**HERMITE-HADAMARD-TYPE INEQUALITIES FOR GENERALIZED CONVEX**
**FUNCTIONS**

MIHÁLY BESSENYEI INSTITUTE OFMATHEMATICS

UNIVERSITY OFDEBRECEN

H-4010 DEBRECEN, PF. 12 HUNGARY

besse@math.klte.hu

*Received 21 June, 2008; accepted 14 July, 2008*
*Communicated by P.S. Bullen*

ABSTRACT. The aim of the present paper is to extend the classical Hermite-Hadamard inequal- ity to the case when the convexity notion is induced by a Chebyshev system.

*Key words and phrases: Hermite–Hadamard inequality, generalized convexity, Beckenbach families, Chebyshev systems,*
Markov–Krein theory.

*2000 Mathematics Subject Classification. Primary 26A51, 26B25, 26D15.*

**1. I****NTRODUCTION**

Let I be a real interval, that is, a nonempty, connected and bounded subset of R. An
n-dimensional Chebyshev system on I consists of a set of real valued continuous functions
ω_{1}, . . . , ω_{n}and is determined by the property that eachnpoints ofI×Rwith distinct first coor-
dinates can uniquely be interpolated by a linear combination of the functions. More precisely,
we have the following

* Definition 1.1. Let* I ⊂ R

*be a real interval and*ω

_{1}, . . . , ω

_{n}: I → R

*be continuous func-*

*tions. Denote the column vector whose components are*ω

_{1}, . . . , ω

_{n}

*in turn by*ωωωωωωωωω, that is, ωωωωωωωωω :=

(ω_{1}, . . . , ω_{n}). We say that ωωωωωωωωω*is a Chebyshev system over*I*if, for all elements*x_{1} <· · ·< x_{n}*of*
I, the following inequality holds:

ωωωωωωωωω(x_{1}) · · · ωωωωωωωωω(x_{n})
>0.

In fact, it suffices to assume that the determinant above is nonvanishing whenever the argu-
mentsx_{1}, . . . , x_{n}are pairwise distinct points of the domain. Indeed, Bolzano’s theorem guar-
antees that its sign is constant if the arguments are supposed to be in an increasing order, hence
the componentsω_{1}, . . . , ω_{n}can always be rearranged such that ωωωωωωωωωfulfills the requirement of the
definition. However, considering Chebyshev systems as vectors of functions instead of sets of
functions is widely accepted in the technical literature and also turns out to be very convenient
in our investigations.

This research has been supported by the Hungarian Scientific Research Fund (OTKA) Grants NK–68040.

185-08

Without claiming completeness, let us list some important and classical examples of Cheby-
shev systems. In each example ωωωωωωωωωis defined on an arbitraryI ⊂Rexcept for the last one where
I ⊂]− ^{π}_{2},^{π}_{2}[.

• polynomial system: ωωωωωωωωω(x) := (1, x, . . . , x^{n});

• exponential system: ωωωωωωωωω(x) := (1,expx, . . . ,expnx);

• hyperbolic system: ωωωωωωωωω(x) := (1,coshx,sinhx, . . . ,coshnx,sinhnx);

• trigonometric system: ωωωωωωωωω(x) := (1,cosx,sinx, . . . ,cosnx,sinnx).

We make no attempt here to present an exhaustive account of the theory of Chebyshev sys- tems, but only mention that, motivated by some results of A.A. Markov, the first systematic in- vestigation of the geometric theory of Chebyshev systems was done by M. G. Krein. However, let us note that Chebyshev systems play an important role, sometimes indirectly, in numerous fields of mathematics, for example, in the theory of approximation, numerical analysis and the theory of inequalities. The books [16] and [15] contain a rich literature and bibliography of the topics for the interested reader. The notion of convexity can also be extended by applying Chebyshev systems:

* Definition 1.2. Let*ωωωωωωωωω= (ω

_{1}, . . . , ω

_{n})

*be a Chebyshev system over the real interval*I. A function f :I →R

*is said to be generalized convex with respect to*ωωωωωωωωω

*if, for all elements*x

_{0}<· · ·< x

_{n}

*of*I

*, it satisfies the inequality*

(−1)^{n}

f(x_{0}) · · · f(x_{n})
ωωω

ωωωωωω(x0) · · · ωωωωωωωωω(xn)

≥0.

There are other alternatives to express thatf is generalized convex with respect toωωωωωωωωω, for ex-
ample,f*is generalized* ωωωωωωωωω-convex or simply ωωωωωωωωω-convex. If the underlyingn-dimensional Cheby-
shev system can uniquely be identified from the context, we briefly say that f *is generalized*
n-convex.

If ωωωωωωωωω is the polynomial Chebyshev system, the definition leads to the notion of higher-order
monotonicity which was introduced and studied by T. Popoviciu in a sequence of papers [20, 22,
21, 24, 23, 27, 29, 25, 30, 28, 26, 31, 33, 32, 34, 35]. A summary of these results can be found
in [36] and [17]. For the sake of uniform terminology, throughout the this paper Popoviciu’s
setting is called polynomial convexity. That is, a functionf :I →R*is said to be polynomially*
n-convex if, for all elementsx_{0} <· · ·< x_{n}ofI, it satisfies the inequality

(−1)^{n}

f(x_{0}) . . . f(x_{n})
1 . . . 1
x_{0} . . . x_{n}

... . .. ...
x^{n−1}_{0} . . . x^{n−1}_{n}

≥0.

Observe that polynomially2-convex functions are exactly the “standard” convex ones. The case,
when the “generalized” convexity notion is induced by the special two dimensional Chebyshev
systemω_{1}(x) := 1andω_{2}(x) := x, is termed standard setting and standard convexity, respec-
tively.

The integral average of any standard convex functionf : [a, b] → Rcan be estimated from the midpoint and the endpoints of the domain as follows:

f

a+b 2

≤ 1 b−a

Z b a

f(x)dx≤ f(a) +f(b)

2 .

This is the well known Hadamard’s inequality ([11]) or, as it is quoted for historical reasons (see [12] and also [18] for interesting remarks), the Hermite–Hadamard-inequality.

The aim of this paper is to verify analogous inequalities for generalized convex functions, that is, to give lower and upper estimations for the integral average of the function using certain base points of the domain. Of course, the base points are supposed to depend only on the underlying Chebyshev system of the induced convexity.

For this purpose, we shall follow an inductive approach since it seems to have more advan- tages than the deductive one. First of all, it makes the original motivations clear; on the other hand, it allows us to use the most suitable mathematical tools. Hence sophisticated proofs that sometimes occur when using a deductive approach can also be avoided.

SECTION 2 investigates the case of polynomial convexity. The base points of the Hermite–

Hadamard-type inequalities turn out to be the zeros of certain orthogonal polynomials. The main tools of the section are based on some methods of numerical analysis, like the Gauss quadrature formula and Hermite-interpolation. A smoothing technique and two theorems of Popoviciu are also crucial.

In SECTION 3 we present Hermite–Hadamard-type inequalities for generalized 2-convex functions. The most important auxiliary result of the proof is a characterization theorem which, in the standard setting, reduces to the well known characterization properties of convex func- tions. Another theorem of the section establishes a tight relationship between standard and generalized2-convexity. This result has important regularity consequences and is also essential in verifying Hermite–Hadamard-type inequalities.

The general case is studied in SECTION 4. The main results guarantee only the existence and also the uniqueness of the base points of the Hermite–Hadamard-type inequalities but offer no explicit formulae for determining them. The main tool of the section is the Krein–Markov theory of moment spaces induced by Chebyshev systems. In some special cases (when the di- mension of the underlying Chebyshev systems are “small”), an elementary alternative approach is also presented.

SECTION 5 is devoted to showing that, at least in the two dimensional case and requiring weak regularity conditions, Hermite–Hadamard-type inequalities are not merely the conse- quences of generalized convexity, but they also characterize it.

Specializing the members of Chebyshev systems, several applications and examples are pre- sented for concrete Hermite–Hadamard-type inequalities in both the cases of polynomial con- vexity and generalized2-convexity. As a simple consequence, the classical Hermite–Hadamard inequality is among the corollaries in each case as well.

The results of this paper can be found in [3, 4, 5, 6, 7] and [1]. In what follows, we present them without any further references to the mentioned papers.

**2. P****OLYNOMIAL** **C****ONVEXITY**

The main results of this section state Hermite–Hadamard-type inequalities for polynomially
convex functions. Let us recall that a functionf : I → R*is said to be polynomially*n-convex
if, for all elementsx_{0} <· · ·< x_{n}ofI, it satisfies the inequality

(−1)^{n}

f(x_{0}) . . . f(x_{n})
1 . . . 1
x_{0} . . . x_{n}

... . .. ...
x^{n−1}_{0} . . . x^{n−1}_{n}

≥0.

In order to determine the base points and coefficients of the inequalities, Gauss-type quadrature formulae are applied. Then, using the remainder term of the Hermite-interpolation, the main

results follow immediately for “sufficiently smooth” functions due to the next two theorems of Popoviciu:

* Theorem A. ([17, Theorem 1. p. 387]) Assume that* f : I → R

*is continuous and*n

*times*

*differentiable on the interior of*I. Then,f

*is polynomially*n-convex if and only iff

^{(n)}≥ 0

*on*

*the interior of*I.

* Theorem B. ([17, Theorem 1. p. 391]) Assume that*f : I → R

*is polynomially*n-convex and n≥2. Then,f

*is*(n−2)

*times differentiable and*f

^{(n−2)}

*is continuous on the interior of*I.

To drop the regularity assumptions, a smoothing technique is developed that guarantees the approximation of polynomially convex functions with smooth polynomially convex ones.

**2.1. Orthogonal polynomials and basic quadrature formulae. In what follows,**ρdenotes a
*positive, locally integrable function (briefly: weight function) on an interval*I. The polynomials
P andQ*are said to be orthogonal on*[a, b]⊂ I*with respect to the weight function*ρor simply
ρ-orthogonal on[a, b]if

hP, Qi_{ρ} :=

Z b a

P Qρ= 0.

A system of polynomials is called a ρ-orthogonal polynomial system on [a, b] ⊂ I if each member of the system is ρ-orthogonal to the others on [a, b]. Define the moments ofρ by the formulae

µ_{k} :=

Z b a

x^{k}ρ(x)dx (k = 0,1,2, . . .).

Then, then^{th}degree member of theρ-orthogonal polynomial system on[a, b]has the following
representation via the moments ofρ:

P_{n}(x) :=

1 µ_{0} · · · µn−1

x µ_{1} · · · µ_{n}
... ... . .. ...
x^{n} µ_{n} · · · µ2n−1

.

Clearly, it suffices to show that P_{n} is ρ-orthogonal to the special polynomials 1, x, . . . , x^{n−1}.
Indeed, fork = 1, . . . , n, the first and the(k+ 1)^{st} columns of the determinanthP_{n}(x), x^{k−1}i_{ρ}
are linearly dependent according to the definition of the moments.

In fact, the moments and the orthogonal polynomials depend heavily on the interval [a, b].

Therefore, we use the notionsµk;[a,b] andPn;[a,b]instead ofµk andPn above when we want to or have to emphasize the dependence on the underlying interval.

Throughout this section, the following property of the zeros of orthogonal polynomials plays
a key role (see [39]). Let P_{n} denote the n^{th} degree member of the ρ-orthogonal polynomial
system on[a, b]. Then,P_{n}hasnpairwise distinct zerosξ_{1} <· · ·< ξ_{n}in]a, b[.

Let us consider the following Z b

a

f ρ=

n

X

k=1

c_{k}f(ξ_{k}),
(2.1)

Z b a

f ρ=c_{0}f(a) +

n

X

k=1

c_{k}f(ξ_{k}),
(2.2)

Z b a

f ρ=

n

X

k=1

c_{k}f(ξ_{k}) +c_{n+1}f(b),
(2.3)

Z b a

f ρ=c_{0}f(a) +

n

X

k=1

c_{k}f(ξ_{k}) +c_{n+1}f(b).

(2.4)

Gauss-type quadrature formulae where the coefficients and the base points are to be determined so that (2.1), (2.2), (2.3) and (2.4) are exact whenf is a polynomial of degree at most2n−1, 2n,2nand2n+ 1, respectively. The subsequent four theorems investigate these cases.

* Theorem 2.1. Let*P

_{n}

*be the*n

^{th}

*degree member of the orthogonal polynomial system on*[a, b]

*with respect to the weight function* ρ. Then (2.1) is exact for polynomialsf *of degree at most*
2n−1*if and only if*ξ_{1}, . . . , ξ_{n}*are the zeros of*P_{n}*, and*

(2.5) c_{k}=

Z b a

P_{n}(x)

(x−ξ_{k})P_{n}^{0}(ξ_{k})ρ(x)dx.

*Furthermore,*ξ1, . . . , ξn*are pairwise distinct elements of*]a, b[, andck ≥0*for all*k = 1, . . . , n.

This theorem follows easily from well known results in numerical analysis (see [13], [14], [39]). For the sake of completeness, we provide a proof.

*Proof. First assume that*ξ_{1}, . . . , ξ_{n}are the zeros of the polynomialP_{n}and, for allk = 1, . . . , n,
denote the primitive Lagrange-interpolation polynomials byL_{k}: [a, b]→R. That is,

Lk(x) :=

P_{n}(x)

(x−ξ_{k})P_{n}^{0}(ξ_{k}) ifx6=ξ_{k}

1 ifx=ξ_{k}.

IfQ is a polynomial of degree at most2n−1, then, using the Euclidian algorithm, Qcan be
written in the formQ=P P_{n}+RwheredegP,degR ≤n−1. The inequalitydegP ≤n−1
implies theρ-orthogonality ofP andP_{n}:

Z b a

P Pnρ= 0.

On the other hand,degR ≤n−1yields thatRis equal to its Lagrange-interpolation polyno- mial:

R=

n

X

k=1

R(ξ_{k})L_{k}.

Therefore, considering the definition of the coefficientsc_{1}, . . . , c_{n} in formula (2.5), we obtain
that

Z b a

Qρ= Z b

a

P Pnρ+ Z b

a

Rρ=

n

X

k=1

R(ξk) Z b

a

Lkρ

=

n

X

k=1

ckR(ξk) =

n

X

k=1

ck P(ξk)Pn(ξk) +R(ξk)

=

n

X

k=1

ckQ(ξk).

That is, the quadrature formula (2.1) is exact for polynomials of degree at most2n−1.

Conversely, assume that (2.1) is exact for polynomials of degree at most2n−1. Define the
polynomialQby the formulaQ(x) := (x−ξ_{1})· · ·(x−ξ_{n})and letP be a polynomial of degree
at mostn−1. Then,degP Q≤2n−1, and thus

Z b a

P Qρ=c1P(ξ1)Q(ξ1) +· · ·+cnP(ξn)Q(ξn) = 0.

ThereforeQisρ-orthogonal toP. The uniqueness ofP_{n}implies thatP_{n} =a_{n}Q, andξ_{1}, . . . , ξ_{n}
are the zeros ofP_{n}. Furthermore, (2.1) is exact if we substitute f := L_{k}andf :=L^{2}_{k}, respec-
tively. The first substitution gives (2.5), while the second one shows the nonnegativity of c_{k}.

For further details, consult the book [39, p. 44].

* Theorem 2.2. Let*P

_{n}

*be the*n

^{th}

*degree member of the orthogonal polynomial system on*[a, b]

*with respect to the weight function*ρa(x) := (x−a)ρ(x). Then (2.2) is exact for polynomialsf
*of degree at most*2n*if and only if*ξ1, . . . , ξn*are the zeros of*Pn*, and*

c_{0} = 1
P_{n}^{2}(a)

Z b a

P_{n}^{2}(x)ρ(x)dx,
(2.6)

ck = 1
ξ_{k}−a

Z b a

(x−a)P_{n}(x)

(x−ξ_{k})P_{n}^{0}(ξ_{k})ρ(x)dx.

(2.7)

*Furthermore,*ξ_{1}, . . . , ξ_{n}*are pairwise distinct elements of*]a, b[, andc_{k} ≥0*for all*k = 0, . . . , n.

*Proof. Assume that the quadrature formula (2.2) is exact for polynomials of degree at most*2n.

IfP is a polynomial of degree at most2n−1, then Z b

a

P ρ_{a}=
Z b

a

(x−a)P(x)ρ(x)dx=c_{1}(ξ_{1}−a)P(ξ_{1}) +· · ·+c_{n}(ξ_{n}−a)P(ξ_{n}).

Applying Theorem 2.1 to the weight functionρ_{a}and the coefficients
ca;k :=ck(ξk−a),

we get thatξ_{1}, . . . , ξ_{n} are the zeros ofP_{n}and, for allk = 1, . . . , n, the coefficientsc_{a;k} can be
computed using formula (2.5). Therefore,

ck(ξk−a) = Z b

a

P_{n}(x)

(x−ξ_{k})P_{n}^{0}(ξ_{k})ρa(x)dx=
Z b

a

(x−a)P_{n}(x)

(x−ξ_{k})P_{n}^{0}(ξ_{k})ρ(x)dx.

Substitutingf :=P_{n}^{2} into (2.1), we obtain that
c_{0} = 1

P_{n}^{2}(a)
Z b

a

P_{n}^{2}ρ.

Thus (2.6) and (2.7) are valid, andc_{k} ≥0fork = 0,1, . . . , n.

Conversely, assume that ξ_{1}, . . . , ξ_{n} are the zeros of the orthogonal polynomial P_{n}, and the
coefficientsc_{1}, . . . , c_{n}are given by the formula (2.7). Define the coefficientc_{0} byc_{0} = Rb

a ρ−
(c_{1} +· · ·+c_{n}). IfP is a polynomial of degree at most2n, then there exists a polynomial Q
withdegQ≤2n−1such that

P(x) = (x−a)Q(x) +P(a).

Indeed, the polynomialP(x)−P(a)vanishes at the pointx=a, hence it is divisible by(x−a).

Applying Theorem 2.1 again to the weight functionρ_{a},
Z b

a

Qρa=ca;1Q(ξ1) +· · ·+ca;nQ(ξn)

holds. Thus, using the definition ofc_{0}, the representation of the polynomialP and the quadra-
ture formula above, we have that

Z b a

P(x)ρ(x)dx= Z b

a

(x−a)Q(x) +P(a)

ρ(x)dx

=

n

X

k=1

c_{k}(ξ_{k}−a)Q(ξ_{k}) +

n

X

k=0

P(a)c_{k}

=c_{0}P(a) +

n

X

k=1

c_{k} (ξ_{k}−a)Q(ξ_{k}) +P(a)

=c_{0}P(a) +

n

X

k=1

c_{k}P(ξ_{k}),

which yields that the quadrature formula (2.2) is exact for polynomials of degree at most 2n.

Therefore, substitutingf :=P_{n}^{2}into (2.2), we get formula (2.6).

* Theorem 2.3. Let*Pn

*be the*n

^{th}

*degree member of the orthogonal polynomial system on*[a, b]

*with respect to the weight function*ρ^{b}(x) := (b−x)ρ(x). Then (2.3) is exact for polynomialsf
*of degree at most*2n*if and only if*ξ_{1}, . . . , ξ_{n}*are the zeros of*P_{n}*, and*

c_{k}= 1
b−ξ_{k}

Z b a

(b−x)P_{n}(x)

(x−ξ_{k})P_{n}^{0}(ξ_{k})ρ(x)dx,
(2.8)

c_{n+1} = 1
P_{n}^{2}(b)

Z b a

P_{n}^{2}(x)ρ(x)dx.

(2.9)

*Furthermore,*ξ1, . . . , ξn*are pairwise distinct elements of*]a, b[, andck ≥0*for all*k = 1, . . . , n+

1.

*Hint. Applying a similar argument to the previous one to the weight function*ρ^{b}, we obtain the

statement of the theorem.

* Theorem 2.4. Let*Pn

*be the*n

^{th}

*degree member of the orthogonal polynomial system on*[a, b]

*with respect to the weight function*ρ^{b}_{a}*. Then (2.4) is exact for polynomials*f *of degree at most*
2n+ 1*if and only if*ξ_{1}, . . . , ξ_{n}*are the zeros of*P_{n}*, and*

c_{0} = 1

(b−a)P_{n}^{2}(a)
Z b

a

(b−x)P_{n}^{2}(x)ρ(x)dx,
(2.10)

ck = 1

(b−ξ_{k})(ξ_{k}−a)
Z b

a

(b−x)(x−a)P_{n}(x)

(x−ξ_{k})P_{n}^{0}(ξ_{k}) ρ(x)dx,
(2.11)

c_{n+1} = 1
(b−a)P_{n}^{2}(b)

Z b a

(x−a)P_{n}^{2}(x)ρ(x)dx.

(2.12)

*Furthermore,*ξ_{1}, . . . , ξ_{n}*are pairwise distinct elements of*]a, b[, andc_{k} ≥0*for all*k = 0, . . . , n+

1.

*Proof. Assume that the quadrature formula (2.4) is exact for polynomials of degree at most*
2n+ 1. IfP is a polynomial of degree at most2n−1, then

Z b a

P ρ^{b}_{a} =
Z b

a

(b−x)(x−a)P(x)ρ(x)dx

=c_{1}(b−ξ_{1})(ξ_{1}−a)P(ξ_{1}) +· · ·+c_{n}(b−ξ_{n})(ξ_{n}−a)P(ξ_{n}).

Applying Theorem 2.1 to the weight functionρ^{b}_{a}and the coefficients
c_{a,b;k} :=c_{k}(b−ξ_{k})(ξ_{k}−a),

we get thatξ_{1}, . . . , ξ_{n}are the zeros ofP_{n}and, for allk = 1, . . . , n, the coefficientsc_{a,b;k}can be
computed using formula (2.5). Therefore,

c_{k}(b−ξ_{k})(ξ_{k}−a) =
Z b

a

P_{n}(x)

(x−ξk)P_{n}^{0}(ξk)ρ^{b}_{a}(x)dx

= Z b

a

(b−x)(x−a)P_{n}(x)

(x−ξ_{k})P_{n}^{0}(ξ_{k}) ρ(x)dx.

Substitutingf := (b−x)P_{n}^{2}(x)andf := (x−a)P_{n}^{2}(x)into (2.1), we obtain that

c0 = 1

(b−a)P_{n}^{2}(a)
Z b

a

(b−x)P_{n}^{2}(x)ρ(x)dx,
c_{n+1} = 1

(b−a)P_{n}^{2}(b)
Z b

a

(x−a)P_{n}^{2}(x)ρ(x)dx.

Thus (2.10), (2.11) and (2.12) are valid, furthermore,c_{k}≥0fork = 0, . . . , n+ 1.

Conversely, assume that ξ_{1}, . . . , ξ_{n} are the zeros of P_{n}, and the coefficients c_{1}, . . . , c_{n} are
given by the formula (2.11). Define the coefficientsc_{0} andc_{n+1} by the equations

Z b a

(b−x)ρ(x)dx=c_{0}(b−a) +

n

X

k=1

c_{k}(b−ξ_{k}),
Z b

a

(x−a)ρ(x)dx=

n

X

k=1

c_{k}(ξ_{k}−a) +c_{n+1}(b−a).

IfP is a polynomial of degree at most2n+ 1, then there exists a polynomialQwithdegQ≤ 2n−1such that

(b−a)P(x) = (b−x)(x−a)Q(x) + (x−a)P(b) + (b−x)P(a).

Indeed, the polynomial(b−a)P(x)−(x−a)P(b)−(b−x)P(a)is divisible by(b−x)(x−a) sincex=aandx=bare its zeros. Applying Theorem 2.1 again,

Z b a

Qρ^{b}_{a}=c_{a,b;1}Q(ξ_{1}) +· · ·+c_{a,b;n}Q(ξ_{n})

holds. Thus, using the definition ofc_{0} andc_{n+1}, the representation of the polynomialP and the
quadrature formula above, we have that

(b−a) Z b

a

P(x)ρ(x)dx

= Z b

a

(b−x)(x−a)Q(x) + (x−a)P(b) + (b−x)P(a)

ρ(x)dx

=

n

X

k=1

c_{k}(b−ξ_{k})(ξ_{k}−a)Q(ξ_{k})
+P(b)

Z b a

(x−a)ρ(x)dx+P(a) Z b

a

(b−x)ρ(x)dx

=

n

X

k=1

c_{k}(b−ξ_{k})(ξ_{k}−a)Q(ξ_{k})
+c_{0}(b−a)P(a) +

n

X

k=1

c_{k}(b−ξ_{k})P(a)
+

n

X

k=1

c_{k}(ξ_{k}−a)P(b) +c_{n+1}(b−a)P(b)

=

n

X

k=1

c_{k} (b−ξ_{k})(ξ_{k}−a)Q(ξ_{k}) + (ξ_{k}−a)P(b) + (b−ξ_{k})P(a)
+c_{0}(b−a)P(a) +c_{n+1}(b−a)P(b)

=c_{0}(b−a)P(a) +

n

X

k=1

c_{k}(b−a)P(ξ_{k}) +c_{n+1}(b−a)P(b),

which yields that the quadrature formula (2.4) is exact for polynomials of degree at most2n+ 1.

Therefore, substitutingf := (b−x)P_{n}^{2}(x)andf := (x−a)P_{n}^{2}(x)into (2.4), formulae (2.10)

and (2.12) follow.

Letf : [a, b] → Rbe a differentiable function, x_{1}, . . . , x_{n} be pairwise distinct elements of
[a, b], and1≤r ≤nbe a fixed integer. We denote the Hermite interpolation polynomial byH,
which satisfies the following conditions:

H(x_{k}) =f(x_{k}) (k = 1, . . . , n),
H^{0}(x_{k}) =f^{0}(x_{k}) (k = 1, . . . , r).

We recall thatdegH =n+r−1. From a well known result, (see [13, Sec. 5.3, pp. 230-231]), for allx∈[a, b]there existsθsuch that

(2.13) f(x)−H(x) = ωn(x)ωr(x)

(n+r)! f^{(n+r)}(θ),
where

ω_{k}(x) = (x−x_{1})· · ·(x−x_{k}).

**2.2. An approximation theorem. It is well known that there exists a function** ϕ which pos-
sesses the following properties:

(i) ϕ :R→R+ isC^{∞}, i. e., it is infinitely many times differentiable;

(ii) suppϕ ⊂[−1,1];

(iii) R

Rϕ = 1.

Usingϕ, one can define the functionϕ_{ε}for allε >0by the formula
ϕ_{ε}(x) = 1

εϕx ε

(x∈R).

Then, as it can easily be checked,ϕεsatisfies the following conditions:

(i’) ϕ_{ε}:R→R+isC^{∞};
(ii’) suppϕε⊂[−ε, ε];

(iii’) R

Rϕε= 1.

LetI ⊂ R be a nonempty open interval, f : I → Rbe a continuous function, and choose
ε >0. Denote the convolution off andϕ_{ε}byf_{ε}, that is

f_{ε}(x) :=

Z

R

f(y)ϕ¯ _{ε}(x−y)dy (x∈R)

wheref(y) =¯ f(y)ify ∈I, otherwisef¯(y) = 0. Let us recall, thatf_{ε}→f uniformly asε →0
on each compact subinterval of I, and f_{ε} is infinitely many times differentiable on R. These
important results can be found for example in [40, p. 549].

* Theorem 2.5. Let* I ⊂ R

*be an open interval,*f : I → R

*be a polynomially*n-convex con-

*tinuous function. Then, for all compact subintervals*[a, b] ⊂ I, there exists a sequence of

*polynomially*n-convex andC

^{∞}

*functions*(fk)

*which converges uniformly to*f

*on*[a, b].

*Proof. Choose*a, b∈I andε_{0} >0such that the inclusion[a−ε_{0}, b+ε_{0}]⊂I holds. We show
that the functionτ_{ε}f : [a, b]→Rdefined by the formula

τ_{ε}f(x) :=f(x−ε)

is polynomiallyn-convex on[a, b]for0< ε < ε0. Leta ≤x0 <· · ·< xn ≤b andk ≤ n−1 be fixed. By induction, we are going to verify the identity

(2.14)

τ_{ε}f(x_{0}) · · · τ_{ε}f(x_{n})
1 · · · 1
x_{0} · · · x_{n}

... . .. ...
x^{k−1}_{0} · · · x^{k−1}_{n}

x^{k}_{0} · · · x^{k}_{n}
... . .. ...
x^{n−1}_{0} · · · x^{n−1}_{n}

=

τ_{ε}f(x_{0}) · · · τ_{ε}f(x_{n})

1 · · · 1

x_{0}−ε · · · x_{n}−ε
... . .. ...
(x_{0}−ε)^{k−1} · · · (x_{n}−ε)^{k−1}

x^{k}_{0} · · · x^{k}_{n}
... . .. ...
x^{n−1}_{0} · · · x^{n−1}_{n}

.

Ifk = 1, then this equation obviously holds. Assume, for a fixed positive integerk ≤ n−2, that (2.14) remains true. The binomial theorem implies the identity

x^{k} =
k

0

ε^{k}+
k

1

ε^{k−1}(x−ε) +· · ·+
k

k

(x−ε)^{k}.

That is,(x−ε)^{k}is the linear combination of the elements1, x−ε, . . . ,(x−ε)^{k}andx^{k}. Therefore,
adding the appropriate linear combination of the2^{nd}, . . . ,(k+ 1)^{st} rows to the (k+ 2)^{nd} row,
we arrive at the equation

τ_{ε}f(x_{0}) · · · τ_{ε}f(x_{n})

1 · · · 1

x_{0}−ε · · · x_{n}−ε
... . .. ...
(x_{0}−ε)^{k−1} · · · (x_{n}−ε)^{k−1}

x^{k}_{0} · · · x^{k}_{n}
x^{k+1}_{0} · · · x^{k+1}_{n}

... . .. ...
x^{n−1}_{0} · · · x^{n−1}_{n}

=

τ_{ε}f(x_{0}) · · · τ_{ε}f(x_{n})

1 · · · 1

x_{0}−ε · · · x_{n}−ε
... . .. ...
(x_{0}−ε)^{k−1} · · · (x_{n}−ε)^{k−1}

(x_{0}−ε)^{k} · · · (x_{n}−ε)^{k}
x^{k+1}_{0} · · · x^{k+1}_{n}

... . .. ...
x^{n−1}_{0} · · · x^{n−1}_{n}

.

Hence formula (2.14) holds for all fixed positivek whenever 1 ≤ k ≤ n−1. The particular
casek = n−1gives the polynomialn-convexity ofτ_{ε}f. Applying a change of variables and

the previous result, we get that

(−1)^{n}

f_{ε}(x_{0}) · · · f_{ε}(x_{n})
1 · · · 1
x_{0} · · · x_{n}

... . .. ...
x^{n−1}_{0} · · · x^{n−1}_{n}

= Z

R

(−1)^{n}

f(t)ϕ¯ _{ε}(x_{0}−t) · · · f¯(t)ϕ_{ε}(x_{n}−t)

1 · · · 1

x_{0} · · · x_{n}

... . .. ...

x^{n−1}_{0} · · · x^{n−1}_{n}

dt

= Z

R

(−1)^{n}

f(x¯ 0−s) · · · f¯(xn−s)
1 · · · 1
x_{0} · · · x_{n}

... . .. ...
x^{n−1}_{0} · · · x^{n−1}_{n}

ϕ_{ε}(s)ds

= Z

R

(−1)^{n}

τ_{s}f(x_{0}) · · · τ_{s}f(x_{n})
1 · · · 1
x_{0} · · · x_{n}

... . .. ...
x^{n−1}_{0} · · · x^{n−1}_{n}

ϕ_{ε}(s)ds ≥0,

which shows the polynomialn-convexity off_{ε}on[a, b]for0< ε < ε_{0}.

To complete the proof, choose a positive integern_{0} such that the relation _{n}^{1}

0 < ε_{0} holds. If
we defineε_{k}andf_{k}byε_{k} := _{n}^{1}

0+k andf_{k} :=f_{ε}_{k} fork ∈N, then0< ε_{k} < ε_{0}, and thus(f_{k})^{∞}_{k=1}

satisfies the requirements of the theorem.

**2.3. Hermite–Hadamard-type inequalities. In the sequel, we shall need two additional aux-**
iliary results. The first one investigates the convergence properties of the zeros of orthogonal
polynomials.

* Lemma 2.1. Let*ρ

*be a weight function on*[a, b], and(a

_{j})

*be strictly monotone decreasing,*(b

_{j})

*be strictly monotone increasing sequences such that*aj →a,bj → b

*and*a1 < b1

*. Denote the*

*zeros of*P

_{m;j}

*by*ξ

_{1;j}, . . . , ξ

_{m;j}

*, where*P

_{m;j}

*is the*m

^{th}

*degree member of the*ρ|

_{[a}

_{j}

_{,b}

_{j}

_{]}

*-orthogonal*

*polynomial system on*[aj, bj], and denote the zeros ofPm

*by*ξ1, . . . , ξm

*, where*Pm

*is the*m

^{th}

*degree member of the*ρ-orthogonal polynomial system on[a, b]. Then,

j→∞lim ξk;j =ξk (k= 1, . . . , n).

*Proof. Observe first that the mapping* (a, b) 7→ µ_{k;[a,b]} is continuous, therefore µ_{k;[a}_{j}_{,b}_{j}_{]} →
µ_{k;[a,b]}henceP_{m;j} → P_{m}pointwise according to the representation of orthogonal polynomials.

Takeε >0such that

]ξ_{k}−ε, ξ_{k}+ε[⊂]a, b[,

]ξk−ε, ξk+ε[∩]ξl−ε, ξl+ε[=∅ (k 6=l, k, l ∈ {1, . . . , m}).

The polynomialP_{m} changes its sign on ]ξ_{k} −ε, ξ_{k} +ε[since it is of degree m and it has m
pairwise distinct zeros; therefore, due to the pointwise convergence,P_{m;j} also changes its sign
on the same interval up to an index. That is, for sufficiently largej,ξ_{k;j} ∈]ξ_{k}−ε, ξ_{k}+ε[.

The other auxiliary result investigates the one-sided limits of polynomially n-convex func- tions at the endpoints of the domain. Let us note that its first assertion involves, in fact, two cases according to the parity of the convexity.

* Lemma 2.2. Let*f : [a, b]→R

*be a polynomially*n-convex function. Then, (i) (−1)

^{n}f(a)≥lim sup

_{t→a+0}(−1)

^{n}f(t);

(ii) f(b)≥lim sup_{t→b−0}f(t).

*Proof. It suffices to restrict the investigations to the even case of assertion* (i) only since the
proofs of the other ones are completely the same. For the sake of brevity, we shall use the
notation f_{+}(a) := lim sup_{t→a+0}f(t). Take the elements x_{0} := a < x_{1} := t < · · · < x_{n} of
[a, b]. Then, the (even order) polynomial convexity off implies

f(a) f(t) f(x_{2}) . . . f(x_{n})

1 1 1 . . . 1

a t x_{2} . . . x_{n}
... ... ... . .. ...
a^{n−1} t^{n−1} x^{n−1}_{2} . . . x^{n−1}_{n}

≥0.

Therefore, taking the limsup ast→a+ 0, we obtain that

f(a) f+(a) f(x2) . . . f(xn)

1 1 1 . . . 1

a a x_{2} . . . x_{n}
... ... ... . .. ...
a^{n−1} a^{n−1} x^{n−1}_{2} . . . x^{n−1}_{n}

≥0.

The adjoint determinants of the elements f(x_{2}), . . . , f(x_{n}) in the first row are equal to zero
since their first and second columns coincide; on the other hand,f(a)andf_{+}(a)have the same
(positive) Vandermonde-type adjoint determinant. Hence, applying the expansion theorem on
the first row, we obtain the desired inequality

f(a)−f_{+}(a)≥0.

The main results concern the cases of odd and even order polynomial convexity separately in the subsequent two theorems.

* Theorem 2.6. Let*ρ : [a, b] → R

*be a positive integrable function. Denote the zeros of*Pm

*by*ξ

_{1}, . . . , ξ

_{m}

*where*P

_{m}

*is the*m

^{th}

*degree member of the orthogonal polynomial system on*[a, b]

*with respect to the weight function* (x− a)ρ(x), and denote the zeros of Q_{m} *by* η_{1}, . . . , η_{m}
*where*Q_{m} *is the*m^{th}*degree member of the orthogonal polynomial system on*[a, b]*with respect*
*to the weight function*(b−x)ρ(x). Define the coefficientsα_{0}, . . . , α_{m} *and*β_{1}, . . . , β_{m+1} *by the*
*formulae*

α0 := 1
P_{m}^{2}(a)

Z b a

P_{m}^{2}(x)ρ(x)dx,
α_{k} := 1

ξ_{k}−a
Z b

a

(x−a)P_{m}(x)

(x−ξ_{k})P_{m}^{0} (ξ_{k})ρ(x)dx

*and*

β_{k}:= 1
b−ηk

Z b a

(b−x)Q_{m}(x)

(x−ηk)Q^{0}_{m}(ηk)ρ(x)dx,
β_{m+1} := 1

Q^{2}_{m}(b)
Z b

a

Q^{2}_{m}(x)ρ(x)dx.

*If a function* f : [a, b] → R *is polynomially* (2m + 1)-convex, then it satisfies the following
*Hermite–Hadamard-type inequality*

α_{0}f(a) +

m

X

k=1

α_{k}f(ξ_{k})≤
Z b

a

f ρ≤

m

X

k=1

β_{k}f(η_{k}) +β_{m+1}f(b).

*Proof. First assume that* f is(2m + 1) times differentiable. Then, according to Theorem A,
f^{(2m+1)} ≥0on]a, b[. LetH be the Hermite interpolation polynomial determined by the condi-
tions

H(a) =f(a),
H(ξ_{k}) =f(ξ_{k}),
H^{0}(ξ_{k}) =f^{0}(ξ_{k}).

By the remainder term (2.13) of the Hermite interpolation, ifxis an arbitrary element of]a, b[, then there existsθ∈]a, b[such that

f(x)−H(x) = (x−a)(x−ξ_{1})^{2}· · ·(x−ξ_{m})^{2}

(2m+ 1)! f^{(2m+1)}(θ).

That is, f ρ ≥ Hρon[a, b]due to the nonnegativity off^{(2m+1)} and the positivity ofρ. On the
other hand,H is of degree2m, therefore Theorem 2.2 yields that

Z b a

f ρ≥ Z b

a

Hρ=α_{0}H(a) +

m

X

k=1

α_{k}H(ξ_{k}) =α_{0}f(a) +

m

X

k=1

α_{k}f(ξ_{k}).

For the general case, letf be an arbitrary polynomially(2m+ 1)-convex function. Without
loss of generality we may assume thatm≥1; in this case,f is continuous (see Theorem B). Let
(a_{j})and(b_{j})be sequences fulfilling the requirements of Lemma 2.1. According to Theorem 2.5,
there exists a sequence ofC^{∞}, polynomially(2m+1)-convex functions(f_{i;j})such thatf_{i;j} →f
uniformly on [a_{j}, b_{j}] as i → ∞. Denote the zeros ofP_{m;j} by ξ_{1;j}, . . . , ξ_{m;j} where P_{m;j} is the
m^{th} degree member of the orthogonal polynomial system on[a_{j}, b_{j}]with respect to the weight
function(x−a)ρ(x). Define the coefficientsα_{0;j}, . . . , α_{m;j} analogously toα_{0}, . . . , α_{m}with the
help ofP_{m;j}. Then,ξ_{k;j} → ξ_{k}due to Lemma 2.1, and henceα_{k;j} → α_{k}asj → ∞. Applying
the previous step of the proof on the smooth functions(f_{i;j}), it follows that

α_{0;j}f_{i;j}(a_{j}) +

m

X

k=1

α_{k;j}f_{i;j}(ξ_{k;j})≤
Z bj

aj

f_{i;j}ρ.

Taking the limitsi→ ∞and thenj → ∞, we get the inequality
α_{0}

lim inf

t→a+0 f(t) +

m

X

k=1

α_{k}f(ξ_{k})≤
Z b

a

f ρ.

This, together with Lemma 2.2, gives the left hand side inequality to be proved. The proof of the right hand side inequality is analogous, therefore it is omitted.

The second main result offers Hermite–Hadamard-type inequalities for even-order polynomi- ally convex functions. In this case, the symmetrical structure disappears: the lower estimation involves none of the endpoints, while the upper estimation involves both of them.

* Theorem 2.7. Let*ρ : [a, b] → R

*be a positive integrable function. Denote the zeros of*Pm

*by*ξ1, . . . , ξm

*where*Pm

*is the*m

^{th}

*degree member of the orthogonal polynomial system on*[a, b]

*with respect to the weight function*ρ(x), and denote the zeros ofQ_{m−1} *by*η_{1}, . . . , η_{m−1} *where*
Q_{m−1}*is the*(m−1)^{st}*degree member of the orthogonal polynomial system on*[a, b]*with respect*
*to the weight function*(b−x)(x−a)ρ(x). Define the coefficientsα_{1}, . . . , α_{m} *and*β_{0}, . . . , β_{m+1}
*by the formulae*

α_{k}:=

Z b a

P_{m}(x)

(x−ξ_{k})P_{m}^{0} (ξ_{k})ρ(x)dx
*and*

β_{0} = 1

(b−a)Q^{2}_{m−1}(a)
Z b

a

(b−x)Q^{2}_{m−1}(x)ρ(x)dx,

β_{k} = 1

(b−η_{k})(ξ_{k}−a)
Z b

a

(b−x)(x−a)Qm−1(x)

(x−η_{k})Q^{0}_{m−1}(η_{k}) ρ(x)dx,

β_{m+1} = 1

(b−a)Q^{2}_{m−1}(b)
Z b

a

(x−a)Q^{2}_{m−1}(x)ρ(x)dx.

*If a function*f : [a, b]→R*is polynomially*(2m)-convex, then it satisfies the following Hermite–

*Hadamard-type inequality*

m

X

k=1

α_{k}f(ξ_{k})≤
Z b

a

f ρ≤β_{0}f(a) +

m−1

X

k=1

β_{k}f(η_{k}) +β_{m}f(b).

*Proof. First assume that*f isn = 2mtimes differentiable. Thenf^{(2m)} ≥ 0on]a, b[according
to Theorem B. Consider the Hermite interpolation polynomialH that interpolates the function
f in the zeros ofP_{m}in the following manner:

H(ξ_{k}) =f(ξ_{k}),
H^{0}(ξ_{k}) =f^{0}(ξ_{k}).

By the remainder term (2.13) of the Hermite interpolation, ifxis an arbitrary element of]a, b[, then there existsθ∈]a, b[such that

f(x)−H(x) = (x−ξ_{1})^{2}· · ·(x−ξ_{m})^{2}

(2m)! f^{(2m)}(θ).

Hencef ρ≥Hρon[a, b]due to the nonnegativity off^{(2m)}and the positivity ofρ. On the other
hand,H is of degree2m−1, therefore Theorem 2.1 yields the left hand side of the inequality
to be proved:

Z b a

f ρ≥ Z b

a

Hρ=

m

X

k=1

α_{k}H(ξ_{k}) =

m

X

k=1

α_{k}f(ξ_{k}).

Now consider the Hermite interpolation polynomial H that interpolates the function f at the zeros ofQm−1and at the endpoints of the domain in the following way:

H(a) =f(a),
H(η_{k}) =f(η_{k}),
H^{0}(η_{k}) =f^{0}(η_{k}),

H(b) =f(b).

By the remainder term (2.13) of the Hermite interpolation, ifxis an arbitrary element of]a, b[, then there exists aθ ∈]a, b[such that

f(x)−H(x) = (x−a)(x−b)(x−η_{1})^{2}· · ·(x−ηm−1)^{2}

(2m)! f^{(2m)}(θ).

The factors of the right hand side are nonnegative except for the factor(x−b)which is negative, hencef ρ≤ Hρ. On the other hand, His of degree 2m−1, therefore Theorem 2.4 yields the right hand side inequality to be proved:

Z b a

f ρ≤ Z b

a

Hρ=β_{0}H(a) +

m−1

X

k=1

β_{k}H(η_{k}) +β_{m}H(b)

=β0f(a) +

m−1

X

k=1

βkf(ηk) +βmf(b).

From this point, an analogous argument to the corresponding part of the previous proof gives the statement of the theorem without any differentiability assumptions on the functionf.

Specializing the weight functionρ≡1, the roots of the inequalities can be obtained as convex combinations of the endpoints of the domain. The coefficients of the convex combinations are the zeros of certain orthogonal polynomials on[0,1]in both cases. Observe that interchanging the role of the endpoints in any side of the inequality concerning the odd order case, we obtain the other side of the inequality.

* Theorem 2.8. Let, for*m≥0, the polynomialP

_{m}

*be defined by the formula*

Pm(x) :=

1 ^{1}_{2} · · · _{m+1}^{1}
x ^{1}_{3} · · · _{m+2}^{1}
*...* *...* *. ..* *...*
x^{m} _{m+2}^{1} · · · _{2m+1}^{1}

.

*Then,*Pm *has*m*pairwise distinct zeros*λ1, . . . , λm *in*]0,1[. Define the coefficientsα0, . . . , αm

*by*

α_{0} := 1
P_{m}^{2}(0)

Z 1 0

P_{m}^{2}(x)dx,
α_{k} := 1

λ_{k}
Z 1

0

xP_{m}(x)

(x−λ_{k})P_{m}^{0} (λ_{k})dx.

*If a function* f : [a, b] → R *is polynomially* (2m + 1)-convex, then it satisfies the following
*Hermite–Hadamard-type inequality*

α0f(a) +

m

X

k=1

αkf (1−λk)a+λkb

≤ 1 b−a

Z b a

f(x)dx

≤

m

X

k=1

αkf λka+ (1−λk)b

+α0f(b).

*Proof. Apply Theorem 2.6 in the particular setting when*a:= 0,b:= 1and the weight function
is ρ ≡ 1. Then, as simple calculations show, P_{m} is exactly the m^{th} degree member of the
orthogonal polynomial system on [0,1]with respect to the weight function ρ(x) = x(see the
beginning of this section). Therefore,P_{m}hasmpairwise distinct zeros0< λ_{1} <· · ·< λ_{m} <1.

Moreover, the coefficientsα_{0}, . . . , α_{m} have the form above. Define the functionF : [0,1]→R
by the formula

F(t) := f (1−t)a+tb .

It is easy to check thatF is polynomially(2m+1)-convex on the interval[0,1]. Hence, applying Theorem 2.6 and the previous observations, it follows that

Z 1 0

F(t)dt≥α_{0}F(0) +

m

X

k=1

α_{k}F(λ_{k})

=α_{0}f(a) +

m

X

k=1

α_{k}f (1−λ_{k})a+λ_{k}b
.

On the other hand, to complete the proof of the left hand side inequality, observe that 1

b−a Z b

a

f(x)dx= Z 1

0

F(t)dt.

For verifying the right hand side one, define the functionϕ : [a, b]→Rby the formula ϕ(x) :=−f(a+b−x).

Then,ϕis polynomially(2m+ 1)-convex on[a, b]. The previous inequality applied onϕgives
the upper estimation of the Hermite–Hadamard-type inequality forf.
* Theorem 2.9. Let, for*m≥1, the polynomialsP

_{m}

*and*Q

_{m−1}

*be defined by the formulae*

P_{m}(x) :=

1 1 · · · _{m}^{1}
x ^{1}_{2} · · · _{m+1}^{1}

*...* *...* *. ..* *...*
x^{m} _{m+1}^{1} · · · _{2m}^{1}

,

Qm−1(x) :=

1 _{2·3}^{1} · · · _{m(m+1)}^{1}
x _{3·4}^{1} · · · _{(m+1)(m+2)}^{1}

*...* *...* *. ..* *...*
x^{m−1} _{(m+1)(m+2)}^{1} · · · _{(2m−1)2m}^{1}

.

*Then,*P_{m} *has*m*pairwise distinct zeros*λ_{1}, . . . , λ_{m} *in*]0,1[*and*Qm−1 *has*m−1*pairwise dis-*
*tinct zeros*µ_{1}, . . . , µm−1*in*]0,1[, respectively. Define the coefficientsα_{1}, . . . , α_{m}*and*β_{0}, . . . , β_{m}
*by*

α_{k} :=

Z 1 0

P_{m}(x)

(x−λ_{k})P_{m}^{0} (λ_{k})dx

*and*

β_{0} := 1
Q^{2}_{m−1}(0)

Z 1 0

(1−x)Q^{2}_{m−1}(x)dx,
β_{k}:= 1

(1−µ_{k})µ_{k}
Z 1

0

x(1−x)Qm−1(x)
(x−µ_{k})Q^{0}_{m−1}(µ_{k})dx,
β_{m} := 1

Q^{2}_{m−1}(1)
Z 1

0

xQ^{2}_{m−1}(x)dx.

*If a function*f : [a, b]→R*is polynomially*(2m)-convex, then it satisfies the following Hermite–

*Hadamard-type inequality*

m

X

k=1

α_{k}f (1−λ_{k})a+λ_{k}b

≤ 1 b−a

Z b a

f(x)dx

≤β_{0}f(a) +

m−1

X

k=1

β_{k}f (1−µ_{k})a+µ_{k}b

+β_{m}f(b).

*Proof. Substitute* a := 0, b := 1and ρ ≡ 1 into Theorem 2.7. Then, P_{m} is exactly the m^{th}
degree member of the orthogonal polynomial system on the interval [0,1] with respect to the
weight functionρ(x) = 1; similarly,Qm−1 is the(m−1)^{st} degree member of the orthogonal
polynomial system on the interval[0,1]with respect to the weight function ρ(x) = (1−x)x.

Therefore, Q_{m} has m pairwise distinct zeros 0 < λ_{1} < · · · < λ_{m} < 1and Qm−1 has m−1
pairwise distinct zeros0 < µ_{1} < · · · < µm−1 < 1. Moreover, the coefficientsα_{1}, . . . , α_{m} and
β_{0}, . . . , β_{m} have the form above. To complete the proof, apply Theorem 2.7 on the function
F : [0,1]→Rdefined by the formula

F(t) := f (1−t)a+tb .

**2.4. Applications. In the particular setting when**m = 1, Theorem 2.8 reduces to the classical
Hermite–Hadamard inequality:

* Corollary 2.1. If* f : [a, b] → R

*is a polynomially*2-convex (i.e. convex) function, then the

*following inequalities hold*

f

a+b 2

≤ 1 b−a

Z b a

f(x)dx≤ f(a) +f(b)

2 .

In the subsequent corollaries we present Hermite–Hadamard-type inequalities in those cases when the zeros of the polynomials in Theorem 2.8 and Theorem 2.9 can explicitly be computed.

* Corollary 2.2. If* f : [a, b] → R

*is a polynomially*3-convex function, then the following in-

*equalities hold*

1

4f(a) + 3 4f

a+ 2b 3

≤ 1 b−a

Z b a

f(x)dx≤ 3 4f

2a+b 3

+ 1

4f(b).

* Corollary 2.3. If* f : [a, b] → R

*is a polynomially*4-convex function, then the following in-

*equalities hold*

1

2f 3 +√ 3

6 a+3−√ 3

6 b

! + 1

2f 3−√ 3

6 a+ 3 +√ 3

6 b

!

≤ 1 b−a

Z b a

f(x)dx≤ 1

6f(a) + 2 3f

a+b 2

+1

6f(b).

* Corollary 2.4. If* f : [a, b] → R

*is a polynomially*5-convex function, then the following in-

*equalities hold*

1

9f(a) + 16 +√ 6

36 f 4 +√ 6

10 a+6−√ 6 10 b

!

+16−√ 6

36 f 4−√ 6

10 a+6 +√ 6 10 b

!

≤ 1 b−a

Z b a

f(x)dx

≤ 16−√ 6

36 f 6 +√ 6

10 a+ 4−√ 6 10 b

!

+16 +√ 6

36 f 6−√ 6

10 a+4 +√ 6 10 b

! +1

9f(b).

In some other cases analogous statements can be formulated applying Theorem 2.9. For
simplicity, instead of writing down these corollaries explicitly, we shall present a list which
contains the zeros of P_{n} (denoted by λ_{k}), the coefficients α_{k} for the left hand side inequality,
also the zeros ofQ_{n}(denoted byµ_{k}), and the coefficientsβ_{k} for the right hand side inequality,
respectively.

**Case**n = 6
The zeros ofP_{3}:

5−√ 15

10 , 1

2, 5 +√ 15 10 ; the corresponding coefficients:

5 18, 4

9, 5
18.
The zeros ofQ_{2}:

5−√ 5

10 , 5 +√ 5 10 ; the corresponding coefficients:

1 12, 5

12, 5 12, 1

12.
**Case**n = 8

The zeros ofP4:

1 2−

p525 + 70√ 30

70 , 1

2 −

p525−70√ 30

70 ,

1 2+

p525−70√ 30

70 , 1

2+

p525 + 70√ 30

70 ;

the corresponding coefficients:

1 4 −

√30 72 , 1

4+

√30 72 , 1

4 +

√30 72 , 1

4 −

√30
72 .
The zeros ofQ_{3}:

1 2 −

√21 14 , 1

2, 1 2+

√21 14 ; the corresponding coefficients:

1

20, 49

180, 16

45, 49 180, 1

20.
**Case**n = 10

The zeros ofP_{5}:

1 2−

p245 + 14√ 70

42 , 1

2 −

p245−14√ 70

42 ,

1 2, 1

2 +

p245−14√ 70

42 , 1

2+

p245 + 14√ 70

42 ;

the corresponding coefficients:

322−13√ 70

1800 , 322 + 13√ 70

1800 , 64

225, 322 + 13√ 70

1800 , 322−13√ 70 1800 . The zeros ofQ4:

1 2 −

p147 + 42√ 7

42 , 1

2 −

p147−42√ 7

42 ,

1 2+

p147−42√ 7

42 , 1

2+

p147 + 42√ 7

42 ;

the corresponding coefficients:

1

30, 14−√ 7

60 , 14 +√ 7

60 , 14 +√ 7

60 , 14−√ 7

60 , 1

30.
**Case**n = 12(right hand side inequality)

The zeros ofQ_{5}:

1 2−

p495 + 66√ 15

66 , 1

2 −

p495−66√ 15

66 ,

1 2, 1

2 +

p495−66√ 15

66 , 1

2+

p495 + 66√ 15

66 ;

the corresponding coefficients:

1

42, 124−7√ 15

700 , 124 + 7√ 15

700 , 128

525, 124 + 7√

15

700 , 124−7√ 15

700 , 1

42.

During the investigations of the higher–order cases above, we can use the symmetry of the zeros of the orthogonal polynomials with respect to 1/2, and therefore the calculations lead to solving linear or quadratic equations. The first case where “casus irreducibilis” appears is n = 7; similarly, this is the reason for presenting only the right hand side inequality for polynomially12-convex functions.