HERMITE-HADAMARD-TYPE INEQUALITIES FOR GENERALIZED CONVEX FUNCTIONS

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Hermite-Hadamard-type Inequalities Mihály Bessenyei vol. 9, iss. 3, art. 63, 2008

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HERMITE-HADAMARD-TYPE INEQUALITIES FOR GENERALIZED CONVEX FUNCTIONS

MIHÁLY BESSENYEI

Institute of Mathematics University of Debrecen

H-4010 Debrecen, Pf. 12, Hungary EMail:besse@math.klte.hu

Received: 21 June, 2008

Accepted: 14 July, 2008

Communicated by: P.S. Bullen

2000 AMS Sub. Class.: Primary 26A51, 26B25, 26D15.

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

Abstract: The aim of the present paper is to extend the classical Hermite-Hadamard inequality to the case when the convexity notion is induced by a Chebyshev system.

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

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1. Introduction

Let I be a real interval, that is, a nonempty, connected and bounded subset of R. Ann-dimensional Chebyshev system onI consists of a set of real valued continuous functionsω₁, . . . , ω_nand is determined by the property that eachnpoints ofI×R with distinct first coordinates can uniquely be interpolated by a linear combination of the functions. More precisely, we have the following

Definition 1.1. LetI ⊂Rbe a real interval andω1, . . . , ωn :I →Rbe continuous functions. Denote the column vector whose components areω1, . . . , ωnin turn byωωωωωωωωω, that is, ωωωωωωωωω := (ω₁, . . . , ω_n). We say that ωωωωωωωωω is a Chebyshev system overI if, for all elementsx₁ <· · ·< x_nofI, the following inequality holds:

ωωωωωωωωω(x₁) · · · ωωωωωωωωω(x_n) >0.

In fact, it suffices to assume that the determinant above is nonvanishing whenever the arguments x₁, . . . , x_n are pairwise distinct points of the domain. Indeed, Bolzano’s theorem guarantees that its sign is constant if the arguments are supposed to be in an increasing order, hence the componentsω₁, . . . , ω_n can always be rear- ranged such that ωωωωωωωωω fulfills 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.

Without claiming completeness, let us list some important and classical examples of Chebyshev systems. In each example ωωωωωωωωω is defined on an arbitraryI ⊂ Rexcept for the last one whereI ⊂]−^π₂,^π₂[.

• polynomial system: ωωωωωωωωω(x) := (1, x, . . . , xⁿ);

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

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

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

We make no attempt here to present an exhaustive account of the theory of Cheby- shev systems, but only mention that, motivated by some results of A.A. Markov, the first systematic investigation 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 = (ω₁, . . . , ω_n)be a Chebyshev system over the real interval I. A functionf :I →Ris said to be generalized convex with respect to ωωωωωωωωωif, for all elementsx₀ <· · ·< x_nofI, it satisfies the inequality

(−1)ⁿ

f(x0) · · · f(xn) ωωω

ωωωωωω(x0) · · · ωωωωωωωωω(xn)

≥0.

There are other alternatives to express that f is generalized convex with respect to ωωωωωωωωω, for example,f is generalizedωωωωωωωωω-convex or simply ωωωωωωωωω-convex. If the underlying n-dimensional Chebyshev system can uniquely be identified from the context, we briefly say thatf is generalizedn-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 → Ris said to be polynomiallyn-convex if, for all

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elementsx₀ <· · ·< x_nofI, it satisfies the inequality

(−1)ⁿ

f(x₀) . . . f(x_n) 1 . . . 1 x0 . . . xn

... . .. ... xⁿ⁻¹₀ . . . xⁿ⁻¹_n

≥0.

Observe that polynomially 2-convex functions are exactly the “standard” convex ones. The case, when the “generalized” convexity notion is induced by the special two dimensional Chebyshev systemω₁(x) := 1andω₂(x) :=x, is termed standard setting and standard convexity, respectively.

The integral average of any standard convex function f : [a, b] → R can 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], [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 advantages than the deductive one. First of all, it makes the original motiva- tions 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.

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SECTION2investigates 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 charac- terization theorem which, in the standard setting, reduces to the well known char- acterization properties of convex functions. Another theorem of the section estab- lishes 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 dimension of the underlying Chebyshev systems are “small”), an elementary alternative approach is also presented.

SECTION5 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 consequences of generalized convexity, but they also characterize it.

Specializing the members of Chebyshev systems, several applications and examples are presented for concrete Hermite–Hadamard-type inequalities in both the cases of polynomial convexity and generalized 2-convexity. As a simple conse- quence, 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.

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2. Polynomial Convexity

The main results of this section state Hermite–Hadamard-type inequalities for polynomially convex functions. Let us recall that a function f : I → R is said to be polynomiallyn-convex if, for all elements x₀ < · · · < x_n of I, it satisfies the inequality

(−1)ⁿ

f(x₀) . . . f(x_n) 1 . . . 1 x0 . . . xn

... . .. ... xⁿ⁻¹₀ . . . xⁿ⁻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 thatf : I → Ris continuous and n times differentiable on the interior ofI. Then,f is polynomiallyn-convex if and only iff⁽ⁿ⁾ ≥0on the interior ofI.

Theorem B. ([17, Theorem 1. p. 391]) Assume that f : I → R is polynomially n-convex andn≥2. Then,fis(n−2)times differentiable andf⁽ⁿ⁻²⁾is continuous on the interior ofI.

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

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2.1. Orthogonal polynomials and basic quadrature formulae

In what follows, ρ denotes a positive, locally integrable function (briefly: weight function) on an intervalI. The polynomialsP andQ are said to be orthogonal on [a, b]⊂I with respect to the weight functionρor simplyρ-orthogonal on[a, b]if

hP, Qi_ρ:=

Z b a

P Qρ= 0.

A system of polynomials is called aρ-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 ofρby the formulae

µ_k :=

Z b a

x^kρ(x)dx (k = 0,1,2, . . .).

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

P_n(x) :=

1 µ₀ · · · µn−1

x µ1 · · · µn

... ... . .. ... xⁿ µn · · · µ2n−1

.

Clearly, it suffices to show that P_n is ρ-orthogonal to the special polynomials1, x, . . . , xⁿ⁻¹. Indeed, for k = 1, . . . , n, the first and the (k + 1)^st columns of the determinanthP_n(x), x^k−1i_ρare 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 notionsµ_k;[a,b]andP_n;[a,b]instead ofµ_kandP_nabove when we want to or have to emphasize the dependence on the underlying interval.

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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 has n pairwise distinct zeros ξ₁ <· · ·< ξ_nin]a, b[.

Let us consider the following Z b

a

f ρ=

n

X

k=1

c_kf(ξ_k), (2.1)

Z b a

f ρ=c₀f(a) +

n

X

k=1

c_kf(ξ_k), (2.2)

Z b a

f ρ=

n

X

k=1

c_kf(ξ_k) +c_n+1f(b), (2.3)

Z b a

f ρ=c0f(a) +

n

X

k=1

ckf(ξk) +cn+1f(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. LetP_nbe then^thdegree member of the orthogonal polynomial system on[a, b]with respect to the weight functionρ. Then (2.1) is exact for polynomialsf of degree at most2n−1if and only ifξ1, . . . , ξnare the zeros ofPn, and

(2.5) c_k =

Z b a

Pn(x)

(x−ξ_k)P_n⁰(ξ_k)ρ(x)dx.

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Furthermore, ξ₁, . . . , ξ_n are pairwise distinct elements of]a, b[, and c_k ≥ 0for 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 thatξ₁, . . . , ξ_nare the zeros of the polynomialP_nand, for allk = 1, . . . , n, denote the primitive Lagrange-interpolation polynomials by L_k : [a, b] → R. That is,

L_k(x) :=







P_n(x)

(x−ξ_k)P_n⁰(ξ_k) ifx6=ξ_k

1 ifx=ξ_k.

IfQis a polynomial of degree at most 2n−1, then, using the Euclidian algorithm, Q can be written in the form Q = P P_n +R where degP,degR ≤ n −1. The inequalitydegP ≤n−1implies theρ-orthogonality ofP andP_n:

Z b a

P Pnρ= 0.

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

R=

n

X

k=1

R(ξ_k)L_k.

Therefore, considering the definition of the coefficientsc₁, . . . , c_n in formula (2.5),

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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

c_kR(ξ_k) =

n

X

k=1

c_k P(ξ_k)P_n(ξ_k) +R(ξ_k)

=

n

X

k=1

c_kQ(ξ_k).

That is, the quadrature formula (2.1) is exact for polynomials of degree at most 2n−1.

Conversely, assume that (2.1) is exact for polynomials of degree at most2n−1.

Define the polynomialQby the formulaQ(x) := (x−ξ1)· · ·(x−ξn)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) +· · ·+c_nP(ξ_n)Q(ξ_n) = 0.

ThereforeQisρ-orthogonal toP. The uniqueness ofP_nimplies thatP_n=a_nQ, and ξ₁, . . . , ξ_nare the zeros ofP_n. Furthermore, (2.1) is exact if we substitutef := L_k andf := L²_k, respectively. The first substitution gives (2.5), while the second one shows the nonnegativity ofc_k. For further details, consult the book [39, p. 44].

Theorem 2.2. LetP_nbe then^thdegree member of the orthogonal polynomial system on[a, b]with respect to the weight functionρ_a(x) := (x−a)ρ(x). Then (2.2) is exact for polynomialsf of degree at most2nif and only ifξ₁, . . . , ξ_nare the zeros ofP_n, and

c₀ = 1 P_n²(a)

Z b a

P_n²(x)ρ(x)dx, (2.6)

c_k = 1 ξ_k−a

Z b a

(x−a)P_n(x)

(2.7)

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Furthermore, ξ₁, . . . , ξ_n are pairwise distinct elements of]a, b[, and c_k ≥ 0for all k = 0, . . . , n.

Proof. Assume that the quadrature formula (2.2) is exact for polynomials of degree at most2n. 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=c₁(ξ₁−a)P(ξ₁) +· · ·+c_n(ξ_n−a)P(ξ_n).

Applying Theorem2.1to the weight functionρaand the coefficients c_a;k :=c_k(ξ_k−a),

we get thatξ₁, . . . , ξ_nare the zeros ofP_nand, for allk = 1, . . . , n, the coefficients c_a;kcan be computed using formula (2.5). Therefore,

c_k(ξ_k−a) = Z b

a

P_n(x)

(x−ξ_k)P_n⁰(ξ_k)ρ_a(x)dx= Z b

a

(x−a)P_n(x)

Substitutingf :=P_n²into (2.1), we obtain that c₀ = 1

P_n²(a) Z b

a

P_n²ρ.

Thus (2.6) and (2.7) are valid, andc_k ≥0fork = 0,1, . . . , n.

Conversely, assume thatξ₁, . . . , ξ_nare the zeros of the orthogonal polynomialP_n, and the coefficientsc₁, . . . , c_nare given by the formula (2.7). Define the coefficient c₀ byc₀ = Rb

a ρ−(c₁ +· · ·+c_n). IfP is a polynomial of degree at most2n, then there exists a polynomialQwithdegQ≤2n−1such that

P(x) = (x−a)Q(x) +P(a).

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Indeed, the polynomialP(x)−P(a)vanishes at the pointx=a, hence it is divisible by(x−a). Applying Theorem2.1again to the weight functionρ_a,

Z b a

Qρ_a=c_a;1Q(ξ₁) +· · ·+c_a;nQ(ξ_n)

holds. Thus, using the definition ofc₀, the representation of the polynomial P and the quadrature 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

=c₀P(a) +

n

X

k=1

c_k (ξ_k−a)Q(ξ_k) +P(a)

=c₀P(a) +

n

X

k=1

c_kP(ξ_k),

which yields that the quadrature formula (2.2) is exact for polynomials of degree at most2n. Therefore, substitutingf :=P_n² into (2.2), we get formula (2.6).

Theorem 2.3. LetP_nbe then^thdegree member of the orthogonal polynomial system on[a, b]with respect to the weight functionρ^b(x) := (b−x)ρ(x). Then (2.3) is exact for polynomialsf of degree at most2nif and only ifξ1, . . . , ξnare the zeros ofPn,

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and

c_k= 1 b−ξ_k

Z b a

(b−x)P_n(x)

(x−ξ_k)P_n⁰(ξ_k)ρ(x)dx, (2.8)

c_n+1 = 1 P_n²(b)

Z b a

P_n²(x)ρ(x)dx.

(2.9)

Furthermore, ξ₁, . . . , ξ_n are pairwise distinct elements of]a, b[, and c_k ≥ 0for all k = 1, . . . , n+ 1.

Hint. Applying a similar argument to the previous one to the weight functionρ^b, we obtain the statement of the theorem.

Theorem 2.4. LetP_nbe then^thdegree member of the orthogonal polynomial system on[a, b]with respect to the weight functionρ^b_a. Then (2.4) is exact for polynomialsf of degree at most2n+ 1if and only ifξ₁, . . . , ξ_nare the zeros ofP_n, and

c₀ = 1

(b−a)P_n²(a) Z b

a

(b−x)P_n²(x)ρ(x)dx, (2.10)

c_k = 1

(b−ξ_k)(ξ_k−a) Z b

a

(b−x)(x−a)Pn(x)

(x−ξ_k)P_n⁰(ξ_k) ρ(x)dx, (2.11)

c_n+1 = 1 (b−a)P_n²(b)

Z b a

(x−a)P_n²(x)ρ(x)dx.

(2.12)

Furthermore, ξ₁, . . . , ξ_n are pairwise distinct elements of]a, b[, and c_k ≥ 0for all k = 0, . . . , n+ 1.

Proof. Assume that the quadrature formula (2.4) is exact for polynomials of degree

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at most2n+ 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

=c1(b−ξ1)(ξ1−a)P(ξ1) +· · ·+cn(b−ξn)(ξn−a)P(ξn).

Applying Theorem2.1to the weight functionρ^b_aand the coefficients c_a,b;k :=c_k(b−ξ_k)(ξ_k−a),

we get thatξ₁, . . . , ξ_nare the zeros ofP_nand, for allk = 1, . . . , n, the coefficients c_a,b;kcan be computed using formula (2.5). Therefore,

c_k(b−ξ_k)(ξ_k−a) = Z b

a

P_n(x)

(x−ξ_k)P_n⁰(ξ_k)ρ^b_a(x)dx

= Z b

a

(b−x)(x−a)P_n(x)

(x−ξ_k)P_n⁰(ξ_k) ρ(x)dx.

Substitutingf := (b−x)P_n²(x)andf := (x−a)P_n²(x)into (2.1), we obtain that

c₀ = 1

(b−a)P_n²(a) Z b

a

(b−x)P_n²(x)ρ(x)dx, c_n+1 = 1

(b−a)P_n²(b) Z b

a

(x−a)P_n²(x)ρ(x)dx.

Thus (2.10), (2.11) and (2.12) are valid, furthermore,ck ≥0fork= 0, . . . , n+ 1.

Conversely, assume thatξ1, . . . , ξnare the zeros ofPn, and the coefficientsc1, . . . , cn

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are given by the formula (2.11). Define the coefficientsc₀ andc_n+1by the equations Z b

a

(b−x)ρ(x)dx=c₀(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 polynomialQwith degQ≤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 Theorem2.1again,

Z b a

Qρ^b_a =c_a,b;1Q(ξ₁) +· · ·+c_a,b;nQ(ξ_n)

holds. Thus, using the definition ofc₀andc_n+1, the representation of the polynomial P 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

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=

n

X

k=1

c_k(b−ξ_k)(ξ_k−a)Q(ξ_k) +c₀(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

ck (b−ξk)(ξk−a)Q(ξk) + (ξk−a)P(b) + (b−ξk)P(a) +c₀(b−a)P(a) +c_n+1(b−a)P(b)

=c₀(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_n²(x)andf := (x−a)P_n²(x)into (2.4), formulae (2.10) and (2.12) follow.

Let f : [a, b] → R be a differentiable function, x₁, . . . , 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), H⁰(x_k) = f⁰(x_k) (k = 1, . . . , r).

We recall thatdegH =n+r−1. From a well known result, (see [13, Sec. 5.3, pp.

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230-231]), for allx∈[a, b]there existsθ such that (2.13) f(x)−H(x) = ω_n(x)ω_r(x)

(n+r)! f^(n+r)(θ), where

ω_k(x) = (x−x₁)· · ·(x−x_k).

2.2. An approximation theorem

It is well known that there exists a functionϕwhich possesses 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_εfor allε >0by the formula ϕ_ε(x) = 1

εϕx ε

(x∈R).

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

(i’) ϕε:R→R⁺isC^∞; (ii’) suppϕ_ε⊂[−ε, ε];

(iii’) R

Rϕ_ε= 1.

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LetI ⊂Rbe a nonempty open interval,f :I →Rbe a continuous function, and chooseε >0. Denote the convolution off andϕ_εbyf_ε, that is

f_ε(x) :=

Z

R

f(y)ϕ¯ _ε(x−y)dy (x∈R)

where f¯(y) = f(y) if y ∈ I, otherwise f(y) = 0. Let us recall, that¯ f_ε → f uniformly as ε → 0 on each compact subinterval of I, and f_ε is infinitely many times differentiable onR. 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 continuous function. Then, for all compact subintervals [a, b] ⊂ I, there exists a sequence of polynomiallyn-convex andC^∞functions(f_k)which converges uniformly tof on[a, b].

Proof. Choosea, b∈Iandε₀ >0such that the inclusion[a−ε₀, b+ε₀]⊂Iholds.

We show that the functionτεf : [a, b]→Rdefined by the formula τ_εf(x) := f(x−ε)

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

(2.14)

τ_εf(x₀) · · · τ_εf(x_n) 1 · · · 1 x₀ · · · x_n

... . .. ... x^k−1₀ · · · x^k−1_n

x^k₀ · · · x^k_n ... . .. ... xⁿ⁻¹₀ · · · xⁿ⁻¹_n

=

τ_εf(x₀) · · · τ_εf(x_n)

1 · · · 1

x₀−ε · · · x_n−ε ... . .. ... (x₀−ε)^k−1 · · · (x_n−ε)^k−1

x^k₀ · · · x^k_n ... . .. ... xⁿ⁻¹₀ · · · xⁿ⁻¹_n

.

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Ifk = 1, then this equation obviously holds. Assume, for a fixed positive integer k ≤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)^kis the linear combination of the elements1, x−ε, . . . ,(x−ε)^kand x^k. Therefore, adding the appropriate linear combination of the 2^nd, . . . ,(k+ 1)^st rows to the(k+ 2)^nd row, we arrive at the equation

τ_εf(x₀) · · · τ_εf(x_n)

1 · · · 1

x₀−ε · · · x_n−ε ... . .. ... (x₀−ε)^k−1 · · · (x_n−ε)^k−1

x^k₀ · · · x^k_n x^k+1₀ · · · x^k+1_n

... . .. ... xⁿ⁻¹₀ · · · xⁿ⁻¹_n

=

τ_εf(x₀) · · · τ_εf(x_n)

1 · · · 1

x₀−ε · · · x_n−ε ... . .. ... (x₀−ε)^k−1 · · · (x_n−ε)^k−1

(x₀ −ε)^k · · · (x_n−ε)^k x^k+1₀ · · · x^k+1_n

... . .. ... xⁿ⁻¹₀ · · · xⁿ⁻¹_n

.

Hence formula (2.14) holds for all fixed positivek whenever1 ≤ k ≤ n−1. The particular case k = n −1 gives the polynomial n-convexity of τ_εf. Applying a change of variables and the previous result, we get that

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(−1)ⁿ

f_ε(x₀) · · · f_ε(x_n) 1 · · · 1 x₀ · · · x_n

... . .. ... xⁿ⁻¹₀ · · · xⁿ⁻¹_n

= Z

R

(−1)ⁿ

f(t)ϕ¯ _ε(x₀−t) · · · f¯(t)ϕ_ε(x_n−t)

1 · · · 1

x₀ · · · x_n

... . .. ...

xⁿ⁻¹₀ · · · xⁿ⁻¹_n

dt

= Z

R

(−1)ⁿ

f(x¯ ₀−s) · · · f¯(x_n−s) 1 · · · 1 x₀ · · · x_n

... . .. ... xⁿ⁻¹₀ · · · xⁿ⁻¹_n

ϕε(s)ds

= Z

R

(−1)ⁿ

τ_sf(x₀) · · · τ_sf(x_n) 1 · · · 1 x₀ · · · x_n

... . .. ... xⁿ⁻¹₀ · · · xⁿ⁻¹_n

ϕ_ε(s)ds≥0,

which shows the polynomialn-convexity off_εon[a, b]for0< ε < ε₀. To complete the proof, choose a positive integern₀ such that the relation _n¹

0 < ε₀ holds. If we defineε_k andf_kbyε_k:= _n¹

0+k andf_k :=f_ε_k fork ∈N, then0< ε_k <

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ε₀, and thus(f_k)^∞_k=1satisfies the requirements of the theorem.

2.3. Hermite–Hadamard-type inequalities

In the sequel, we shall need two additional auxiliary results. The first one investigates the convergence properties of the zeros of orthogonal polynomials.

Lemma 2.1. Letρbe a weight function on[a, b], and(a_j)be strictly monotone de- creasing,(b_j)be strictly monotone increasing sequences such thata_j → a, b_j → b anda₁ < b₁. Denote the zeros of P_m;j byξ_1;j, . . . , ξ_m;j, whereP_m;j is them^th de- gree member of theρ|_[a_j_,b_j_]-orthogonal polynomial system on[a_j, b_j], and denote the zeros ofP_m byξ₁, . . . , ξ_m, whereP_m is them^th degree member of theρ-orthogonal polynomial system on[a, b]. Then,

j→∞lim ξ_k;j =ξ_k (k= 1, . . . , n).

Proof. Observe first that the mapping(a, b)7→µ_k;[a,b]is continuous, thereforeµ_k;[a_j_,b_j_]

→µ_k;[a,b]henceP_m;j →P_mpointwise according to the representation of orthogonal polynomials. Takeε >0such that

]ξ_k−ε, ξ_k+ε[⊂]a, b[,

]ξ_k−ε, ξ_k+ε[∩]ξ_l−ε, ξ_l+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 degreemand it hasmpairwise 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 functions 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.

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Lemma 2.2. Letf : [a, b]→Rbe a polynomiallyn-convex function. Then, (i) (−1)ⁿf(a)≥lim sup_t→a+0(−1)ⁿf(t);

(ii) f(b)≥lim sup_t→b−0f(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 notationf₊(a) := lim sup_t→a+0f(t). Take the elementsx₀ :=a <

x1 := t < · · · < xn of [a, b]. Then, the (even order) polynomial convexity of f implies

f(a) f(t) f(x2) . . . f(xn)

1 1 1 . . . 1

a t x₂ . . . x_n ... ... ... . .. ... aⁿ⁻¹ tⁿ⁻¹ xⁿ⁻¹₂ . . . xⁿ⁻¹_n

≥0.

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

f(a) f₊(a) f(x₂) . . . f(x_n)

1 1 1 . . . 1

a a x2 . . . xn

... ... ... . .. ... aⁿ⁻¹ aⁿ⁻¹ xⁿ⁻¹₂ . . . xⁿ⁻¹_n

≥0.

The adjoint determinants of the elementsf(x₂), . . . , f(x_n)in the first row are equal to zero since their first and second columns coincide; on the other hand, f(a) and f₊(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.

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

Theorem 2.6. Letρ: [a, b]→Rbe a positive integrable function. Denote the zeros ofP_m byξ₁, . . . , ξ_m whereP_m is them^th degree member of the orthogonal polyno- mial system on[a, b]with respect to the weight function(x−a)ρ(x), and denote the zeros ofQm byη1, . . . , ηm where Qm 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 coefficientsα₀, . . . , α_mandβ₁, . . . , β_m+1 by the formulae

α0 := 1 P_m²(a)

Z b a

P_m²(x)ρ(x)dx, α_k := 1

ξ_k−a Z b

a

(x−a)P_m(x)

(x−ξ_k)P_m⁰ (ξ_k)ρ(x)dx and

β_k:= 1 b−ηk

Z b a

(b−x)Q_m(x)

(x−ηk)Q⁰_m(ηk)ρ(x)dx, β_m+1 := 1

Q²_m(b) Z b

a

Q²_m(x)ρ(x)dx.

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

α₀f(a) +

m

X

k=1

α_kf(ξ_k)≤ Z b

a

f ρ≤

m

X

k=1

β_kf(η_k) +β_m+1f(b).

Proof. First assume that f is (2m + 1) times differentiable. Then, according to TheoremA, f^(2m+1) ≥ 0on ]a, b[. Let H be the Hermite interpolation polynomial

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determined by the conditions

H(a) =f(a), H(ξ_k) =f(ξ_k), H⁰(ξ_k) =f⁰(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−ξ₁)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 of ρ. On the other hand,His of degree2m, therefore Theorem2.2yields that

Z b a

f ρ≥ Z b

a

Hρ=α₀H(a) +

m

X

k=1

α_kH(ξ_k) =α₀f(a) +

m

X

k=1

α_kf(ξ_k).

For the general case, letf be an arbitrary polynomially(2m+1)-convex function.

Without loss of generality we may assume that m ≥ 1; in this case, f is continuous (see TheoremB). Let(a_j)and(b_j)be sequences fulfilling the requirements of Lemma2.1. According to Theorem2.5, there exists a sequence ofC^∞, polynomially (2m+ 1)-convex functions(f_i;j)such thatf_i;j →f uniformly on[a_j, b_j]asi→ ∞.

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 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 toα₀, . . . , α_m with the help of Pm;j. Then, ξk;j → ξk due to Lemma 2.1, and hence αk;j → αk as j → ∞. Applying the previous step of the proof on the smooth functions (fi;j), it

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follows that

α_0;jf_i;j(a_j) +

m

X

k=1

α_k;jf_i;j(ξ_k;j)≤ Z bj

aj

f_i;jρ.

Taking the limitsi→ ∞and thenj → ∞, we get the inequality α₀

lim inf

t→a+0 f(t)

+

m

X

k=1

α_kf(ξ_k)≤ Z b

a

f ρ.

This, together with Lemma2.2, 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 polynomially convex functions. In this case, the symmetrical structure dis- appears: 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 of Q_m−1 by η₁, . . . , 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 coefficients α₁, . . . , α_m and β₀, . . . , β_m+1 by the formulae

α_k :=

Z b a

P_m(x)

(x−ξk)P_m⁰ (ξk)ρ(x)dx

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and

β₀ = 1

(b−a)Q²_m−1(a) Z b

a

(b−x)Q²_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)Q⁰_m−1(η_k) ρ(x)dx,

β_m+1 = 1

(b−a)Q²_m−1(b) Z b

a

(x−a)Q²_m−1(x)ρ(x)dx.

If a functionf : [a, b]→Ris polynomially(2m)-convex, then it satisfies the follow- ing Hermite–Hadamard-type inequality

m

X

k=1

α_kf(ξ_k)≤ Z b

a

f ρ≤β₀f(a) +

m−1

X

k=1

β_kf(η_k) +β_mf(b).

Proof. First assume thatf isn = 2mtimes differentiable. Then f^(2m) ≥0on]a, b[

according to Theorem B. Consider the Hermite interpolation polynomial H that interpolates the functionf in the zeros ofP_min the following manner:

H(ξ_k) =f(ξ_k), H⁰(ξ_k) =f⁰(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−ξ₁)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,His of degree2m−1, therefore Theorem2.1yields the left hand

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side of the inequality to be proved:

Z b a

f ρ≥ Z b

a

Hρ=

m

X

k=1

α_kH(ξ_k) =

m

X

k=1

α_kf(ξ_k).

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

H(a) =f(a), H(η_k) =f(η_k), H⁰(η_k) =f⁰(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−η1)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, H is of degree2m−1, therefore Theorem2.4yields the right hand side inequality to be proved:

Z b a

f ρ≤ Z b

a

Hρ=β0H(a) +

m−1

X

k=1

βkH(ηk) +βmH(b)

=β₀f(a) +

m−1

X

k=1

β_kf(η_k) +β_mf(b).