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

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. INTRODUCTION

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 ω₁, . . . , ω_nand 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 ω₁, . . . , ω_n : I → R be continuous func- tions. Denote the column vector whose components areω₁, . . . , ω_nin turn by ωωωωωωωωω, that is, ωωωωωωωωω :=

(ω₁, . . . , ω_n). We say that ωωωωωωωωωis a Chebyshev system overIif, for all elementsx₁ <· · ·< x_nof I, the following inequality holds:

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

In fact, it suffices to assume that the determinant above is nonvanishing whenever the argu- mentsx₁, . . . , x_nare 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 componentsω₁, . . . , ω_ncan 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 ⊂]− ^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);

• 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. Letωωωωωωωωω= (ω₁, . . . , ω_n)be a Chebyshev system over the real intervalI. A function f :I →Ris said to be generalized convex with respect to ωωωωωωωωωif, for all elementsx₀ <· · ·< x_n ofI, it satisfies the inequality

(−1)ⁿ

f(x₀) · · · 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,fis 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 →Ris said to be polynomially n-convex if, for all elementsx₀ <· · ·< x_nofI, it satisfies the inequality

(−1)ⁿ

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

... . .. ... xⁿ⁻¹₀ . . . xⁿ⁻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 systemω₁(x) := 1andω₂(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. POLYNOMIAL CONVEXITY

The main results of this section state Hermite–Hadamard-type inequalities for polynomially convex functions. Let us recall that a functionf : I → Ris said to be polynomiallyn-convex if, for all elementsx₀ <· · ·< x_nofI, it satisfies the inequality

(−1)ⁿ

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

... . .. ... 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 that f : I → R is 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 thatf : I → Ris polynomiallyn-convex and n≥2. Then,f is(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.

2.1. Orthogonal polynomials and basic quadrature formulae. In what follows,ρdenotes a positive, locally integrable function (briefly: weight function) on an intervalI. The polynomials P andQare said to be orthogonal on[a, b]⊂ Iwith respect to the weight functionρor 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^kρ(x)dx (k = 0,1,2, . . .).

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

P_n(x) :=

1 µ₀ · · · µn−1

x µ₁ · · · µ_n ... ... . .. ... xⁿ µ_n · · · µ2n−1

.

Clearly, it suffices to show that P_n is ρ-orthogonal to the special polynomials 1, x, . . . , xⁿ⁻¹. Indeed, fork = 1, . . . , n, the first and the(k+ 1)^st columns of the determinanthP_n(x), x^k−1i_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_nhasnpairwise distinct zerosξ₁ <· · ·< 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

c_kf(ξ_k) +c_n+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_n be then^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−1if and only ifξ₁, . . . , ξ_nare the zeros ofP_n, and

(2.5) c_k=

Z b a

P_n(x)

(x−ξ_k)P_n⁰(ξ_k)ρ(x)dx.

Furthermore,ξ1, . . . , ξnare pairwise distinct elements of]a, b[, andck ≥0for allk = 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 byL_k: [a, b]→R. That is,

Lk(x) :=







P_n(x)

(x−ξ_k)P_n⁰(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 coefficientsc₁, . . . , 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−ξ₁)· · ·(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_nimplies thatP_n =a_nQ, andξ₁, . . . , ξ_n are the zeros ofP_n. Furthermore, (2.1) is exact if we substitute f := L_kandf :=L²_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. LetP_n be then^th degree 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 P_n²(a)

Z b a

P_n²(x)ρ(x)dx, (2.6)

ck = 1 ξ_k−a

Z b a

(x−a)P_n(x)

(x−ξ_k)P_n⁰(ξ_k)ρ(x)dx.

(2.7)

Furthermore,ξ₁, . . . , ξ_nare pairwise distinct elements of]a, b[, andc_k ≥0for allk = 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 Theorem 2.1 to the weight functionρ_aand the coefficients ca;k :=ck(ξk−a),

we get thatξ₁, . . . , ξ_n are the zeros ofP_nand, 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_n⁰(ξ_k)ρa(x)dx= Z b

a

(x−a)P_n(x)

(x−ξ_k)P_n⁰(ξ_k)ρ(x)dx.

Substitutingf :=P_n² into (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 that ξ₁, . . . , ξ_n are the zeros of the orthogonal polynomial P_n, and the coefficientsc₁, . . . , c_nare given by the formula (2.7). Define the coefficientc₀ byc₀ = Rb

a ρ− (c₁ +· · ·+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 ofc₀, 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

=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 most 2n.

Therefore, substitutingf :=P_n²into (2.2), we get formula (2.6).

Theorem 2.3. LetPn be then^th degree 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, . . . , ξ_nare the zeros ofP_n, 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, . . . , ξnare pairwise distinct elements of]a, b[, andck ≥0for allk = 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. LetPn be then^th degree 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 most 2n+ 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)

ck = 1

(b−ξ_k)(ξ_k−a) Z b

a

(b−x)(x−a)P_n(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_nare pairwise distinct elements of]a, b[, andc_k ≥0for allk = 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

=c₁(b−ξ₁)(ξ₁−a)P(ξ₁) +· · ·+c_n(b−ξ_n)(ξ_n−a)P(ξ_n).

Applying Theorem 2.1 to 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 coefficientsc_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−ξk)P_n⁰(ξk)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,c_k≥0fork = 0, . . . , n+ 1.

Conversely, assume that ξ₁, . . . , ξ_n are the zeros of P_n, and the coefficients c₁, . . . , c_n are given by the formula (2.11). Define the coefficientsc₀ andc_n+1 by 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 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;1Q(ξ₁) +· · ·+c_a,b;nQ(ξ_n)

holds. Thus, using the definition ofc₀ 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) +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

c_k (b−ξ_k)(ξ_k−a)Q(ξ_k) + (ξ_k−a)P(b) + (b−ξ_k)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.

Letf : [a, b] → Rbe 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. 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−x₁)· · ·(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_εfor 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_εbyf_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 → Rbe a polynomially n-convex con- tinuous 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∈I andε₀ >0such that the inclusion[a−ε₀, b+ε₀]⊂I holds. We show that the functionτ_εf : [a, b]→Rdefined by the formula

τ_εf(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_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

.

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)^kis the linear combination of the elements1, x−ε, . . . ,(x−ε)^kandx^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_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 whenever 1 ≤ k ≤ n−1. The particular casek = n−1gives the polynomialn-convexity ofτ_εf. Applying a change of variables and

the previous result, we get that

(−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¯(xn−s) 1 · · · 1 x₀ · · · x_n

... . .. ... xⁿ⁻¹₀ · · · xⁿ⁻¹_n

ϕ_ε(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_kandf_kbyε_k := _n¹

0+k andf_k :=f_εk fork ∈N, then0< ε_k < ε₀, 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. Letρbe a weight function on[a, b], and(a_j)be strictly monotone decreasing,(b_j) be strictly monotone increasing sequences such thataj →a,bj → b anda1 < b1. Denote the zeros ofP_m;j byξ_1;j, . . . , ξ_m;j, whereP_m;j is them^th degree member of theρ|_[aj,bj]-orthogonal polynomial system on [aj, bj], and denote the zeros ofPm byξ1, . . . , ξm, where Pm is them^th 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→ µ_k;[a,b] is continuous, therefore µ_k;[aj,bj] → 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[,

]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. 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 notation f₊(a) := lim sup_t→a+0f(t). Take the elements x₀ := a < x₁ := t < · · · < x_n of [a, b]. Then, the (even order) polynomial convexity off implies

f(a) f(t) f(x₂) . . . f(x_n)

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(xn)

1 1 1 . . . 1

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

≥0.

The adjoint determinants of the elements f(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)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. Letρ : [a, b] → Rbe a positive integrable function. Denote the zeros ofPm by ξ₁, . . . , 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 η₁, . . . , η_m whereQ_m is them^thdegree member of the orthogonal polynomial system on[a, b]with respect to the weight function(b−x)ρ(x). Define the coefficientsα₀, . . . , α_m andβ₁, . . . , 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] → R is 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 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), 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 ofρ. On the other hand,H is of degree2m, therefore Theorem 2.2 yields 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 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 toα₀, . . . , α_mwith the help ofP_m;j. Then,ξ_k;j → ξ_kdue to Lemma 2.1, and henceα_k;j → α_kasj → ∞. Applying the previous step of the proof on the smooth functions(f_i;j), it 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 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. Letρ : [a, b] → Rbe a positive integrable function. Denote the zeros ofPm by ξ1, . . . , ξm where Pm is the m^th degree member of the orthogonal polynomial system on [a, b]

with respect to the weight functionρ(x), and denote the zeros ofQ_m−1 byη₁, . . . , η_m−1 where Q_m−1is the(m−1)^stdegree member of the orthogonal polynomial system on[a, b]with respect to the weight function(b−x)(x−a)ρ(x). Define the coefficientsα₁, . . . , α_m andβ₀, . . . , β_m+1 by the formulae

α_k:=

Z b a

P_m(x)

(x−ξ_k)P_m⁰ (ξ_k)ρ(x)dx 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 following 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. Thenf^(2m) ≥ 0on]a, b[according to Theorem B. Consider the Hermite interpolation polynomialH that interpolates the function f 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,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_kH(ξ_k) =

m

X

k=1

α_kf(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), 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, 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=β₀H(a) +

m−1

X

k=1

β_kH(η_k) +β_mH(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, form≥0, the polynomialP_m be defined by the formula

Pm(x) :=

1 ¹₂ · · · _m+1¹ x ¹₃ · · · _m+2¹ ... ... . .. ... x^m _m+2¹ · · · _2m+1¹

.

Then,Pm hasmpairwise distinct zerosλ1, . . . , λm in]0,1[. Define the coefficientsα0, . . . , αm

by

α₀ := 1 P_m²(0)

Z 1 0

P_m²(x)dx, α_k := 1

λ_k Z 1

0

xP_m(x)

(x−λ_k)P_m⁰ (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 whena:= 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_mhasmpairwise distinct zeros0< λ₁ <· · ·< λ_m <1.

Moreover, the coefficientsα₀, . . . , 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≥α₀F(0) +

m

X

k=1

α_kF(λ_k)

=α₀f(a) +

m

X

k=1

α_kf (1−λ_k)a+λ_kb .

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, form≥1, the polynomialsP_m andQ_m−1 be defined by the formulae

P_m(x) :=

1 1 · · · _m¹ x ¹₂ · · · _m+1¹

... ... . .. ... x^m _m+1¹ · · · _2m¹

,

Qm−1(x) :=

1 _2·3¹ · · · _m(m+1)¹ x _3·4¹ · · · _(m+1)(m+2)¹

... ... . .. ... x^m−1 _(m+1)(m+2)¹ · · · _(2m−1)2m¹

.

Then,P_m hasmpairwise distinct zerosλ₁, . . . , λ_m in]0,1[andQm−1 hasm−1pairwise dis- tinct zerosµ₁, . . . , µm−1in]0,1[, respectively. Define the coefficientsα₁, . . . , α_mandβ₀, . . . , β_m by

α_k :=

Z 1 0

P_m(x)

(x−λ_k)P_m⁰ (λ_k)dx

and

β₀ := 1 Q²_m−1(0)

Z 1 0

(1−x)Q²_m−1(x)dx, β_k:= 1

(1−µ_k)µ_k Z 1

0

x(1−x)Qm−1(x) (x−µ_k)Q⁰_m−1(µ_k)dx, β_m := 1

Q²_m−1(1) Z 1

0

xQ²_m−1(x)dx.

If a functionf : [a, b]→Ris polynomially(2m)-convex, then it satisfies the following Hermite–

Hadamard-type inequality

m

X

k=1

α_kf (1−λ_k)a+λ_kb

≤ 1 b−a

Z b a

f(x)dx

≤β₀f(a) +

m−1

X

k=1

β_kf (1−µ_k)a+µ_kb

+β_mf(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 < λ₁ < · · · < λ_m < 1and Qm−1 has m−1 pairwise distinct zeros0 < µ₁ < · · · < µm−1 < 1. Moreover, the coefficientsα₁, . . . , α_m and β₀, . . . , 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 whenm = 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.

Casen = 6 The zeros ofP₃:

5−√ 15

10 , 1

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

5 18, 4

9, 5 18. The zeros ofQ₂:

5−√ 5

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

1 12, 5

12, 5 12, 1

12. Casen = 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 ofQ₃:

1 2 −

√21 14 , 1

2, 1 2+

√21 14 ; the corresponding coefficients:

1

20, 49

180, 16

45, 49 180, 1

20. Casen = 10

The zeros ofP₅:

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. Casen = 12(right hand side inequality)

The zeros ofQ₅:

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.