## Abstract

The classical Hermite–Hadamard inequality, under some regularity assumptions, characterizes convexity of real functions. The aim of this paper is to establish connections between the stability forms of the functional inequalities related to Jensen convexity, convexity and the Hermite–Hadamard inequality.

## 1. Introduction

Let *X* be a normed space and *D* be a convex subset of *X*. It is well known (see Hadamard 1893; Kuczma 1985; Mitrinović & Lacković 1985; Dragomir & Pearce 2000; Niculescu & Persson 2006) that convex functions *f* : *D*→ satisfy the so-called (lower) Hermite–Hadamard inequality(1.1)in other words, the integral average of the values of the function *f* over a segment [*x*,*y*] is not smaller than the value of the function at the midpoint of that segment. The converse is also known to be true (cf. Niculescu & Persson 2003, 2006), i.e. if a continuous *f* satisfies (1.1), then it is also convex.

More generally, it is easy to see that the *ϵ*-convexity of *f* (cf. Hyers & Ulam 1952), i.e. the validity ofyields the following *ϵ*-Hermite–Hadamard inequality(1.2)Concerning the reversed implication, Nikodem *et al*. (2007) have recently shown that the *ϵ*-Hermite–Hadamard inequality (1.2) does not imply the *cϵ*-convexity of *f* (with any *c*>0).

In this paper, we investigate the connection between the stability forms of the functional inequalities related to Jensen convexity, convexity and the Hermite–Hadamard inequality when the stability term is not a constant but depends on the closeness of the variables *x* and *y*. In other words, we consider functions *f* : *D*→ satisfying(1.3)(1.4)and(1.5)where *δ*_{J}, δ_{H} : [0,∞[→, and *δ*_{C} : [0,1]×[0,∞[→ are given functions called the stability terms. The main results of this paper establish connections between these terms. In the case when the stability terms are constructed from power functions of ‖*x*−*y*‖, the connection between inequalities (1.3) and (1.5) has been investigated in our recent papers (Házy & Páles 2004, 2005; Házy 2005, 2007).

## 2. Auxiliary results

Let *C*([0,1]) denote the Banach space of continuous real-valued functions equipped with the usual supremum norm ‖.‖. For *φ*∈*C*([0,1]), define *Tφ* : [0,1]→ by

Then one can easily see that *T* : *C*([0,1])→*C*([0,1]) is a monotone (with respect to the pointwise ordering) continuous linear operator and ‖*T*‖≤1. It is also easy to check that *T* is symmetry preserving, i.e. if *φ* is symmetric (with respect to 1/2), then *Tφ* is also symmetric.

Define the function *ϱ*:[0,1]→Furthermore, for a non-negative parameter *p*, introduce the norm ‖.‖_{p} and the subspace *C*_{p}([0,1]) of *C*([0,1]) byObserve that ‖.‖_{0} coincides with the supremum norm ‖.‖ on *C*_{0}([0,1]). For every *p*>0 and *φ*∈*C*_{p}([0,1]), we also have the inequality(2.1)

The subsequent lemmas play basic roles in our investigation.

*For* 0≤*p*, *the pair* (*C*_{p}([0,1]),‖.‖_{p}) *is a Banach space. Furthermore, for p*≤*q*, *C*_{q}([0,1]) *is a closed subspace of the space* (*C*_{p}([0,1]),‖.‖_{p}) *and, for φ*∈*C*_{q}([0,1]),(2.2)

It is easy to see that *C*_{p}([0,1]) is a normed space with ‖.‖_{p}. If *p*=0, then (*C*_{0}([0,1]),‖.‖_{0}) is a closed subspace of (*C*([0,1]),‖.‖) and hence the completeness of *C*_{0}([0,1]) is obvious.

To prove that (*C*_{p}([0,1]),‖.‖_{p}) is complete if *p*>0, let (*φ*_{n}) be a Cauchy sequence in (*C*_{p}([0,1]),‖.‖_{p}). Then, for every *ϵ*>0, there exists *n*_{0}∈ such that, for every *k*, *n*≥*n*_{0},which, by (2.1), implies, for every *k*, *n*≥*n*_{0},(2.3)Hence, for fixed *t*∈[0,1], the sequence (*φ*_{n}(*t*)) is a Cauchy sequence. Thus, the limit lim_{n→∞}*φ*_{n}(*t*)≕*φ*(*t*) exists for all *t*∈[0,1]. Upon taking the limit *n*→∞ in (2.3), it follows that, for *k*≥*n*_{0},(2.4)On compact subintervals of the open unit interval ]0,1[, the function *ϱ*^{p} is bounded from above; therefore, by (2.4), *φ*_{k} converges uniformly to *φ* on each compact subinterval of ]0,1[. Hence, *φ* is continuous on ]0,1[. Since *p*>0, we have that *ϱ*^{p}(0)=*ϱ*^{p}(1)=0. Hence the inequality (2.4) yields that lim_{t→0}*φ*(*t*)=0=*φ*(0) and lim_{t→0}*φ*(*t*)=0=*φ*(1), showing that *φ* is also continuous at the endpoints of the interval [0,1].

It follows from (2.4), for *k*≥*n*_{0}, thatTherefore, *φ*∈*C*_{p}([0,1]) and ‖*φk*−*φ*‖_{p}→0 as *k*→∞. Thus, the sequence (*φ*_{n}) converges in *C*_{p}([0,1]), showing that *C*_{p}([0,1]) is a Banach space.

Let *p*≤*q* be arbitrary and *φ*∈*C*_{p}([0,1]). Then, by the inequality *ϱ*(*t*)≤1, we havewhich proves (2.2) and *φ*∈*C*_{p}([0,1]). Using inequality (2.2), it also follows that *C*_{q}([0,1]) is a closed subspace of (*C*_{p}([0,1]),‖.‖_{p}). ▪

*Let p*≥0. *Then T* : (*C*_{p}([0,1]),‖.‖_{p})→(*C*_{p}([0,1]),‖.‖_{p}) *is a continuous linear map with* . *In particular, if* 0<*p*<1, *then T is a contraction on* (*C*_{p}([0,1]),‖.‖_{p}).

Applying (2.1), for 0<*s*≤1/2, we obtain the following estimate:and the same can be derived for 1/2≤*s*<1. Thus,(2.5)proving ‖*T*‖_{p}≤γ_{p}. This also shows that *Tφ*∈*C*_{p}([0,1]) whenever *φ*∈*C*_{p}([0,1]).

To see that *T* is a contraction on *C*_{p}([0,1]) whenever 0<*p*<1, it suffices to prove that *γ*_{p}<1, i.e.(2.6)The function *h*(*p*)≔2^{p}−*p*−1 is obviously strictly convex (by the second derivative test) and *h*(0)=*h*(1)=0. Thus, *h*(*p*)<(1−*p*)*h*(0)+*ph*(1)=0 for 0<*p*<1, proving (2.6). ▪

*Let φ*∈*C*_{0}([0,1]). *Then the sequence* (*T*^{n}*φ*) *converges to* 0.

Let *ϵ*>0 be arbitrary. By the Weierstrass approximation theorem, there exists a polynomial *P*:[0,1]→ such that . Then,is also a polynomial satisfying *Q*(0)=0=*Q*(1) andWe show that *Q*∈*C*_{p}([0,1]) whenever *p*∈[0,1]. Indeed, by the Lagrange mean value theorem, for , we haveand a similar inequality is valid for . Thus,

Fix *p* in the open interval ]0,1[ arbitrarily. Since the operator *T* is a contraction on the space *C*_{p}([0,1]), it follows that there exists an index *n*_{0}∈ such that, for *n*≥*n*_{0}, . Thus, using the estimate ‖*T*‖≤1 and also (2.2), for *n*≥*n*_{0}, we getTherefore, the sequence (*T*^{n}*φ*) converges to 0. ▪

The statement of the following lemma can be deduced from known results related to semicontinuous set-valued functions. For the reader's convenience, we provide a direct proof.

*Let φ* : [0,1]→ *be upper semicontinuous with φ*(0)=*φ*(1)=0. *Then, for all ϵ*>0, *there exists ψ*∈*C*_{0}([0,1]) *such that φ*≤*ψ*+*ϵ*.

Let *ϵ*>0 be fixed. Since the function *φ* is upper semicontinuous, thus, for all *p*∈[0,1], there exists an open neighbourhood *U*_{p} of *p*, such that *φ*(*t*)<*φ*(*p*)+*ϵ* for every *t*∈*U*_{p}. We may assume that(2.7)

Since the interval [0,1] is compact, there exist points *p*_{0}=0<*p*_{1}<⋯<*p*_{n−1}=1 such thatThen there exists a partition of unity *λ*_{0}, *λ*_{1},…,*λ*_{n−1}, *λ*_{n} : [0,1]→[0,1] dominated by the family of open sets {*U*_{p0}, *U*_{p1},…,*U*_{pn−1}, *U*_{pn}}, i.e. the functions *λ*_{0}, *λ*_{1},…,*λ*_{n−1}, *λ*_{n} are continuous, for every *t*∈[0,1], and supp *λ*_{i}⊆*U*_{pi}, i.e. *λ*_{i}(*t*)>0 yields *t*∈*U*_{pi} for all *i*.

Define the function *ψ*:[0,1]→ byThe function *ψ* is obviously continuous. Using (2.7), it follows that *λ*_{0}(0)=1, *λ*_{1}(0)=…=*λ*_{n}(0)=0 and *λ*_{0}(1)=…=*λ*_{n−1}(1)=0, *λ*_{n}(1)=1. Hence *ψ*(0)=*φ*(0)=0 and *ψ*(1)=*φ*(1)=0, proving that *ψ*∈*C*_{0}([0,1]). On the other hand, for every *t*∈[0,1],which implies *ψ*+*ϵ*≥*φ*. ▪

## 3. Main results

First, we establish an implication from inequality (1.3) to (1.4). For the sake of convenience, we introduce the following notation:

*Let δ*_{J} : [0,∞[→ *be a non-negative locally Lebesgue integrable function. Assume that f*∈*Λ*(*D*) *satisfies the approximate Jensen inequality* (*1.3*). *Then f also satisfies the approximate Hermite–Hadamard inequality* (*1.4*) *with δ*_{H} : [0,∞[→ *defined by*(3.1)

Let *x*,*y*∈*D* be fixed. If *x*=*y*, then (1.4) holds with equality. Therefore, we may assume that *x*≠*y*. Using (1.3), for *t*∈[0,1], we obtainIntegrating this inequality with respect to *t* over the interval [0,1], we get(3.2)Using the substitution *s*=(1−2*t*)‖*x*−*y*‖ in the last term, we obtainwhence, by (3.2), inequality (1.4) follows. ▪

In the particular case when *δ*_{J} is a linear combination of power functions with non-negative exponents, i.e. if *δ*_{J} is of the form(3.3)where *μ*_{J} is a non-negative Borel measure on the interval [0,∞[, we get the following result.

*Let μ*_{J} *be a non-negative Borel measure on* [0,∞[. *Assume that f*∈*Λ*(*D*) *satisfies*(3.4)*Then*(3.5)

Inequality (3.4) is equivalent to (1.3) with *δ*_{J} defined by (3.3). Thus, by theorem 3.1, inequality (1.4) holds with *δ*_{H} given bywhence (3.5) follows. (One can also see that *δ*_{H}≤*δ*_{J}.) ▪

As an immediate consequence, theorem 3.2 reduces to the following corollary when the measure *μ*_{J} is concentrated at a point *p* with *μ*_{J}({*p*})=*ϵ*.

*Let ϵ*≥0 *and p*≥0. *Assume that f*∈*Λ*(*D*) *satisfies**Then*

In the next theorem, we establish an implication from the inequality (1.4) to (1.3) and (1.5).

*Let δ*_{H} : [0,∞[→ *be an arbitrary function. Assume that f* : *D*→ *is an upper semicontinuous function satisfying the approximate Hermite–Hadamard inequality* (*1.4*). *Then f*∈*Λ*(*D*) *and f satisfies the approximate Jensen inequality* (*1.3*) *whenever* *δ*_{J} : [0,∞[→ *is a non-negative increasing solution of the functional inequality*(3.6)*More generally, f also satisfies the approximate convexity inequality* (*1.5*) *with the stability term δ*_{C} : [0,1]×[0,∞[→ *defined by*(3.7)

Let *x*,*y*∈*D* be fixed. If *x*=*y*, then (1.3) follows from *δ*_{J}(0)≥0. Thus, we may assume that *x*≠*y*. Denote by *φ*_{x,y} the function defined byLet *ϵ*>0 be fixed. Since *φ*_{x,y}(0)=*φ*_{x,y}(1)=0 and *φ*_{x,y} is upper semicontinuous, then, by lemma 2.4, there exists a function *ψ*∈*C*_{0}([0,1]) such that *φ*_{x,y}≤*ψ*+*ϵ*. It follows from this inequality and (1.4) that *φ*_{x,y} is Lebesgue integrable over [0,1]. Hence, *f*∈*Λ*(*D*).

We show, by induction on *n*, that(3.8)for all *x*,*y*∈*D* and *s*∈[0,1].

For *n*=0, (3.8) follows from *φ*_{x,y}≤*ψ*+*ϵ* and the non-negativity of *δ*_{J}.

Assume that (3.8) is true for some *n*∈. Let *s*∈[0,1/2]. First using that *f* satisfies (1.4), then applying the inductive assumption (3.8) and the monotonicity of *δ*_{J}, and finally using (3.6) with *t*:=2*s*‖*x*−*y*‖, we get thatwhich proves (3.8) for *n*+1 if *s*∈[0,1/2]. An analogous argument yields the same inequality for *s*∈[1/2,1].

To complete the proof of the theorem, take the limit *n*→∞ and *ϵ*→0 in (3.8). Using lemma 2.3, we get (1.5). The substitution *t*=1/2 in (1.5) also yields (1.3). ▪

In the particular case when *δ*_{H} is a linear combination of power functions with exponents from the open interval ]0,1[, i.e. if *δ*_{H} is of the form(3.9)where *μ*_{H} is a non-negative Borel measure on the interval ]0,1[, we get the following result.

*Let μ*_{H} *be a non-negative Borel measure on* ]0,1[ *such that*(3.10)*Assume that f* : *D*→ *is an upper semicontinuous function, satisfying the approximate Hermite–Hadamard inequality*(3.11)*Then f*∈*Λ*(*D*) *and, for all x*, *y*∈*D*,(3.12)*More generally, for all x*, *y*∈*D and s*∈[0,1],(3.13)

Let *δ*_{H} be defined by (3.9). Then (3.11) is equivalent to (1.4). To deduce (3.13) using theorem 3.4, it suffices to show that the function *δ*_{J} defined byis non-negative, increasing and satisfies (3.6).

The non-negativity of *δ*_{J} follows from the non-negativity of the measure *μ*_{H} and from the inequality 2^{p}<*p*+1 for 0<*p*<1. Observe also that *δ*_{J} is finite-valued since (3.10) holds and *u*^{p} is between 1 and *u* for all *p*∈[0,1]. For *p*≥0, the function *u*↦*u*^{p} is increasing, whence it follows that *δ*_{J} is increasing too.

To see that (3.6) holds, let *t*>0. Then we havewhich shows that (3.6) holds with equality. ▪

In the special case when the measure *μ*_{H} is concentrated at a point *p* with *μ*_{H}({*p*})=*ϵ*, we can formulate the following consequence of theorem 3.5.

*Let ϵ*≥0 *and p*∈]0,1[. *Assume that f* : *D*→ *is an upper semicontinuous function satisfying the approximate Hermite–Hadamard inequality**Then, for all x*, *y*∈*D*,*More generally, for all x*, *y*∈*D and s*∈[0,1],

In order to get estimates also in the case when the measure *μ*_{H} is not supported in the interval ]0,1[, we need the following lemma.

*Let Φ* :]0,∞[×[0,1]→[0,∞[ *be defined by**Then, for each fixed p*>0, *we have Φ*(*p*,*t*)<∞ *for all t*∈[0,1] *and the function Φ*_{p}=(*p*,.) *fulfils the equality*(3.14)*Furthermore, for* 0<*p*<1,(3.15)*and, for p*=1,(3.16)*where the function σ* : [0,1]→ *and the constant c*∈ *are defined by*(3.17)*and*

By the definition of ‖.‖_{p} and lemma 2.2, we haveIf 0<*p*<1, then we getHence, for all *t*∈[0,1], we obtain , which proves (3.15).

If *p*≥1, then, by the monotonicity of the operator *T*, for an arbitrarily fixed 0<*p*_{0}<1, we have (*T*^{n}*ϱ*^{p})(t)≤(*T*^{n}*ϱ*^{p0})(*t*), thereforeWith an easy calculation, we getwhich proves (3.14).

To show that the estimate (3.16) holds, we first prove that the constant *c* and the function *σ* satisfy, for *t*∈[0,1], the inequality(3.18)In view of the symmetry of *σ*, it suffices to show that (3.18) holds for . For these values of *t*, we have that *σ*(*t*)=−*t*ln(*t*) andOne can check the inequality . Therefore, (3.18) is equivalent to the inequality(3.19)We show that the left-hand side of this inequality is bounded from above on . Indeed, for , we have . On the other hand, *ψ* is continuous on the closed interval , hence it is bounded from above here. In fact, one can get the numerical upper bound 21.48991 for *ψ*. Therefore, (3.19) holds with proving (3.18).

Finally, we prove that the inequality (3.18) yields (3.16). Applying the linear operators *T*, *T*^{2}, …, *T*^{n} to the inequality (3.18) and adding up the inequalities so obtained, we get(3.20)Since *σ*∈*C*_{0}([0,1]), using lemma 2.3 and taking the limit *n*→∞ in (3.20), the function *T*^{n+1}(*cσ*) converges to 0 and *ϱ*(*t*)+(*Tϱ*)(*t*)+(*T*^{2}*ϱ*)(*t*)+⋯+(*T*^{n}*ϱ*)(*t*) tends to *Φ*(1,*t*), which shows that the inequality (3.16) holds. ▪

The statement of theorem 3.5 is strengthened by the following result.

*Let μ*_{H} *be a non-negative Borel measure on* ]0,∞[ *and let Φ* : [0,1]×]0,∞[→[0,∞[ *be the function introduced in* *lemma 3.7*. *Assume that f* : *D*→ *is an upper semicontinuous function, satisfying the Hermite–Hadamard-type stability inequality*(3.21)*Then, for all x*, *y*∈*D and s*∈[0,1],(3.22)

The proof of this result follows the same pattern as that of theorem 3.4.

Let *x*, *y*∈*D* be fixed. We may assume that *x*≠*y*. Denote by *φ*_{x,y} the function defined byLet *ϵ*>0 be fixed. Then, by lemma 2.4, there exists a function *ψ*∈*C*_{0}([0,1]) such that *φ*_{x,y}≤*ψ*+*ϵ*. We show, by induction on *n*, that(3.23)for all *x*, *y*∈*D* and *s*∈[0,1].

For *n*=0, (3.23) follows from *φ*_{x,y}≤*ψ*+*ϵ* and the non-negativity of *Φ*.

Assume that (3.23) is true for some *n*∈. Let *s*∈[0,1/2]. First using that *f* satisfies (3.21) and then applying the inductive assumption (3.23), we getwhich proves (3.23) for *n*+1 if *s*∈[0,1/2]. An analogous argument yields the same inequality for *s*∈[1/2,1].

To complete the proof of inequality (3.22), take the limit *n*→∞ and *ϵ*→0 in (3.23) and apply lemma 2.3. ▪

In the special case when the measure *μ*_{H} is of the form *ϵδ*_{p}, we can deduce the following consequence of theorem 3.8.

*Let ϵ*≥0 *and p*>0. *Assume that f* : *D*→ *is an upper semicontinuous function, satisfying the Hermite–Hadamard-type stability inequality**Then, for all x, y*∈*D and s*∈[0,1],

## Acknowledgments

This research was supported by the Hungarian Scientific Research Fund (OTKA) grants K-62316 and NK-68040.

## Footnotes

- Received July 12, 2008.
- Accepted October 6, 2008.

- © 2008 The Royal Society