e λ y = H ( x ) .
Hence, for all x , y I with x < y , we have:
H ( x ) f ( x ) H ( y ) f ( y ) = H ( x ) ( f ( x ) + f ( y ) ) = e λ y e λ x λ e λ x + e λ y f ( x ) + f ( y ) ,
which shows that (28) is equivalent to (27). □
Remark 1. 
Passing to the limit as λ 0 + , (27) reduces to the standard Hermite–Hadamard inequality (2).
Similarly, using Theorem 2 (or replacing f by f in Corollary 7), we obtain the following characterization of the class of functions G λ defined by (24).
Corollary 8. 
Let f C 2 ( I ) . The following statements are equivalent:
(i)
f G λ ;
(ii)
For all x , y I with x < y , it holds that:
x y f ( τ ) d τ e λ y e λ x λ e λ x + e λ y f ( x ) + f ( y ) .

4. Conclusions

The Hermite–Hadamard inequality (Inequality (1)) provides an upper and lower bounds of the (integral) mean of a convex function over a certain interval. Moreover, each of the two sides of (1) provides a characterization of convex functions. In the special case when a function f is twice differentiable in a certain interval I, the convexity of f is equivalent to the differential inequality f 0 in I. Thus, it is natural to ask whether it is possible to obtain a characterization of twice differentiable functions satisfying more general differential inequalities. In this paper, we gave a positive answer to this question for the class of functions f satisfying differential inequalities of the form ( α f ) + β f + γ 0 in I, where α C 1 ( I ) and β , γ C ( I ) . Namely, assuming that for every x , y I with x < y , the Dirichlet boundary value problem:
( α ( z ) H ( z ) ) + β ( z ) H ( z ) = 1 , x < z < y , H ( x ) = H ( y ) = 0
admits a unique nonnegative solution H. We show that the considered differential inequality is equivalent to:
x y f ( τ ) d τ H ( x ) α ( x ) f ( x ) H ( y ) α ( y ) f ( y ) + x y γ ( τ ) H ( τ ) d τ
for every x , y I with x < y . The above inequality is a generalization of the right side of Hermite–Hadamard inequality (1), which can be obtained by taking α = 1 and β = γ = 0 . We also discussed some special cases of α , β and γ , and provided some characterizations in those cases.
In this work, only second-order differential inequalities are investigated. It would be interesting to show whether it is possible to obtain a characterization of functions f satisfying higher-order differential inequalities. For instance, the class of functions f satisfying f 0 in I deserves to be studied.

Author Contributions

All authors contributed equally to this paper. All authors have read and agreed to the published version of the manuscript.

Funding

The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) for funding and supporting this work through Research Partnership Program no RP-21-09-03.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Dragomir, S.S.; Pearce, C.E.M. Selected Topics on Hermite-Hadamard Inequalities and Applications; RGMIA Monographs; Victoria University: Footscray, VIC, Australia, 2000. [Google Scholar]
  2. Dragomir, S.S.; Agarwal, R.P. Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl. Math. Lett. 1998, 11, 91–95. [Google Scholar] [CrossRef]
  3. Guessab, A.; Semisalov, B. Optimal general Hermite-Hadamard-type inequalities in a ball and their applications in multidimensional numerical integration. Appl. Numer. Math. 2021, 170, 83–108. [Google Scholar] [CrossRef]
  4. Pearce, C.E.; Pecarić, J. Inequalities for differentiable mappings with application to special means and quadrature formulae. Appl. Math. Lett. 2000, 13, 51–55. [Google Scholar] [CrossRef]
  5. Sarikaya, M.Z.; Saglam, A.; Yildirim, H. New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are convex and quasi-convex. Int. J. Open Probl. Comput. Sci. Math. (IJOPCM) 2012, 5, 3. [Google Scholar] [CrossRef]
  6. Latif, M.A. On some new inequalities of Hermite-Hadamard type for functions whose derivatives are s-convex in the second sense in the absolute value. Ukr. Math. J. 2016, 67, 1552–1571. [Google Scholar] [CrossRef]
  7. Zhao, D.; Gulshan, G.; Ali, M.A.; Nonlaopon, K. Some new midpoint and trapezoidal-type inequalities for general convex functions in q-calculus. Mathematics 2022, 10, 444. [Google Scholar] [CrossRef]
  8. Kórus, P. An extension of the Hermite-Hadamard inequality for convex and s-convex functions. Aequ. Math. 2019, 93, 527–534. [Google Scholar] [CrossRef]
  9. Samraiz, M.; Perveen, Z.; Rahman, G.; Adil Khan, M.; Nisar, K.S. Hermite-Hadamard fractional inequalities for differentiable functions. Fractal Fract. 2022, 6, 60. [Google Scholar] [CrossRef]
  10. Nasri, N.; Aissaoui, F.; Bouhali, K.; Frioui, A.; Meftah, B.; Zennir, K.; Radwan, T. Fractional weighted midpoint-type inequalities for s-convex functions. Symmetry 2023, 15, 612. [Google Scholar] [CrossRef]
  11. Gulshan, G.; Budak, H.; Hussain, R.; Nonlaopon, K. Some new quantum Hermite-Hadamard type inequalities for s-convex functions. Symmetry 2022, 14, 870. [Google Scholar] [CrossRef]
  12. Niculescu, P.C. The Hermite-Hadamard inequality for log convex functions. Nonlinear Anal. 2012, 75, 662–669. [Google Scholar] [CrossRef]
  13. Dragomir, S. Refinements of the Hermite-Hadamard integral inequality for log-convex functions. Aust. Math. Soc. Gaz. 2000, 28, 129–134. [Google Scholar]
  14. Dragomir, S. New inequalities of Hermite-Hadamard type for log-convex functions. Khayyam J. Math. 2017, 3, 98–115. [Google Scholar]
  15. Dragomir, S.S. Inequalities of Hermite-Hadamard type for h-convex functions on linear spaces. Proyecciones 2015, 34, 323–341. [Google Scholar] [CrossRef]
  16. Breaz, D.; Yildiz, C.; Cotirla, L.; Rahman, G. New Hadamard type inequalities for modified h-convex functions. Fractal Fract. 2023, 7, 216. [Google Scholar] [CrossRef]
  17. Dragomir, S.S.; Toader, G. Some inequalities for m-convex functions. Stud. Univ. Babes Bolyai Math. 1993, 38, 21–28. [Google Scholar]
  18. Dragomir, S.S. On some new inequalities of Hermite-Hadamard type for m-convex functions. Tamkang J. Math. 2002, 33, 1. [Google Scholar] [CrossRef]
  19. Chen, D.; Anwar, M.; Farid, G.; Bibi, W. Inequalities for q-h-integrals via h-convex and m-convex functions. Symmetry 2023, 15, 666. [Google Scholar] [CrossRef]
  20. Qi, Y.; Wen, Q.; Li, G.; Xiao, K.; Wang, S. Discrete Hermite-Hadamard-type inequalities for (s, m)-convex function. Fractals 2022, 30, 2250160. [Google Scholar] [CrossRef]
  21. Zhao, D.F.; An, T.Q.; Ye, G.J.; Liu, W. New Jensen and Hermite-Hadamard type inequalities for h-convex interval-valued functions. J. Inequal. Appl. 2018, 2018, 302. [Google Scholar] [CrossRef]
  22. Rashid, S.; Noor, M.A.; Noor, K.I.; Safdar, F.; Chu, Y.-M. Hermite-Hadamard inequalities for the class of convex functions on time scale. Mathematics 2019, 7, 956. [Google Scholar] [CrossRef]
  23. Budak, H.; Ali, M.A.; Tarhanaci, M. Some new quantum Hermite-Hadamard-like inequalities for coordinated convex functions. J. Optim. Theory Appl. 2020, 186, 899–910. [Google Scholar] [CrossRef]
  24. Samet, B. A convexity concept with respect to a pair of functions. Numer. Funct. Anal. Optim. 2022, 43, 522–540. [Google Scholar] [CrossRef]
  25. Barani, A. Hermite-Hadamard and Ostrowski type inequalities on hemispheres. Mediterr. J. Math. 2016, 13, 4253–4263. [Google Scholar] [CrossRef]
  26. Chen, Y. Hadamard’s inequality on a triangle and on a polygon. Tamkang J. Math. 2004, 35, 247–254. [Google Scholar] [CrossRef]
  27. De la Cal, J.; Cárcamo, J.; Escauriaza, L. A general multidimensional Hermite-Hadamard type inequality. J. Math. Anal. Appl. 2009, 356, 659–663. [Google Scholar] [CrossRef]
  28. Mihailescu, M.; Niculescu, C. An extension of the Hermite-Hadamard inequality through subharmonic functions. Glasg. Math. J. 2007, 49, 509–514. [Google Scholar] [CrossRef]
  29. Niculescu, C.P.; Persson, L.E. Old and new on the Hermite-Hadamard inequality. Real Anal. Exch. 2004, 29, 663–686. [Google Scholar] [CrossRef]
  30. Hardy, G.H.; Littlewood, J.E.; Pólya, G. Inequalities; Cambridge University Press: Cambridge, UK, 1934. [Google Scholar]
  31. Roberts, A.W.; Varberg, D.E. Convex Functions; Academic Press: Cambridge, MA, USA, 1973. [Google Scholar]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Aldawish, I.; Jleli, M.; Samet, B. On Hermite–Hadamard-Type Inequalities for Functions Satisfying Second-Order Differential Inequalities. Axioms 2023, 12, 443. https://doi.org/10.3390/axioms12050443

AMA Style

Aldawish I, Jleli M, Samet B. On Hermite–Hadamard-Type Inequalities for Functions Satisfying Second-Order Differential Inequalities. Axioms. 2023; 12(5):443. https://doi.org/10.3390/axioms12050443

Chicago/Turabian Style

Aldawish, Ibtisam, Mohamed Jleli, and Bessem Samet. 2023. "On Hermite–Hadamard-Type Inequalities for Functions Satisfying Second-Order Differential Inequalities" Axioms 12, no. 5: 443. https://doi.org/10.3390/axioms12050443

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop