1 Statement of the Inequality

1.1 Convex and concave function forms

Let \(f\) be a convex function on a closed interval \([a,b]\) with \(a<b\). The Hermite–Hadamard inequality compares three quantities: the average of \(f\) over the interval, the value at the midpoint, and the average of the endpoint values. For concave functions, the inequalities reverse.

1.2 Endpoint and midpoint bounds

For convex \(f\colon [a,b]\to \mathbb{R}\), \[ f\!\left(\frac{a+b}{2}\right)\ \le\ \frac{1}{b-a}\int_a^b f(x)\,dx\ \le\ \frac{f(a)+f(b)}{2}. \] If \(f\) is concave, then the same expressions satisfy \[ f\!\left(\frac{a+b}{2}\right)\ \ge\ \frac{1}{b-a}\int_a^b f(x)\,dx\ \ge\ \frac{f(a)+f(b)}{2}. \]

1.3 Integral average interpretation

The inequality can be read as a statement about the location of the “mean value” of a convex function: the integral average lies between the midpoint value and the endpoint average. Since convexity forces the graph to sit below its chords and above its tangents, these bounds reflect how curvature constrains where the function’s mass (in an average sense) can be concentrated.

2 Assumptions and Preliminaries

2.1 Convexity on a closed interval

Convexity on \([a,b]\) means that for any \(x,y\in[a,b]\) and any \(\lambda\in[0,1]\), \[ f(\lambda x+(1-\lambda)y)\ \le\ \lambda f(x)+(1-\lambda)f(y). \] This condition is sufficient for the midpoint and chord-based reasoning used in standard proofs.

2.2 Measurability and integrability requirements

To interpret \(\int_a^b f(x)\,dx\), one typically assumes that \(f\) is measurable and integrable on \([a,b]\). For many standard convex functions, these properties follow automatically: convex functions on an interval are locally bounded and are therefore integrable on compact subintervals.

2.3 Equivalent characterizations of convexity (brief)

Convexity admits multiple equivalent viewpoints—such as preservation of convex combinations, supporting hyperplanes/tangents (in appropriate regularity settings), and monotonicity of secant slopes. These perspectives are often used to construct proofs of the inequality or to derive variants.

3 Proof Methods

3.1 Jensen’s inequality approach

A common route uses Jensen’s inequality. For the uniform distribution on \([a,b]\), the convexity of \(f\) yields \[ f\!\left(\mathbb{E}[X]\right)\ \le\ \mathbb{E}[f(X)], \] where \(X\) is uniformly distributed on \([a,b]\). Since \(\mathbb{E}[X]=\frac{a+b}{2}\) and \(\mathbb{E}[f(X)]=\frac{1}{b-a}\int_a^b f(x)\,dx\), this directly produces the left-hand bound \[ f\!\left(\frac{a+b}{2}\right)\ \le\ \frac{1}{b-a}\int_a^b f(x)\,dx. \] For the right-hand bound, one can apply convexity to suitable symmetrizations of \(x\) around the midpoint to reduce the problem to endpoint values.

3.2 Secant-line (chord) geometry argument

Convexity implies that the function lies below its secant line. For each \(x\in[a,b]\), the point \((x,f(x))\) is constrained by the chord through \((a,f(a))\) and \((b,f(b))\). Writing \(x=\lambda a+(1-\lambda)b\) with \(\lambda=\frac{b-x}{b-a}\), one obtains \[ f(x)\ \le\ \lambda f(a)+(1-\lambda)f(b). \] Integrating this inequality over \(x\in[a,b]\) gives the upper bound \[ \frac{1}{b-a}\int_a^b f(x)\,dx\ \le\ \frac{f(a)+f(b)}{2}. \]

3.3 Tangent-line and supporting hyperplane viewpoint

When one interprets convexity via supporting lines, one can bound the function from below by affine expressions derived from supporting hyperplanes at relevant points. In one dimension, tangents or subgradients provide inequalities of the form \[ f(x)\ \ge\ f(c)+m(x-c) \] for appropriate \(c\) and slope \(m\). Combining such lower bounds with symmetry around \(c=\frac{a+b}{2}\) leads to the midpoint inequality, after averaging over the interval.

3.4 Weighted averaging and rearrangement-style proofs

Some proofs use weighted convexity: instead of integrating a pointwise chord bound, one averages convex combinations of function values at symmetric points \(a+b-x\) and \(x\). By constructing an inequality of the form \[ f(x)+f(a+b-x)\ \le\ f(a)+f(b) \] (or its reverse), and then integrating over half the interval, one recovers the endpoint average bound. Other rearrangement-style arguments formalize how convexity controls averages under pairing and ordering.

4 Variants and Generalizations

4.1 Weighted Hermite–Hadamard inequalities

There are formulations where the average of \(f\) is taken with respect to non-uniform weights on \([a,b]\). Under suitable conditions, convexity bounds the weighted mean between expressions involving endpoint values, the value at a weighted “center,” or integrals of the weights against affine majorants/minorants of \(f\).

4.2 Hermite–Hadamard for symmetric intervals

For intervals symmetric about the origin, \([-h,h]\), the inequality can be written more compactly: \[ f(0)\ \le\ \frac{1}{2h}\int_{-h}^{h} f(x)\,dx\ \le\ \frac{f(-h)+f(h)}{2} \] for convex \(f\). Symmetry simplifies many applications, especially in estimating integrals with sample points arranged around a center.

4.3 Multidimensional extensions (convex on convex sets)

In higher dimensions, the inequality generalizes to convex functions on convex compact sets \(K\subset\mathbb{R}^n\). Rather than endpoints and midpoints, one uses averages of \(f\) over \(K\) and comparisons with averages over boundary pieces or with values at barycenters. The precise form depends on the geometric structure of \(K\) and the notion of “center” used, but the guiding idea remains: convexity bounds integral averages by simpler geometric evaluations.

4.4 Reversed inequalities for concave functions

If \(f\) is concave on \([a,b]\), then all inequalities reverse direction. Equivalently, applying the Hermite–Hadamard inequality to \(-f\) transforms the convex case into the concave case, yielding the corresponding reversed bounds.

5 Connections to Other Results

5.1 Relationship to Jensen’s inequality

Hermite–Hadamard can be viewed as a concrete, integral version of Jensen’s inequality for the uniform measure on an interval, with additional information extracted from the geometry of chords. Jensen gives the midpoint bound directly, while the endpoint bound reflects convexity’s control of secants and how this control survives averaging.

In numerical integration, the midpoint rule approximates \(\int_a^b f\) by \((b-a)f(\frac{a+b}{2})\), and the trapezoidal rule uses \(\frac{b-a}{2}(f(a)+f(b))\). For convex \(f\), Hermite–Hadamard implies that the true integral average lies between these two approximations (up to scaling by \(b-a\)). This provides a qualitative error ordering without requiring derivatives.

5.3 Connections with Ostrowski-type inequalities

Ostrowski-type inequalities also estimate the deviation between \(f\)’s values at points and its integral mean. While they typically involve additional smoothness or bounded derivatives, Hermite–Hadamard sits adjacent in spirit: both quantify how function behavior is reflected in integral averages, with convexity playing the role of a structural regularity condition.

5.4 Convexity and integral inequalities in analysis

The inequality is part of a larger family of convex integral inequalities, including results involving majorization, convex ordering, and inequalities derived from the shape of graphs. Its strength is that it requires only convexity and a single interval, leading to robust estimates in approximation theory and in the analysis of averages.

6 Applications

6.1 Error bounds for numerical integration

When one knows that an integrand is convex (or concave) on \([a,b]\), Hermite–Hadamard yields a priori bounds on the integral average using only the midpoint and endpoints. This can inform stopping criteria, adaptive schemes, or sanity checks in computation where full derivative information is unavailable.

6.2 Bounds for means generated by convex functions

Many classical means can be expressed through convex functions and integral averages. Hermite–Hadamard then provides inequalities between different mean-like quantities—often comparing an “instantaneous” evaluation (midpoint) to an “endpoint-driven” average.

6.3 Inequalities in probability via convexity

In probability, convexity interacts naturally with expectations. If a random variable is restricted to an interval and the function applied to it is convex, then expectation bounds analogous to Hermite–Hadamard can be interpreted as constraints on \(\mathbb{E}[f(X)]\) given limited information about the distribution’s support and symmetry. The uniform case on an interval corresponds most directly to the classical inequality.

6.4 Estimating integrals using limited samples

The inequality is valuable when only three evaluations of \(f\) are feasible: \(f(a)\), \(f(\frac{a+b}{2})\), and \(f(b)\). The integral average is then trapped between two computable quantities, giving a certified range for \(\int_a^b f\).

7 Examples

7.1 Quadratic and exponential functions

For \(f(x)=x^2\) on \([a,b]\), the function is convex, so \[ \left(\frac{a+b}{2}\right)^2 \le \frac{1}{b-a}\int_a^b x^2\,dx \le \frac{a^2+b^2}{2}. \] For \(f(x)=e^x\), convexity ensures the same ordering, and the bounds can be compared numerically or analytically against \(\frac{e^b-e^a}{b-a}\).

7.2 Piecewise-linear convex functions

A piecewise-linear convex function has the property that it lies below its chords and above its breakpoints in a controlled way. On intervals that align with the breakpoints, the integral average can often be computed exactly, making it straightforward to verify the midpoint and endpoint bounds and to observe how sharpness depends on whether the graph is affine.

7.3 Sharpness checks with equality cases

Consider affine functions \(f(x)=mx+c\). Substituting into the inequality shows both bounds become equal: \[ f\!\left(\frac{a+b}{2}\right)=\frac{1}{b-a}\int_a^b f(x)\,dx=\frac{f(a)+f(b)}{2}. \] For strictly convex functions, the inequalities are typically strict unless the interval degenerates or the function behaves affinely on the whole interval.

8 Equality and Sharpness

8.1 Conditions for equality at the endpoints/midpoint

Equality in the left inequality, \(f(\frac{a+b}{2}) = \frac{1}{b-a}\int_a^b f\), occurs precisely when the convexity constraint forces \(f\) to align with an affine function on \([a,b]\). Similarly, equality on the right inequality requires the chord bound to be tight everywhere, again implying affine behavior on the entire interval.

8.2 When the bounds become tight (affine functions)

Both inequalities are simultaneously tight for affine functions, as the graph then coincides with its chords and with any secant or tangent structure. In that case, convexity is “borderline”: the function does not curve, so averaging does not create any gap between midpoint, endpoint average, and integral average.

8.3 Discussion of best possible constants in common variants

In standard variants (for example, weighted or symmetric versions), the constants that appear are often inherited from the uniform-measure or chord-average geometry. The sharpness analysis typically parallels the affine-function argument: any improvement in a constant would contradict the equality behavior of affine functions. As a result, the classical midpoint and endpoint coefficients are essentially best possible within their natural forms.

9 Further Reading and References

9.1 Classical sources

Hermite–Hadamard inequalities appear in classical treatments of convexity and inequalities, often in chapters devoted to integral inequalities and approximation methods. Many early texts present them alongside Jensen’s inequality and related convexity-based bounds.

9.2 Survey-style treatments of convex integral inequalities

Survey literature on convex integral inequalities expands on weighted versions, multidimensional generalizations, and connections to majorization and convex ordering. These treatments often organize results by the type of convexity (ordinary, Jensen-convex, or generalized variants) and by the geometry of the domain.

9.3 Modern extensions in inequality theory

More recent work extends the Hermite–Hadamard framework to other convexity notions, to refined remainder terms, and to inequalities suited for numerical methods. Extensions may include sharper error estimates when additional regularity is available, as well as bounds under alternative notions of convexity.