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 in- equality to the case when the convexity notion is induced by a Chebyshev sys- tem.
Acknowledgement: This research has been supported by the Hungarian Scientific Research Fund (OTKA) Grants NK–68040.
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Contents
1 Introduction 3
2 Polynomial Convexity 7
2.1 Orthogonal polynomials and basic quadrature formulae . . . 8
2.2 An approximation theorem . . . 18
2.3 Hermite–Hadamard-type inequalities . . . 22
2.4 Applications . . . 33
3 Generalized2-Convexity 38 3.1 Characterizations via generalized lines . . . 38
3.2 Connection with standard convexity . . . 52
3.3 Hermite–Hadamard-type inequalities . . . 56
3.4 Applications . . . 59
4 Generalized Convexity Induced by Chebyshev Systems 63 4.1 Characterizations and regularity properties . . . 64
4.2 Moment spaces induced by Chebyshev systems . . . 67
4.3 Hermite–Hadamard-type inequalities . . . 72
4.4 An alternative approach in a particular case . . . 78
5 Characterizations via Hermite–Hadamard Inequalities 86 5.1 Further properties of generalized lines . . . 86
5.2 Hermite–Hadamard-type inequalities and(ω1, ω2)-convexity . . . . 91
Hermite-Hadamard-type Inequalities Mihály Bessenyei vol. 9, iss. 3, art. 63, 2008
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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, . . . , ωnand 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, . . . , ωn). We say that ωωωωωωωωω is a Chebyshev system overI if, for all elementsx1 <· · ·< xnofI, the following inequality holds:
ωωωωωωωωω(x1) · · · ωωωωωωωωω(xn) >0.
In fact, it suffices to assume that the determinant above is nonvanishing when- ever the arguments x1, . . . , xn 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, . . . , ωn 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 sys- tems:
Definition 1.2. Let ωωωωωωωωω = (ω1, . . . , ωn)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 <· · ·< xnofI, 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 con- vexity. That is, a functionf : I → Ris said to be polynomiallyn-convex if, for all
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elementsx0 <· · ·< xnofI, it satisfies the inequality
(−1)n
f(x0) . . . f(xn) 1 . . . 1 x0 . . . xn
... . .. ... xn−10 . . . xn−1n
≥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 ap- proach 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 ex- amples 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 poly- nomially convex functions. Let us recall that a function f : I → R is said to be polynomiallyn-convex if, for all elements x0 < · · · < xn of I, it satisfies the in- equality
(−1)n
f(x0) . . . f(xn) 1 . . . 1 x0 . . . xn
... . .. ... xn−10 . . . xn−1n
≥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” func- tions 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 guar- antees the approximation of polynomially convex functions with smooth polynomi- ally 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, Qiρ:=
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
µk :=
Z b a
xkρ(x)dx (k = 0,1,2, . . .).
Then, the nth degree member of the ρ-orthogonal polynomial system on [a, b] has the following representation via the moments ofρ:
Pn(x) :=
1 µ0 · · · µn−1
x µ1 · · · µn
... ... . .. ... xn µn · · · µ2n−1
.
Clearly, it suffices to show that Pn 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 determinanthPn(x), xk−1iρare linearly dependent according to the definition of the moments.
In fact, the moments and the orthogonal polynomials depend heavily on the inter- val[a, b]. Therefore, we use the notionsµk;[a,b]andPn;[a,b]instead ofµkandPnabove 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 poly- nomials plays a key role (see [39]). Let Pn denote the nth degree member of the ρ-orthogonal polynomial system on [a, b]. Then, Pn has n pairwise distinct zeros ξ1 <· · ·< ξnin]a, b[.
Let us consider the following Z b
a
f ρ=
n
X
k=1
ckf(ξk), (2.1)
Z b a
f ρ=c0f(a) +
n
X
k=1
ckf(ξk), (2.2)
Z b a
f ρ=
n
X
k=1
ckf(ξk) +cn+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. LetPnbe thenthdegree 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) ck =
Z b a
Pn(x)
(x−ξk)Pn0(ξk)ρ(x)dx.
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Furthermore, ξ1, . . . , ξn are pairwise distinct elements of]a, b[, and ck ≥ 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, . . . , ξnare the zeros of the polynomialPnand, for allk = 1, . . . , n, denote the primitive Lagrange-interpolation polynomials by Lk : [a, b] → R. That is,
Lk(x) :=
Pn(x)
(x−ξk)Pn0(ξk) ifx6=ξk
1 ifx=ξk.
IfQis a polynomial of degree at most 2n−1, then, using the Euclidian algorithm, Q can be written in the form Q = P Pn +R where degP,degR ≤ n −1. The inequalitydegP ≤n−1implies theρ-orthogonality ofP andPn:
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(ξk)Lk.
Therefore, considering the definition of the coefficientsc1, . . . , cn 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
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 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) +· · ·+cnP(ξn)Q(ξn) = 0.
ThereforeQisρ-orthogonal toP. The uniqueness ofPnimplies thatPn=anQ, and ξ1, . . . , ξnare the zeros ofPn. Furthermore, (2.1) is exact if we substitutef := Lk andf := L2k, respectively. The first substitution gives (2.5), while the second one shows the nonnegativity ofck. For further details, consult the book [39, p. 44].
Theorem 2.2. LetPnbe thenthdegree member of the orthogonal polynomial system on[a, b]with respect to the weight functionρa(x) := (x−a)ρ(x). Then (2.2) is exact for polynomialsf of degree at most2nif and only ifξ1, . . . , ξnare the zeros ofPn, and
c0 = 1 Pn2(a)
Z b a
Pn2(x)ρ(x)dx, (2.6)
ck = 1 ξk−a
Z b a
(x−a)Pn(x)
(x−ξk)Pn0(ξk)ρ(x)dx.
(2.7)
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Furthermore, ξ1, . . . , ξn are pairwise distinct elements of]a, b[, and ck ≥ 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 ρa= Z b
a
(x−a)P(x)ρ(x)dx=c1(ξ1−a)P(ξ1) +· · ·+cn(ξn−a)P(ξn).
Applying Theorem2.1to the weight functionρaand the coefficients ca;k :=ck(ξk−a),
we get thatξ1, . . . , ξnare the zeros ofPnand, for allk = 1, . . . , n, the coefficients ca;kcan be computed using formula (2.5). Therefore,
ck(ξk−a) = Z b
a
Pn(x)
(x−ξk)Pn0(ξk)ρa(x)dx= Z b
a
(x−a)Pn(x)
(x−ξk)Pn0(ξk)ρ(x)dx.
Substitutingf :=Pn2into (2.1), we obtain that c0 = 1
Pn2(a) Z b
a
Pn2ρ.
Thus (2.6) and (2.7) are valid, andck ≥0fork = 0,1, . . . , n.
Conversely, assume thatξ1, . . . , ξnare the zeros of the orthogonal polynomialPn, and the coefficientsc1, . . . , cnare given by the formula (2.7). Define the coefficient c0 byc0 = Rb
a ρ−(c1 +· · ·+cn). 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 functionρa,
Z b a
Qρa=ca;1Q(ξ1) +· · ·+ca;nQ(ξn)
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
ck(ξk−a)Q(ξk) +
n
X
k=0
P(a)ck
=c0P(a) +
n
X
k=1
ck (ξk−a)Q(ξk) +P(a)
=c0P(a) +
n
X
k=1
ckP(ξk),
which yields that the quadrature formula (2.2) is exact for polynomials of degree at most2n. Therefore, substitutingf :=Pn2 into (2.2), we get formula (2.6).
Theorem 2.3. LetPnbe thenthdegree member of the orthogonal polynomial system on[a, b]with respect to the weight functionρb(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
ck= 1 b−ξk
Z b a
(b−x)Pn(x)
(x−ξk)Pn0(ξk)ρ(x)dx, (2.8)
cn+1 = 1 Pn2(b)
Z b a
Pn2(x)ρ(x)dx.
(2.9)
Furthermore, ξ1, . . . , ξn are pairwise distinct elements of]a, b[, and ck ≥ 0for all k = 1, . . . , n+ 1.
Hint. Applying a similar argument to the previous one to the weight functionρb, we obtain the statement of the theorem.
Theorem 2.4. LetPnbe thenthdegree member of the orthogonal polynomial system on[a, b]with respect to the weight functionρba. Then (2.4) is exact for polynomialsf of degree at most2n+ 1if and only ifξ1, . . . , ξnare the zeros ofPn, and
c0 = 1
(b−a)Pn2(a) Z b
a
(b−x)Pn2(x)ρ(x)dx, (2.10)
ck = 1
(b−ξk)(ξk−a) Z b
a
(b−x)(x−a)Pn(x)
(x−ξk)Pn0(ξk) ρ(x)dx, (2.11)
cn+1 = 1 (b−a)Pn2(b)
Z b a
(x−a)Pn2(x)ρ(x)dx.
(2.12)
Furthermore, ξ1, . . . , ξn are pairwise distinct elements of]a, b[, and ck ≥ 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 ρba = 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 functionρbaand the coefficients ca,b;k :=ck(b−ξk)(ξk−a),
we get thatξ1, . . . , ξnare the zeros ofPnand, for allk = 1, . . . , n, the coefficients ca,b;kcan be computed using formula (2.5). Therefore,
ck(b−ξk)(ξk−a) = Z b
a
Pn(x)
(x−ξk)Pn0(ξk)ρba(x)dx
= Z b
a
(b−x)(x−a)Pn(x)
(x−ξk)Pn0(ξk) ρ(x)dx.
Substitutingf := (b−x)Pn2(x)andf := (x−a)Pn2(x)into (2.1), we obtain that
c0 = 1
(b−a)Pn2(a) Z b
a
(b−x)Pn2(x)ρ(x)dx, cn+1 = 1
(b−a)Pn2(b) Z b
a
(x−a)Pn2(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 andcn+1by the equations Z b
a
(b−x)ρ(x)dx=c0(b−a) +
n
X
k=1
ck(b−ξk), Z b
a
(x−a)ρ(x)dx=
n
X
k=1
ck(ξk−a) +cn+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
Qρba =ca,b;1Q(ξ1) +· · ·+ca,b;nQ(ξn)
holds. Thus, using the definition ofc0andcn+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
ck(b−ξk)(ξk−a)Q(ξk) +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
ck(b−ξk)(ξk−a)Q(ξk) +c0(b−a)P(a) +
n
X
k=1
ck(b−ξk)P(a) +
n
X
k=1
ck(ξk−a)P(b) +cn+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) +cn+1(b−a)P(b)
=c0(b−a)P(a) +
n
X
k=1
ck(b−a)P(ξk) +cn+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)Pn2(x)andf := (x−a)Pn2(x)into (2.4), formulae (2.10) and (2.12) follow.
Let f : [a, b] → R be a differentiable function, x1, . . . , xn be pairwise distinct elements of[a, b], and1≤r≤nbe a fixed integer. We denote the Hermite interpo- lation polynomial byH, which satisfies the following conditions:
H(xk) = f(xk) (k = 1, . . . , n), H0(xk) = f0(xk) (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) = ωn(x)ωr(x)
(n+r)! f(n+r)(θ), where
ωk(x) = (x−x1)· · ·(x−xk).
2.2. An approximation theorem
It is well known that there exists a functionϕwhich possesses the following proper- ties:
(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 functionϕεfor allε >0by the formula ϕε(x) = 1
εϕx ε
(x∈R).
Then, as it can easily be checked,ϕεsatisfies the following conditions:
(i’) ϕε:R→R+isC∞; (ii’) suppϕε⊂[−ε, ε];
(iii’) R
Rϕε= 1.
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LetI ⊂Rbe a nonempty open interval,f :I →Rbe a continuous function, and chooseε >0. Denote the convolution off andϕεbyfε, that is
fε(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¯ fε → f uniformly as ε → 0 on each compact subinterval of I, and fε 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(fk)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 τεf(x) := f(x−ε)
is polynomiallyn-convex on[a, b]for0 < ε < ε0. Leta≤ x0 <· · · < xn ≤ band k ≤n−1be fixed. By induction, we are going to verify the identity
(2.14)
τεf(x0) · · · τεf(xn) 1 · · · 1 x0 · · · xn
... . .. ... xk−10 · · · xk−1n
xk0 · · · xkn ... . .. ... xn−10 · · · xn−1n
=
τεf(x0) · · · τεf(xn)
1 · · · 1
x0−ε · · · xn−ε ... . .. ... (x0−ε)k−1 · · · (xn−ε)k−1
xk0 · · · xkn ... . .. ... xn−10 · · · xn−1n
.
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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
xk = k
0
εk+ k
1
εk−1(x−ε) +· · ·+ k
k
(x−ε)k.
That is,(x−ε)kis the linear combination of the elements1, x−ε, . . . ,(x−ε)kand xk. Therefore, adding the appropriate linear combination of the 2nd, . . . ,(k+ 1)st rows to the(k+ 2)nd row, we arrive at the equation
τεf(x0) · · · τεf(xn)
1 · · · 1
x0−ε · · · xn−ε ... . .. ... (x0−ε)k−1 · · · (xn−ε)k−1
xk0 · · · xkn xk+10 · · · xk+1n
... . .. ... xn−10 · · · xn−1n
=
τεf(x0) · · · τεf(xn)
1 · · · 1
x0−ε · · · xn−ε ... . .. ... (x0−ε)k−1 · · · (xn−ε)k−1
(x0 −ε)k · · · (xn−ε)k xk+10 · · · xk+1n
... . .. ... xn−10 · · · xn−1n
.
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 τεf. Applying a change of variables and the previous result, we get that
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(−1)n
fε(x0) · · · fε(xn) 1 · · · 1 x0 · · · xn
... . .. ... xn−10 · · · xn−1n
= Z
R
(−1)n
f(t)ϕ¯ ε(x0−t) · · · f¯(t)ϕε(xn−t)
1 · · · 1
x0 · · · xn
... . .. ...
xn−10 · · · xn−1n
dt
= Z
R
(−1)n
f(x¯ 0−s) · · · f¯(xn−s) 1 · · · 1 x0 · · · xn
... . .. ... xn−10 · · · xn−1n
ϕε(s)ds
= Z
R
(−1)n
τsf(x0) · · · τsf(xn) 1 · · · 1 x0 · · · xn
... . .. ... xn−10 · · · xn−1n
ϕε(s)ds≥0,
which shows the polynomialn-convexity offεon[a, b]for0< ε < ε0. To complete the proof, choose a positive integern0 such that the relation n1
0 < ε0 holds. If we defineεk andfkbyεk:= n1
0+k andfk :=fεk fork ∈N, then0< εk <
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ε0, and thus(fk)∞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 investi- gates the convergence properties of the zeros of orthogonal polynomials.
Lemma 2.1. Letρbe a weight function on[a, b], and(aj)be strictly monotone de- creasing,(bj)be strictly monotone increasing sequences such thataj → a, bj → b anda1 < b1. Denote the zeros of Pm;j byξ1;j, . . . , ξm;j, wherePm;j is themth de- gree member of theρ|[aj,bj]-orthogonal polynomial system on[aj, bj], and denote the zeros ofPm byξ1, . . . , ξm, wherePm is themth degree member of theρ-orthogonal polynomial system on[a, b]. Then,
j→∞lim ξk;j =ξk (k= 1, . . . , n).
Proof. Observe first that the mapping(a, b)7→µk;[a,b]is continuous, thereforeµk;[aj,bj]
→µk;[a,b]hencePm;j →Pmpointwise according to the representation of orthogonal polynomials. Takeε >0such that
]ξk−ε, ξk+ε[⊂]a, b[,
]ξk−ε, ξk+ε[∩]ξl−ε, ξl+ε[=∅ (k 6=l, k, l ∈ {1, . . . , m}).
The polynomialPm changes its sign on]ξk−ε, ξk+ε[since it is of degreemand it hasmpairwise distinct zeros; therefore, due to the pointwise convergence,Pm;j also changes its sign on the same interval up to an index. That is, for sufficiently largej, ξk;j ∈]ξk−ε, ξk+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 supt→a+0(−1)nf(t);
(ii) f(b)≥lim supt→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 supt→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 . . . xn ... ... ... . .. ... an−1 tn−1 xn−12 . . . xn−1n
≥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 x2 . . . xn
... ... ... . .. ... an−1 an−1 xn−12 . . . xn−1n
≥0.
The adjoint determinants of the elementsf(x2), . . . , f(xn)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, ap- plying 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 ofPm byξ1, . . . , ξm wherePm is themth 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 mth degree member of the orthogonal polynomial system on[a, b]with respect to the weight function(b−x)ρ(x). Define the coefficientsα0, . . . , αmandβ1, . . . , βm+1 by the formulae
α0 := 1 Pm2(a)
Z b a
Pm2(x)ρ(x)dx, αk := 1
ξk−a Z b
a
(x−a)Pm(x)
(x−ξk)Pm0 (ξk)ρ(x)dx and
βk:= 1 b−ηk
Z b a
(b−x)Qm(x)
(x−ηk)Q0m(ηk)ρ(x)dx, βm+1 := 1
Q2m(b) Z b
a
Q2m(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
αkf(ξk)≤ Z b
a
f ρ≤
m
X
k=1
βkf(ηk) +βm+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(ξk) =f(ξk), H0(ξk) =f0(ξk).
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−ξm)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
αkH(ξk) =α0f(a) +
m
X
k=1
αkf(ξk).
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 continu- ous (see TheoremB). Let(aj)and(bj)be sequences fulfilling the requirements of Lemma2.1. According to Theorem2.5, there exists a sequence ofC∞, polynomially (2m+ 1)-convex functions(fi;j)such thatfi;j →f uniformly on[aj, bj]asi→ ∞.
Denote the zeros of Pm;j by ξ1;j, . . . , ξm;j where Pm;j is the mth degree member of the orthogonal polynomial system on[aj, bj]with respect to the weight function (x−a)ρ(x). Define the coefficientsα0;j, . . . , αm;j analogously toα0, . . . , αm 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
α0;jfi;j(aj) +
m
X
k=1
αk;jfi;j(ξk;j)≤ Z bj
aj
fi;jρ.
Taking the limitsi→ ∞and thenj → ∞, we get the inequality α0
lim inf
t→a+0 f(t)
+
m
X
k=1
αkf(ξk)≤ 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 esti- mation 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 mth degree member of the orthogonal polynomial system on [a, b] with respect to the weight function ρ(x), and denote the zeros of Qm−1 by η1, . . . , ηm−1 where Qm−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, . . . , αm and β0, . . . , βm+1 by the formulae
αk :=
Z b a
Pm(x)
(x−ξk)Pm0 (ξk)ρ(x)dx
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and
β0 = 1
(b−a)Q2m−1(a) Z b
a
(b−x)Q2m−1(x)ρ(x)dx,
βk = 1
(b−ηk)(ξk−a) Z b
a
(b−x)(x−a)Qm−1(x)
(x−ηk)Q0m−1(ηk) ρ(x)dx,
βm+1 = 1
(b−a)Q2m−1(b) Z b
a
(x−a)Q2m−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
αkf(ξk)≤ Z b
a
f ρ≤β0f(a) +
m−1
X
k=1
βkf(ηk) +βmf(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 ofPmin the following manner:
H(ξk) =f(ξk), H0(ξk) =f0(ξk).
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−ξm)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
αkH(ξk) =
m
X
k=1
αkf(ξk).
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(ηk) =f(ηk), H0(ηk) =f0(ηk),
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
βkf(ηk) +βmf(b).