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Some generalizations of the Hermite–Hadamard integral inequality


In this article we give two possible generalizations of the Hermite–Hadamard integral inequality for the class of twice differentiable functions, where the convexity property of the target function is not assumed in advance. They represent a refinement of this inequality in the case of convex/concave functions with numerous applications.

1 Introduction

A function \(f: I\subset \mathbb{R}\to \mathbb{R}\) is said to be convex on an nonempty interval I if the inequality

$$ f\biggl(\frac{x+y}{2}\biggr)\le \frac{f(x)+f(y)}{2} $$

holds for all \(x,y\in I\).

If inequality (1.1) reverses, then f is said to be concave on I [1].

Let \(f: I\subset \mathbb{R}\to \mathbb{R}\) be a convex function on an interval I and \(a,b\in I\).


$$ f\biggl(\frac{a+b}{2}\biggr)\le \frac{1}{b-a} \int _{a}^{b} f(t)\,dt\le \frac{f(a)+f(b)}{2}. $$

This double inequality is known in the literature as the Hermite–Hadamard (HH) integral inequality for convex functions.

It has plenty of applications in most different areas of pure and applied mathematics (see [24] and the references therein).

If f is a concave function, then both inequalities in (1.2) hold in the reverse direction, i.e.,

$$ \frac{f(a)+f(b)}{2}\le \frac{1}{b-a} \int _{a}^{b} f(t)\,dt\le f\biggl( \frac{a+b}{2} \biggr). $$

During 130 years of its existence, this inequality has been intensely studied, extended, and generalized by many authors. Some recent trends can be found in [517] and [1823].

As an example we quote an improvement by arbitrary means given in [24].

Let \(f: I\subset \mathbb{R}^{+}\to \mathbb{R}\) be a convex function and \(S=S(a,b)\), \(T=T(a,b)\) be some means of positive numbers \(a, b\in I\).


$$ \frac{1}{b-a} \int _{a}^{b} f(t)\,dt\le \frac{1}{2} f(S)+ \frac{1}{2(b-a)}\bigl[(S-a)f(a)+(b-S)f(b)\bigr]; $$


$$ \frac{1}{b-a} \int _{a}^{b} f(t)\,dt\ge \frac{1}{b-a} \biggl[(T-a)f\biggl( \frac{a+T}{2}\biggr)+(b-T)f\biggl(\frac{T+b}{2} \biggr)\biggr]. $$

For any means S and T, approximations (1.4) and (1.5) are better than original (1.2).

In this article we investigate the possibility of a form of the Hermite-Hadamard inequality for functions that are not necessarily convex/concave on I. This has already been attempted in [25] where the convexity/concavity of the second derivative was shown to be crucial in managing improvements of the HH inequality as a linear combination of its endpoints.

We derive here two forms of the Hermite-Hadamard inequality under the sole condition that the second derivative of the target function f exists locally on an interval I. Thus, \(f\in C^{(2)}(I)\) and, because \(f''\) is continuous on a closed interval \(E:=[a,b]\subset I\), it follows that the quantities \(m=m_{f}(a,b):=\min_{t\in E}f''(t)\) and \(M=M_{f}(a,b):=\max_{t\in E}f''(t)\) exist.

These numbers will play an important role in the sequel.

2 Results and proofs

We begin with an improved variant of the Hermite–Hadamard inequality.

Lemma 2.1

Let \(f: I\subset \mathbb{R}\to \mathbb{R}\) be a convex function on an interval I and \(a,b\in I\).


$$ f\biggl(\frac{a+b}{2}\biggr)\le \frac{1}{b-a} \int _{a}^{b} f(t)\,dt\le \frac{1}{2}\biggl[ \frac{f(a)+f(b)}{2}+f\biggl(\frac{a+b}{2}\biggr)\biggr]. $$

It is shown in [4] that this improvement is best possible of the form

$$ p \frac{f(a)+f(b)}{2}+ q f\biggl(\frac{a+b}{2}\biggr);\quad p,q\ge 0, p+q=1. $$

Our first main result is the following.

Theorem 2.2

Let \(g\in C^{(2)}(E)\) and \(p+q=r+s=1\), \(0\le p\), \(r\le 1/2\).


$$\begin{aligned}& r \frac{g(a)+g(b)}{2} + s g\biggl(\frac{a+b}{2}\biggr)+\bigl(sm-r(m+M)\bigr) \frac{(b-a)^{2}}{24} \\& \quad \le \frac{1}{b-a} \int _{a}^{b} g(t)\,dt \\& \quad \le p \frac{g(a)+g(b)}{2} + q g\biggl(\frac{a+b}{2}\biggr)+\bigl(q M-p(m+M)\bigr) \frac{(b-a)^{2}}{24}, \end{aligned}$$

with \(m:=\min_{x\in [a,b]}g''(x)\), \(M:=\max_{x\in [a,b]}g''(x)\), and \(E:=[a,b]\).


For given \(g\in C^{(2)}(E)\), define an auxiliary function f by \(f(t):= g(t)-m t^{2}/2\). Since \(f''(t)=g''(t)-m\ge 0\), we find out that f is a convex function on E. Therefore, applying the form of Hermite–Hadamard inequality given by (2.1), we obtain

$$\begin{aligned}& g\biggl(\frac{a+b}{2}\biggr)-\frac{m}{2}\biggl(\frac{a+b}{2} \biggr)^{2} \\& \quad \le \frac{1}{b-a} \int _{a}^{b} g(t)\,dt -\frac{m}{2} \frac{b^{3}-a^{3}}{3(b-a)} \\& \quad \le\frac{1}{2}\biggl[\frac{g(a)+g(b)}{2}-\frac{m}{2}\biggl( \frac{a^{2}+b^{2}}{2}\biggr)+g\biggl( \frac{a+b}{2}\biggr)-\frac{m}{2} \biggl(\frac{a+b}{2}\biggr)^{2}\biggr], \end{aligned}$$

that is,

$$ \begin{aligned} g\biggl(\frac{a+b}{2}\biggr)+\frac{m}{24}(b-a)^{2} &\le \frac{1}{b-a} \int _{a}^{b} g(t)\,dt \\ &\le \frac{1}{2}\biggl[ \frac{g(a)+g(b)}{2}+g\biggl(\frac{a+b}{2}\biggr)\biggr]-\frac{m}{48}(b-a)^{2}. \end{aligned} $$

On the other hand, taking the auxiliary function f as \(f(t)=Mt^{2}/2-g(t)\), we see that it is also convex on E.

Applying Lemma 2.1 again, we obtain

$$ \begin{aligned} \frac{1}{2}\biggl[\frac{g(a)+g(b)}{2}+g\biggl( \frac{a+b}{2}\biggr)\biggr]-\frac{M}{48}(b-a)^{2} &\le \frac{1}{b-a} \int _{a}^{b} g(t)\,dt \\ &\le g\biggl(\frac{a+b}{2} \biggr)+ \frac{M}{24}(b-a)^{2}. \end{aligned} $$

Now, for arbitrary \(\alpha , \beta \ge 0\), \(\alpha +\beta =1\), multiplying the right-hand sides of inequalities (2.2) and (2.3) with α and β respectively, we get

$$\begin{aligned} \frac{1}{b-a} \int _{a}^{b} g(t)\,dt &\le \frac{\alpha }{2} \biggl[ \frac{g(a)+g(b)}{2}+g\biggl(\frac{a+b}{2}\biggr)\biggr]- \frac{m}{24}(b-a)^{2}]\\ &\quad {}+\beta \biggl[g\biggl( \frac{a+b}{2} \biggr)+\frac{M}{24}(b-a)^{2}\biggr] \\ &=\frac{\alpha }{2}\biggl(\frac{g(a)+g(b)}{2}\biggr)+(\beta +\alpha /2)g\biggl( \frac{a+b}{2}\biggr)+(\beta M-\alpha m/2)\frac{(b-a)^{2}}{24}. \end{aligned}$$

Similar treating of the left-hand sides of (2.2) and (2.3) involving numbers \(\gamma , \delta \ge 0\), \(\gamma +\delta =1\), gives

$$ \frac{1}{b-a} \int _{a}^{b} g(t)\,dt\ge \frac{\gamma }{2}\biggl( \frac{g(a)+g(b)}{2}\biggr)+(\delta +\gamma /2)g\biggl(\frac{a+b}{2}\biggr)+( \delta m- \gamma M/2)\frac{(b-a)^{2}}{24}. $$

Denoting \(\gamma /2=r\), \(\delta +\gamma /2=s\); \(\alpha /2=p\), \(\beta +\alpha /2=q\), we obtain the required result. □

There are plenty of applications of Theorem 2.2. For instance, an improvement of the assertion from Lemma 2.1 is given in the following.

Corollary 2.3

Let \(f\in C^{(2)}(E)\). Then

$$ f\biggl(\frac{a+b}{2}\biggr)+m\frac{(b-a)^{2}}{24}\le \frac{1}{b-a} \int _{a}^{b} f(t)\,dt \le \frac{1}{2}\biggl[ \frac{f(a)+f(b)}{2}+f\biggl(\frac{a+b}{2}\biggr)\biggr]-m \frac{(b-a)^{2}}{48}. $$


Putting \(r=0\), \(s=1\); \(p=q=1/2\), we get the desired result. Note that \(m\ge 0\) if f is a convex function on E. □

Of great importance in the theory of numerical integration is the so-called Simpson’s rule (cf. [26]).

Lemma 2.4

Let \(f\in C^{(4)}(E)\). Then

$$ \frac{f(a)+f(b)}{6}+\frac{2}{3}f\biggl(\frac{a+b}{2}\biggr)- \frac{1}{b-a} \int _{a}^{b} f(t)\,dt=\frac{f^{(4)}(\xi )}{2880}(b-a)^{4}, \quad a< \xi < b. $$

Therefore, we obtain at once an estimation

$$ \frac{n}{2880}(b-a)^{4}\le \frac{f(a)+f(b)}{6}+\frac{2}{3}f \biggl( \frac{a+b}{2}\biggr)-\frac{1}{b-a} \int _{a}^{b} f(t)\,dt\le \frac{N}{2880}(b-a)^{4}, $$

where \(n=n_{f}(a,b):=\min_{t\in E}f^{(4)}(t)\) and \(N=N_{f}(a,b):=\max_{t\in E}f^{(4)}(t)\).

There is a problem how to apply Simpson’s rule if \(f\notin C^{(4)}(E)\). A possible answer for twice differentiable functions is given in the following.

Corollary 2.5

Let \(f\in C^{(2)}(E)\). Then

$$ \biggl\vert \frac{f(a)+f(b)}{6}+\frac{2}{3}f\biggl(\frac{a+b}{2} \biggr)-\frac{1}{b-a} \int _{a}^{b} f(t)\,dt \biggr\vert \le \frac{1}{72}(M-m) (b-a)^{2}. $$


Putting in Theorem 2.2\(r=p=1/3\); \(s=q=2/3\), we obtain

$$ -(M-m)\frac{(b-a)^{2}}{72}\le \frac{f(a)+f(b)}{6}+\frac{2}{3}f\biggl( \frac{a+b}{2}\biggr)-\frac{1}{b-a} \int _{a}^{b} f(t)\,dt\le (M-m) \frac{(b-a)^{2}}{72}, $$

and the proof follows. □

Another refinement of the Hermite–Hadamard inequality is given in the following.

Corollary 2.6

For \(f\in C^{(2)}(E)\), denote \(M/m=t\ge 1\). Then

$$ \begin{aligned} \frac{1}{t+2}\frac{f(a)+f(b)}{2}+\frac{t+1}{t+2}f\biggl(\frac{a+b}{2} \biggr)&\le \frac{1}{b-a} \int _{a}^{b} f(t)\,dt\\ &\le \frac{t}{2t+1} \frac{f(a)+f(b)}{2}+\frac{t+1}{2t+1}f\biggl(\frac{a+b}{2}\biggr). \end{aligned} $$


Applying Theorem 2.2 with \(r=1/(t+2)\), \(s=(t+1)/(t+2)\); \(p=t/(2t+1)\), \(q=(t+1)/(2t+1)\), we obtain the proof since in this case

$$ sm-r(m+M)=qM-p(m+M)=0. $$


The restriction \(0\le r\), \(p\le 1/2\) is unavoidable in the proof of Theorem 2.2. Nevertheless, the following assertion gives an integral representation which absolutely enlarges the range of p, q.

Lemma 2.7

For \(\phi \in C^{(2)}(E)\) and arbitrary p, q; \(p+q=1\), we have the identity

$$ p \frac{\phi (a)+\phi (b)}{2}+ q \phi \biggl(\frac{a+b}{2}\biggr)-\frac{1}{b-a} \int _{a}^{b} \phi (t)\,dt= \frac{(b-a)^{2}}{16} \int _{0}^{1} t(2p-t) \bigl( \phi ''(x)+\phi ''(y)\bigr)\,dt, $$

where \(x:=a\frac{t}{2}+b(1-\frac{t}{2})\), \(y:=b\frac{t}{2}+a(1-\frac{t}{2})\).

It is not difficult to prove the above relation by a double partial integration of its right-hand side.

Hence, our second main result is given in the following.

Theorem 2.8

Let \(\phi \in C^{(2)}(E)\) and, for \(p\in \mathbb{R}\), denote

$$ p \frac{\phi (a)+\phi (b)}{2}+ (1-p) \phi \biggl(\frac{a+b}{2}\biggr)- \frac{1}{b-a} \int _{a}^{b} \phi (t)\,dt:=T_{\phi }(a,b;p). $$


$$ 1. \quad (3p-1)\frac{(b-a)^{2}}{24}m\le T_{\phi }(a,b;p)\le (3p-1) \frac{(b-a)^{2}}{24}M $$

for \(p\ge \frac{1}{2}\);

$$ 2. \quad \bigl(A(p) m-B(p) M\bigr)\frac{(b-a)^{2}}{6}\le T_{\phi }(a,b;p) \le \bigl(A(p) M-B(p)m\bigr)\frac{(b-a)^{2}}{6}, $$

with \(A(p)=p^{3}\), \(B(p)=(p+1)(p-1/2)^{2}\), and \(0< p <\frac{1}{2}\);

$$ 3. \quad (3p-1)\frac{(b-a)^{2}}{24}M\le T_{\phi }(a,b;p)\le (3p-1) \frac{(b-a)^{2}}{24}m $$

for \(p \le 0\).


We prove only the right-hand side inequalities. The other proofs are analogous.

1. In the case \(p\ge 1/2\), \(0\le t\le 1\), note that \(2p-t\ge 0\); \(\phi ''(x)\), \(\phi ''(y)\le M\). Hence, by Lemma 2.7, we get

$$\begin{aligned} T_{\phi }(a,b;p)&=\frac{(b-a)^{2}}{16} \int _{0}^{1} t(2p-t) \bigl(\phi ''(x)+ \phi ''(y)\bigr)\,dt \le 2M\frac{(b-a)^{2}}{16} \int _{0}^{1} t(2p-t)\,dt \\ &= M\biggl(p-\frac{1}{3}\biggr)\frac{(b-a)^{2}}{8}. \end{aligned}$$

2. For \(0< p<1/2\), write

$$\begin{aligned} T_{\phi }(a,b;p)&=\frac{(b-a)^{2}}{16} \int _{0}^{2p} t(2p-t) \bigl(\phi ''(x)+ \phi ''(y)\bigr)\,dt\\ &\quad {}- \frac{(b-a)^{2}}{16} \int _{2p}^{1} t(t-2p) \bigl(\phi ''(x)+ \phi ''(y)\bigr)\,dt \\ &\le 2M \frac{(b-a)^{2}}{16} \int _{0}^{2p} t(2p-t)\,dt-2m \frac{(b-a)^{2}}{16} \int _{2p}^{1} t(t-2p)\,dt\\ &=\frac{(b-a)^{2}}{8}\biggl[ \frac{4p^{3}}{3}M-\biggl(\frac{1}{3}-p +\frac{4p^{3}}{3}\biggr)m \biggr], \end{aligned}$$

which is equivalent to statement 2.

3. In the case \(p\le 0\), we have \(2p-t\le 0\); \(\phi ''(x)\), \(\phi ''(y)\ge m\). Therefore,

$$\begin{aligned} T_{\phi }(a,b;p)&\le 2m\frac{(b-a)^{2}}{16} \int _{0}^{1} t(2p-t)\,dt \\ &= m\biggl(p-\frac{1}{3}\biggr)\frac{(b-a)^{2}}{8}. \end{aligned}$$


Remark 2.9

The approximations from Theorems 2.2 and 2.8 can be compared if \(r=p\), \(s=q\); \(0\le p\le 1/2\). It is not difficult to see that they coincide for \(p=0\) and \(p=1/2\). In other cases the second approximation is better.

For example, if \(p=1/3\), we obtain an improvement of Corollary 2.5, i.e., another estimation of Simpson’s rule for twice differentiable functions.

Corollary 2.10

Let \(f\in C^{(2)}(E)\). Then

$$ \biggl\vert \frac{f(a)+f(b)}{6}+\frac{2}{3}f\biggl(\frac{a+b}{2} \biggr)-\frac{1}{b-a} \int _{a}^{b} f(t)\,dt \biggr\vert \le \frac{1}{162}(M-m) (b-a)^{2}. $$

We conjecture that the constant \(1/162\) is best possible.

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  1. Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1978)

    MATH  Google Scholar 

  2. Niculescu, C.P., Persson, L.E.: Old and new on the Hermite-Hadamard inequality. Real Anal. Exch. 29(2), 663–685 (2003/4)

    Article  MathSciNet  Google Scholar 

  3. Rostamian Delavar, M., Dragomir, S.S., De La Sen, M.: Hermite-Hadamard’s trapezoid and mid-point type inequalities on a disk. J. Inequal. Appl. 2019, Article ID 105 (2019)

    Article  MathSciNet  Google Scholar 

  4. Simić, S.: Some refinements of Hermite-Hadamard inequality and an open problem. Kragujev. J. Math. 42(3), 349–356 (2018)

    Article  MathSciNet  Google Scholar 

  5. Khan, M.A., Iqbal, A., Suleman, M., Chu, Y.M.: Hermite-Hadamard type inequalities for fractional integrals via Green’s function. J. Inequal. Appl. 2018, Article ID 161 (2018)

    Article  MathSciNet  Google Scholar 

  6. Khan, M.A., Mohammad, N., Nwaeze, E.R., Chu, Y.M.: Quantum Hermite-Hadamard inequality by means of a Green function. Adv. Differ. Equ. 2020, Article ID 99 (2020).

    Article  MathSciNet  Google Scholar 

  7. Awan, M.U., Akhtar, N., Iftikhar, S., Noor, M.A., Chu, Y.M.: New Hermite-Hadamard type inequalities for n-polynomial harmonically convex functions. J. Inequal. Appl. 2020, Article 125, 12 pages (2020)

    Article  MathSciNet  Google Scholar 

  8. Awan, M.U., Talib, S., Chu, Y.M., Noor, M.A., Noor, K.I.: Some new refinements of Hermite-Hadamard-type inequalities involving \(\Psi _{k}\)-Riemann-Liouville fractional integrals and applications. Math. Probl. Eng. 2020, Article ID 3051920 (2020)

    Article  MathSciNet  Google Scholar 

  9. Iqbal, A., Khan, M.A., Ullah, S., Chu, Y.M.: Some new Hermite-Hadamard-type inequalities associated with conformable fractional integrals and their applications. J. Funct. Spaces 2020, Article ID 9845407 (2020)

    MathSciNet  MATH  Google Scholar 

  10. Khurshid, Y., Khan, M.A., Chu, Y.M.: Conformable integral version of Hermite-Hadamard-Fejér inequalities via η-convex functions. AIMS Math. 5(5), 5106–5120 (2020)

    Article  MathSciNet  Google Scholar 

  11. Latif, M.A., Rashid, S., Dragomir, S.S., Chu, Y.M.: Hermite-Hadamard type inequalities for co-ordinated convex and quasi-convex functions and their applications. J. Inequal. Appl. 2019, Article ID 317 (2019).

    Article  Google Scholar 

  12. Rasid, S., Noor, M.A., Noor, K.I., Safdar, F., Chu, Y.M.: Hermite-Hadamard type inequalities for the class of convex functions on time scale. Mathematics 7(10), Article ID 956 (2019).

    Article  Google Scholar 

  13. Hengxiao, Q., Yussouf, M., Mehmood, S., Chu, Y.M., Farid, G.: Fractional integral versions of Hermite-Hadamard type inequality for generalized exponentially convexity. AIMS Math. 5(6), 6030–6042 (2020)

    Article  MathSciNet  Google Scholar 

  14. Iqbal, A., Khan, M.A., Mohammad, N., Nwaeze, E.R., Chu, Y.-M.: Revisiting the Hermite-Hadamard integral inequality via a Green function. AIMS Math. 5(6), 6087–6107 (2020)

    Article  MathSciNet  Google Scholar 

  15. Yung, C.Y., Yussouf, M., Chu, Y.M., Farid, G.: Fractional generalized Hadamard and Fejér-Hadamard inequalities for m-convex function. AIMS Math. 5(6), 6325–6340 (2020).

    Article  MathSciNet  Google Scholar 

  16. Guo, S., Chu, Y.M., Farid, G., Mehmood, S., Nazeer, W.: Fractional Hadamard and Fejér-Hadamard inequalities associated with exponentially \((s, m)\)-convex functions. J. Funct. Spaces 2020, Article ID 2410385 (2020).

    Article  MATH  Google Scholar 

  17. Zhou, S.-S., Rashid, S., Noor, M.A., Noor, K.I., Safdar, F., Chu, Y.-M.: New Hermite-Hadamard type inequalities for exponentially convex functions and applications. AIMS Math. 5(6), 6874–6901 (2020)

    Article  MathSciNet  Google Scholar 

  18. Feng, B., Ghafoor, M., Chu, Y.M., Qureshi, M.I., Feng, X., Yao, C., Qiao, X.: Hermite-Hadamard and Jensen’s type inequalities for modified \((p, h)\)-convex functions. AIMS Math. 5(6), 6959–6971 (2020)

    Article  MathSciNet  Google Scholar 

  19. Khan, Z.A.: Hadamard type fractional differential equations for the system of integral inequalities on time scales. Integral Transforms Spec. Funct. 31(5), 412–423 (2020)

    Article  MathSciNet  Google Scholar 

  20. Khurshid, Y., Khan, M.A., Chu, Y.M., Khan, Z.A.: Hermite-Hadamard Fejeér inequalities for conformal fractional integrals via preinvex functions. J. Funct. Spaces 2019, Article ID 3146210, 1–10 (2019)

    MATH  Google Scholar 

  21. Khan, M.A., Hanif, M., Khan, Z.A.H., Ahmad, K., Chu, Y.M.: Association of Jensen’s inequality for s-convex function with Csiszaár divergence. J. Inequal. Appl. 2019, 162 (2019) 1–14

    Article  Google Scholar 

  22. Khan, Z.A.: Further nonlinear version of inequalities and their applications. Filomat 33(18), 6005–6014 (2019)

    Article  MathSciNet  Google Scholar 

  23. Ullah, S.Z., Khan, M.A., Khan, Z.A., Chu, Y.M.: Integral majorization type inequalities for the functions in the sense of strong convexity. J. Funct. Spaces 2019, 1–12 (2019)

    Article  MathSciNet  Google Scholar 

  24. Simić, S.: Further improvements of Hermite-Hadamard integral inequality. Kragujev. J. Math. 43(2), 259–265 (2019)

    MathSciNet  Google Scholar 

  25. Simić, S., Bandar, B.M.: Some improvements of the Hermite-Hadamard integral inequality. Symmetry 12, Article ID 117 (2020)

    Article  Google Scholar 

  26. Ueberhuber, C.W.: Numerical Computation 2. Springer, Berlin (1997)

    Book  Google Scholar 

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The authors are grateful to the referees for their valuable comments.


This project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.

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Theoretical part, SS; numerical part with examples, BB-M. All authors read and approved the final manuscript.

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Correspondence to Slavko Simić.

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Simić, S., Bin-Mohsin, B. Some generalizations of the Hermite–Hadamard integral inequality. J Inequal Appl 2021, 72 (2021).

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  • Hermite–Hadamard integral inequality
  • Twice differentiable functions
  • Convex functions
  • Simpson’s rule