Using these inequalities, we produce new exponentially convex func- tions

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BY HERMITE POLYNOMIALS AND RELATED RESULTS

G. ARAS-GAZI Ć, V. ˇCULJAK, J. PE ˇCARI Ć and A. VUKELI Ć

Communicated by Luminit¸a Vese

In this note, we consider convex functions of higher order. Using the Hermite’s interpolating polynomials and conditions on the Green’s functions the results concerning for the converse of Jensen’s inequality for signed measure are pre- sented. Using these inequalities, we produce new exponentially convex functions. Finally, we give several examples of the families of functions for which the obtained results can be applied.

AMS 2010 Subject Classification: Primary 26D15, Secondary 26D07, 26A51.

Key words: Green function, Jensen’s inequality, n-convex function, Hermite interpolating polynomial, Cauchy type mean value theorems, n- exponential convexity, exponential convexity, log-convexity, means.

1. INTRODUCTION

Let (Ω,A, µ) be a measure space. The well-known Jensen inequality asserts that

(1.1) f

R

Ωpgdµ R

Ωpdµ(x)

≤ R

Ωpf(g)dµ R

Ωpdµ ,

holds iff is a convex function on intervalI ⊆R, whereg: Ω→I be a function from L^∞(µ) and p : Ω→ R be a nonnegative function from L¹(µ), such that R

Ωpdµ6= 0.

If I = [α, β], where −∞ < α < β <+∞, and function f is continuous, then the converse of the integral Jensen’s inequality states

(1.2)

R

Ωpf(g) dµ R

Ωpdµ ≤f(α)β−¯g

β−α +f(β)g¯−α β−α,

where ¯g=

R

Ωpgdµ R

Ωpdµ (see [8] or [12]).

In [3, 9] and [10] authors obtained generalization for real (signed) measure:

MATH. REPORTS17(67),2(2015), 201–223

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Theorem A 1. Let f : [α, β] → R be convex, and g : [a, b] → [α, β]

integrable with respect to real (signed) measureµ. Ifµis such thatRb

a dµ(t) = 1 and

Z b a

G_L(g(t), s) dµ(t)≤0, ∀s∈[α, β], (1.3)

then it holds

Z b a

f(g(t))dµ(t)≤f(α) β−Rb

ag(t)dµ(t) β−α

!

+f(β) Rb

ag(t)dµ(t)−α β−α

! (1.4)

where Lagrange Green’s function on [α, β]×[α, β]is defined by G_L(t, s) =

( _{(α−s)(t−β)}

α−β , s≤t

−(β−s)(t−α)

β−α, s≥t.

(1.5)

The reverse inequality in (1.3) implies the reverse inequality in (1.4);

Recently, in [11] the following generalization form, M ∈[α, β] and Stielt- jes measure dλis done.

Theorem A 2. Let g : [a, b] → R be a continuous function and [α, β]

be an interval such that the image of g is a subset of [α, β]. Let m, M ∈[α, β]

(m 6= M) be such that g([a, b]) ⊆[m, M] for all t ∈ [a, b]. Let λ: [a, b] → R be a continuous function or a function of bounded variation, and λ(a)6=λ(b).

Then the following two statements are equivalent:

(1) For every continuous convex functionf : [α, β]→R it holds Rb

af(g(t)) dλ(t) Rb

a dλ(t)

≤ M−g

M−mf(m) + g−m M−mf(M) (2) for alls∈[α, β]it holds

Rb

aGL(g(t), s) dλ(t) Rb

a dλ(t)

≤ M −g

M −mGL(m, s) + g−m

M −mGL(M, s) where G_L is the Green functions defined on [α, β]×[α, β] by (1.5), and

g= Rb

ag(x)dλ(x) Rb

adλ(x) .

In the same paper [11], authors established the following generalization of Jensen’s inequality for Stieltjes measure dλ:

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Theorem A 3. Let g: [a, b]→R be a continuous function and[α, β]be an interval such that the image of g is a subset of [α, β]. Let λ: [a, b]→R be a continuous function or a function of bounded variation, and λ(a)6=λ(b)and g∈[α, β]. Then the following two statements are equivalent:

(1) For every continuous convex functionf : [α, β]→R it holds f(g)≤

Rb

af(g(t)) dλ(t) Rb

a dλ(t) (2) for alls∈[α, β]it holds

GL(g, s)≤ Rb

aGL(g(t), s) dλ(t) Rb

a dλ(t) In this note, we consider n-convex functions.

Definition 1.Let f be a real-valued function defined on the segment [a, b]. Thedivided difference of order nof the function f at distinct points x₀, ..., x_n∈[a, b], is defined recursively (see [2, 12]) by

f[xi] =f(xi), (i= 0, . . . , n) and

f[x0, . . . , xn] = f[x1, . . . , xn]−f[x0, . . . , xn−1]

x_n−x₀ .

The valuef[x0, . . . , xn] is independent of the order of the pointsx0, . . . , xn. The definition may be extended to include the case that some (or all) of the points coincide. Assuming that f^(j−1)(x) exists, we define

(1.6) f[x, . . . , x

| {z }

j−times

] = f^(j−1)(x) (j−1)! .

The notion of n-convexitygoes back to Popoviciu ([14]). We follow the definition given by Karlin ([6]):

Definition 2. A function f : [a, b] → R is said to be n-convex on [a, b], n≥0, if for all choices of (n+ 1) distinct points in [a, b], n−th order divided difference of f satisfies

f[x₀, ..., x_n]≥0.

In fact, Popoviciu proved that each continuousn-convex function on [a, b]

is the uniform limit of the sequence of n-convex polynomials. Many related results, as well as some important inequalities due to Favard, Berwald and Steffensen can be found in [7].

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In this note we give a results for Theorem A3 and Theorem A4 for n-convex functions. Using Hermite’s polynomial we obtain identities and from these results we get the conditions, using the corresponding Green functions, on a real signed measure µ such that generalisation of Jensen’s type and the type of converses of Jensen’s inequality are valid. Also, we give mean value theorems using that inequalities. We will introduce the notion ofn-exponentially convex functions and deduce a method of producing n-exponentially convex functions using some known families of functions of the same type.

1.1. HERMITE INTERPOLATING POLYNOMIAL We follow here notations and terminology from the book by [1].

Let −∞< α < β <∞, and α≤α1 < α2... < αr≤β, (r ≥2) be given.

Forf ∈Cⁿ[α, β] a unique polynomialP_H(t) of degree (n−1), exists, fulfilling one of the following conditions:

Hermite conditions:

P_H⁽ⁱ⁾(α_j) =f⁽ⁱ⁾(α_j); 0≤i≤k_j, 1≤j≤r,

r

X

j=1

k_j+r =n, in particular:

Simple Hermite or Osculatory conditions: (n= 2m, r=m, kj = 1 for all j)

P_O(α_j) =f(α_j), P_O⁰ (α_j) =f⁰(α_j), 1≤j≤m, Lagrange conditions: (r, kj = 0 for all j)

PL(αj) =f(αj), 1≤j≤n,

Type (m, n−m) conditions: (r = 2,1 ≤ m ≤ n−1, k₁ = m−1, k₂ = n−m−1)

P_mn⁽ⁱ⁾(α) =f⁽ⁱ⁾(α), 0≤i≤m−1, Pmn⁽ⁱ⁾(β) =f⁽ⁱ⁾(β), 0≤i≤n−m−1,

Two-point Taylor conditions: (n= 2m, r = 2, k₁=k₂ =m−1) P_2T⁽ⁱ⁾(α) =f⁽ⁱ⁾(α), P_2T⁽ⁱ⁾(β) =f⁽ⁱ⁾(β), 0≤i≤m−1.

The associated error |e_H(t)|can be represented in terms of the Green’s functionGH(t, s) for the multipoint boundary value problem

z⁽ⁿ⁾(t) = 0, z⁽ⁱ⁾(α_j) = 0,0≤i≤k_j, 1≤j≤r, that is, the following results holds:

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Theorem A 4. LetF ∈Cⁿ[α, β], and letP_H be its Hermite interpolating polynomial. Then

F(t) =P_H(t) +e_H(t)

=

r

X

j=1 kj

X

i=0

H_ij(t)F⁽ⁱ⁾(α_j) + Z _β

α

G_H(t, s)F⁽ⁿ⁾(s)ds, (1.7)

where H_ij are fundamental polynomials of the Hermite basis defined by

Hij(t) = 1 i!

ω(t) (t−αj)^k^j⁺¹⁻ⁱ

kj−i

X

k=0

1 kj!

(t−αj)^k^j⁺¹ ω(t)

(k) t=αj

(t−αj)^k, (1.8)

where

(1.9) ω(t) =

r

Y

j=1

(t−α_j)^k^j⁺¹, and GH is the Green’s function defined by

(1.10) GH(t, s) =





 Pl

j=1

Pkj

i=0

(αj−s)ⁿ⁻ⁱ⁻¹

(n−i−1)! Hij(t), s≤t

−Pl j=l+1

Pkj

i=0

(αj−s)ⁿ⁻ⁱ⁻¹

(n−i−1)! H_ij(t), s≥t.

for all αl≤s≤αl+1, l= 1, ..., r−1.

2. GENERALIZATION OF JENSEN’S INEQUALITY BY HERMITE POLYNOMIALS

Using Hermite polynomials we obtain the following representation of Jensen’s inequality:

Lemma 1. Let f : [α, β]→R be of class Cⁿ on [α, β], (n≥2). Let µ be a regular, real (signed) Borel measure and g: [a, b]→R be integrable function with respect to measureµsuch thatg([a, b])⊆[α, β]andg=

Rb

ag(t) dµ(t) Rb

a dµ(t) ∈[α, β].

Then it holds f(g)−

Rb

af(g(t)) dµ(t) Rb

a dµ(t) (2.1)

=

r

X

j=1 kj

X

i=0

f⁽ⁱ⁾(α_j) H_ij(g)− Rb

aH_ij(g(t)) dµ(t) Rb

a dµ(t)

!

+ Z β

α

f⁽ⁿ⁾(s) G_H(g, s)− Rb

aGH(g(t), s) dµ(t) R_b

a dµ(t)

! ds,

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where H_ij are defined on[α, β]by (1.8) andG_H is the Green’s function defined by (1.10).

Proof. By Theorem A4 we can represent every function f ∈ Cⁿ[α, β], (n≥2) in the form:

f(x) =

r

X

j=1 kj

X

i=0

H_ij(x)f⁽ⁱ⁾(α_j) + Z β

α

G_H(x, s)f⁽ⁿ⁾(s)ds, (2.2)

and calculatef(g) : f(g) =

r

X

j=1 kj

X

i=0

H_ij(g)f⁽ⁱ⁾(α_j) + Z β

α

G_H(g, s)f⁽ⁿ⁾(s)ds.

The integration of the composition of functionsf◦gfor the real measure µon [a, b] gives

Rb

af(g(t)) dµ(t) Rb

a dµ(t) =

r

X

j=1 kj

X

i=0

f⁽ⁱ⁾(αj) Rb

aHij(g(t)) dµ(t) Rb

a dµ(t) +

Z β α

f⁽ⁿ⁾(s)· Rb

aG_H(g(t), s) dµ(t) Rb

a dµ(t) ds.

Now, we can easily calculate the differencef(g)−

Rb

af(g(t)) dµ(t) Rb

a dµ(t) and obtain the identity (2.1).

Using Lemma 1 we can get the following generalization of Jensen’s inequality by Hermite polynomials:

Theorem 1. Let µ be a regular, real (signed) Borel measure and g : [a, b]→Rbe integrable function with respect to measure µsuch that g([a, b])⊆ [α, β]and g=

Rb

ag(t) dµ(t) Rb

a dµ(t) ∈[α, β].

For any n≥2 if for alls∈[α, β]

(2.3) G_H(g, s)≤

Rb

a dµ(t) then for every n-convex function f : [α, β]→Rit holds

f(g)≤ R_b

af(g(t)) dµ(t) Rb

a dµ(t) +

r

X

j=1 kj

X

i=0

f⁽ⁱ⁾(α_j) H_ij(g)− R_b

aHij(g(t)) dµ(t) Rb

a dµ(t)

! , (2.4)

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whereG_H is the Green’s function (1.10) and functionsH_ij are defined by (1.8).

If the reverse inequality in (2.3) is valid, then the reverse inequality in (2.4) is also valid.

As a consequence of the above results, follow the results for the left-hand side of the generalized Hermite-Hadamard inequality:

Corollary1. Letµbe a regular, real (signed) Borel measure on interval [a, b]⊆[α, β] and x=

Rb axdµ(x) Rb

a dµ(x) ∈[α, β].

(2.5) G_H(x, s)≤

Rb

a GH(x, s) dµ(x) R_b

a dµ(x) then for every n-convex function f : [α, β]→Rit holds

f(x)≤ Rb

af(x) dµ(x) Rb

a dµ(x) +

r

X

j=1 kj

X

i=0

f⁽ⁱ⁾(α_j) H_ij(x)− Rb

aH_ij(x) dµ(x) Rb

a dµ(x)

! , (2.6)

whereG_H is the Green’s function (1.10) and functionsH_ij are defined by (1.8).

3. GENERALIZATION OF THE CONVERSE OF JENSEN’S INEQUALITY BY HERMITE POLYNOMIALS

Let µ be a regular, real (signed) Borel measure and let g : [a, b] → R be integrable with respect to µ such that g([a, b]) ⊆ [m, M] ⊆ [α, β] for all t∈[a, b].For a given function F : [α, β]→Rdenote by LR(F, g, m, M, µ) (3.1) LR(F, g, m, M, µ) =

Rb

aF(g(t)) dµ(t) Rb

a dµ(t)

− M−g

M−mF(m)− g−m

M −mF(M), whereg=

Rb

ag(t) dµ(t) Rb

a dµ(t) .

Using Hermite polynomials we obtain the following representation for left side of the converse of Jensen’s inequality:

Lemma 2. Let f : [α, β]→R be of class Cⁿ on [α, β], (n≥2). Let µ be a regular, real (signed) Borel measure and g: [a, b]→R be integrable function with respect to measureµ such thatg([a, b])⊆[m, M]⊆[α, β]for all t∈[a, b].

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Let g=

Rb

ag(t) dµ(t) Rb

a dµ(t) .Then it holds LR(f, g, m, M, µ) =

r

X

j=1 kj

X

i=0

f⁽ⁱ⁾(α_j)·LR(H_ij, g, m, M, µ) (3.2)

+ Z β

α

f⁽ⁿ⁾(s)·[LR(GH(·, s), g, m, M, µ)] ds where H_ij are defined on[α, β]by (1.8) andG_H is the Green’s function defined by (1.10).

Proof. By using representation of every function f ∈ C⁽ⁿ⁾[α, β] and its Hermite interpolating polynomial in the form (2.2) we calculate f(m) and f(M). The integration of the composition of functionsf◦gfor the real measure µon [a, b] gives

Rb

af(g(t)) dµ(t) Rb

a dµ(t) =

r

X

j=1 kj

X

i=0

f⁽ⁱ⁾(α_j) Rb

a dµ(t) +

Z β α

f⁽ⁿ⁾(s)· Rb

a dµ(t) ds.

Now, we can easily calculate the difference LR(f, g, m, M, µ) and obtain the identity (3.2).

Using Lemma 2 we can get the following generalization of the conversion of Jensen’s inequality by Hermite polynomials:

Theorem 2. Let µ be a regular, real (signed) Borel measure and g : [a, b]→Rbe integrable function with respect to measure µsuch that g([a, b])⊆ [m, M]⊆[α, β]for all t∈[a, b].Let g=

Rb

ag(t) dµ(t) Rb

a dµ(t) . For any n≥2 if for alls∈[α, β]

(3.3)

Rb

a dµ(t)

≤ M−g

M−mG_H(m, s) + g−m

M −mG_H(M, s), then for every n-convex function f : [α, β]→Rit holds

Rb

af(g(t)) dµ(t) Rb

a dµ(t)

≤ M −g

M −mf(m) + g−m M−mf(M) (3.4)

+

r

X

j=1 kj

X

i=0

f⁽ⁱ⁾(α_j)·LR(H_ij, g, m, M, µ),

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where H_ij are defined on[α, β]by (1.8) andG_H is the Green’s function defined by (1.10).

Remark 1.Settingm=αand M =β andRb

adµ(t) = 1 in Theorem 2 we get the following corollary as in [10]:

Corollary 2. Let f : [α, β] → R be n-convex, (n ≥ 2). Let µ be a regular, real (signed) Borel measure and g : [a, b] → R be integrable function with respect to measure µ such thatg([a, b])⊆[α, β].

If (3.5)

Z b a

GH(g(t), s)dµ(t)≤0, ∀s∈[α, β], then it holds

Z b a

f(g(t))dµ(t)≤

r

X

j=1 kj

X

i=0

f⁽ⁱ⁾(α_j)· Z b

a

H_ij(g(t))dµ(t), (3.6)

where the Green’s function GH is defined on [α, β]×[α, β] by (1.10).

It follows the results for the right-hand side of the generalized Hermite- Hadamard inequality:

Rb axdµ(x) Rb

a dµ(x) .

(3.7)

Rb

aG_H(x, s) dµ(x) Rb

a dµ(x)

≤ b−x

b−aG_H(a, s) + x−a

b−aG_H(b, s), then for every n-convex function f : [α, β]→Rit holds

Rb

af(x) dµ(x) Rb

a dµ(x)

≤ b−x

b−af(a) +x−a b−af(b) (3.8)

+

r

X

j=1 kj

X

i=0

f⁽ⁱ⁾(α_j)·LR(H_ij, id, a, b, µ)

where the Green’s function GH is defined on [α, β]×[α, β] by (1.10).

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4.RELATED RESULT FOR TWO-POINT TAYLOR CONDITION

Lemma A 1. Let f ∈Cⁿ[α, β], (n≥2, n= 2m) then it holds f(x) =

m−1

X

i=0 m−1−i

X

k=0

m+k−1 k

·[f⁽ⁱ⁾(α)τ_ik(x) +f⁽ⁱ⁾(β)ν_i(x)]

(4.1)

+ Z β

α

f^(2m)(s)G2T(x, s) ds where τ_ik and ν_ik defined on [α, β] :

τ_ik(x) = (x−α)ⁱ i!

x−β α−β

m x−α β−α

k

, (4.2)

ν_ik(x) = (x−β)ⁱ i!

x−α β−α

m x−β α−β

k

(4.3)

and G_2T is the Green’s function of the two-point Taylor problem:

G_2T(t, s) =

( ₍₋₁₎^m

(2m−1)!p^m(t, s)Pm−1 j=0

m−1+j j

(t−s)^m−1−jq^j(t, s), s≤t

(−1)^m

(2m−1)!q^m(t, s)Pm−1 j=0

m−1+j j

(s−t)^m−1−jp^j(t, s), s≥t (4.4)

and

p(t, s) = (s−α)(β−t)

β−α , q(t, s) =p(s, t), ∀t, s∈[α, β].

4.1. GENERALIZATION OF JENSEN’S INEQUALITY BY HERMITE POLYNOMIALS

FOR TWO-POINT TAYLOR CONDITIONS

Using Hermite polynomials for Two-points Taylor conditions we obtain the following representation of Jensen’s inequality:

Lemma 3. Let f : [α, β] → R be of class C^(2m) on [α, β]. Let µ be a regular, real (signed) Borel measure and g : [a, b] → R be integrable function with respect to measureµsuch thatg([a, b])⊆[α, β]andg=

Rb

ag(t) dµ(t) Rb

a dµ(t) ∈[α, β].

Then it holds f(g)−

Rb

af(g(t)) dµ(t) Rb

a dµ(t) =

m−1

X

i=0 m−1−i

X

k=0

m+k−1 k

·

"

f⁽ⁱ⁾(α)· τ_ik(g)− Rb

aτ_ik(g(t)) dµ(t) R_b

a dµ(t)

! (4.5)

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+f⁽ⁱ⁾(β)· νik(g)− Rb

aν_ik(g(t)) dµ(t) Rb

a dµ(t)

!#

+ Z _β

α

f^(2m)(s)·

"

G_2T(g, s)− R_b

aG_2T(g(t), s) dµ(t) Rb

a dµ(t)

# ds, where τik and νik defined on [α, β] by (4.2) and (4.3) and G2T is the Green’s function of the two-point Taylor problem defined by (4.4).

The following Corollary is a generalization of Jensen’s inequality by Her- mite polynomials for Two-point Taylor conditions:

Corollary 4. Let µ be a regular, real (signed) Borel measure and g : [a, b]→Rbe integrable function with respect to measure µsuch that g([a, b])⊆ [α, β]and g=

Rb

ag(t) dµ(t) Rb

a dµ(t) ∈[α, β]

For any m≥1 if for alls∈[α, β]

(4.6) G_2T(g, s)≤

R_b

aG_2T(g(t), s) dµ(t) Rb

a dµ(t) ,

then for every (2m)-convex function f : [α, β]→R it holds f(g)≤

Rb

af(g(t)) dµ(t) Rb

a dµ(t) +

m−1

X

i=0 m−1−i

X

k=0

m+k−1 k

(4.7) ·

"

f⁽ⁱ⁾(α)· τik(g)−

Rb

aτik(g(t)) dµ(t) Rb

a dµ(t)

!

+f⁽ⁱ⁾(β)· νik(g)− Rb

aνik(g(t)) dµ(t) Rb

a dµ(t)

!#

where G_2T is the Green’s function of the two-point Taylor problem (4.4) and functions τ_ik and ν_ik are defined by (4.2) and (4.3) on[α, β].

Remark 2.For m = 1 in Corollary 4 we obtain Theorem A3 for real (signed) measure:

(1) For every continuous convex function f : [α, β]→Rit holds f(g)≤

Rb

af(g(t)) dµ(t) Rb

a dµ(t) , if and only if

(2) for alls∈[α, β] it holds G_L(g, s)≤

Rb

aGL(g(t), s) dµ(t) R_b

a dµ(t) .

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Furthermore, the statements (1) and (2) are also equivalent if we change the sign of both inequalities.

We use the fact that function GL(·, s), s∈[α, β] is continuous convex on [α, β] and form= 1 for Green’s function it holdsG2T =GL.

It follows the results for the left-hand side of the generalized Hermite- Hadamard inequality:

Rb axdµ(x) Rb

a dµ(x) ∈[α, β].

(4.8) G2T(x, s)≤

Rb

aG_2T(x, s) dµ(x) Rb

a dµ(x) , then for every (2m)-convex function f : [α, β]→R it holds

f(x)≤ Rb

af(x) dµ(x) Rb

a dµ(x) +

m−1

X

i=0 m−1−i

X

k=0

m+k−1 k

(4.9) ·

"

f⁽ⁱ⁾(α)· τik(x)− Rb

aτik(x) dµ(x) Rb

a dµ(x)

!

+f⁽ⁱ⁾(β)· νik(x)− Rb

aνik(x) dµ(x) Rb

a dµ(x)

!#

,

where G2T is the Green’s function of the two-point Taylor problem (4.4) and functions τ_ik and ν_ik are defined by (4.2) and (4.3) on[α, β].

4.2. GENERALIZATION OF THE CONVERSE OF JENSEN’S INEQUALITY BY HERMITE POLYNOMIALS

FOR TWO-POINT TAYLOR CONDITIONS

Using Hermite polynomials for Two-points Taylor conditions we obtain the following representation for left side of the converse of Jensen’s inequality:

Lemma 4. Let f : [α, β] → R be of class C^(2m) on [α, β]. Let µ be a regular, real (signed) Borel measure and g : [a, b] → R be integrable function with respect to measureµ such thatg([a, b])⊆[m, M]⊆[α, β]for all t∈[a, b].

Let g=

Rb

ag(t) dµ(t) Rb

a dµ(t) .Then it holds

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LR(f, g, m, M, µ) (4.10)

=

m−1

X

i=0 m−1−i

X

k=0

m+k−1 k

·h

f⁽ⁱ⁾(α)LR(τ_ik, g, m, M, µ)+f⁽ⁱ⁾(β)LR(ν_ik, g, m, M, µ)i +

Z β α

f^(2m)(s) [LR(G2T(·, s), g, m, M, µ)] ds, where τ_ik and ν_ik defined on [α, β] by (4.2) and (4.3) and G_2T is the Green’s function of the two-point Taylor problem defined by (4.4).

The following Corollary is generalization of the converse of Jensen’s inequality by Hermite polynomials for Two-point Taylor conditions:

Corollary 6. Let µ be a regular, real (signed) Borel measure and g : [a, b]→Rbe integrable function with respect to measure µsuch that g([a, b])⊆ [m, M]⊆[α, β]for all t∈[a, b].Let g=

Rb

ag(t) dµ(t) Rb

a dµ(t) . For any m≥1 if for alls∈[α, β]

(4.11) Rb

aG2T(g(t), s) dµ(t) Rb

a dµ(t)

≤ M−g

M−mG2T(m, s) + g−m

M−mG2T(M, s), then for every (2m)-convex function f : [α, β]→R it holds

Rb

af(g(t)) dµ(t) Rb

a dµ(t)

≤ M −g

M −mf(m) + g−m M−mf(M) (4.12)

+

m−1

X

i=0 m−1−i

X

k=0

m+k−1 k

·h

f⁽ⁱ⁾(α)LR(τik, g, m, M, µ)+f⁽ⁱ⁾(β)LR(νik, g, m, M, µ) i

,

where G_2T is the Green’s function of the two-point Taylor problem (4.4) and functions τ_ik and ν_ik are defined by (4.2) and (4.3) on[α, β].

Remark 3.Form= 1 in Corollary 6 we obtain the result in Theorem A2 for real (signed) measure µ:

(1) For every continuous convex function f : [α, β]→Rit holds Rb

af(g(t)) dµ(t) Rb

a dµ(t) ≤ M−g

M−mf(m) + g−m

M−mf(M), if and only if

(2) for alls∈[α, β] it holds Rb

aGL(g(t), s) dµ(t) R_b

a dµ(t)

≤ M −g

M −mG_L(m, s) + g−m

M −mG_L(M, s).

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Furthermore, the statements (1) and (2) are also equivalent if we change the sign of both inequalities.

We use the fact that function G_L(·, s), s∈[α, β] is continuous convex on [α, β] and form= 1 for Green’s function it holdsG2T =GL.

Remark 4. Settingm=αandM =β andRb

adµ(t) = 1 in Corollary 6 we get the following corollary as in [10]:

Corollary7. Letf : [α, β]→Rbe(2m)-convex. Letµbe a regular, real (signed) Borel measure and g: [a, b]→R be integrable function with respect to measureµ such that g([a, b])⊆[α, β].

If (4.13)

Z b a

G_2T(g(t), s)dµ(t)≤0, ∀s∈[α, β].

then it holds Z b

a

f(g(t))dµ(t)≤

m−1

X

i=0 m−1−i

X

k=0

m+k−1 k

(4.14) ·

"

f⁽ⁱ⁾(α) Z b

a

(g(t)−α)ⁱ i!

g(t)−β α−β

m

g(t)−α β−α

k

dµ(t) +f⁽ⁱ⁾(β)

Z b a

(g(t)−β)ⁱ i!

g(t)−α β−α

m

g(t)−β α−β

k

dµ(t)

# ,

where the Green’s function G2T is defined on [α, β]×[α, β]by (4.4).

It follows the results for the right-hand side of the generalized Hermite- Hadamard inequality:

Rb axdµ(x) Rb

a dµ(x) .

(4.15)

R_b

aG_2T(x, s) dµ(x) Rb

a dµ(x)

≤ b−x

b−aG_2T(a, s) +x−a

b−aG_2T(b, s) then for every (2m)-convex function f : [α, β]→R it holds

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Rb

af(x) dµ(x) Rb

a dµ(x) ≤ b−x

b−af(a) + x−a b−af(b) (4.16)

+

m−1

X

i=0 m−1−i

X

k=0

m+k−1 k

·h

f⁽ⁱ⁾(α)LR(τ_ik, id, a, b, µ) +f⁽ⁱ⁾(β)LR(ν_ik, id, a, b, µ)i ,

where G2T is the Green’s function of the two-point Taylor problem (4.4) and functions τ_ik and ν_ik are defined by (4.2) and (4.3) on[α, β].

5. n-EXPONENTIAL CONVEXITY OF JENSEN’S INEQUALITY BY HERMITE POLYNOMIALS

Motivated by the inequalities (2.4), (3.4), (4.7) and (4.12) we define functionals Φ1(f), Φ2(f),Φ3(f) and Φ4(f) by

Φ₁(f) =f(g)− Rb

af(g(t)) dµ(t) Rb

a dµ(t)

−

r

X

j=1 kj

X

i=0

f⁽ⁱ⁾(α_j) H_ij(g)− Rb

a dµ(t)

! , (5.1)

Φ2(f) = Rb

af(g(t)) dµ(t) Rb

a dµ(t)

− M−g

M−mf(m)− g−m M−mf(M)

−

r

X

j=1 kj

X

i=0

f⁽ⁱ⁾(αj)·LR(Hij, g, m, M, µ), (5.2)

Φ₃(f) =f(g)− Rb

af(g(t)) dµ(t) Rb

a dµ(t)

−

m−1

X

i=0 m−1−i

X

k=0

m+k−1 k

(5.3) ·

"

f⁽ⁱ⁾(α) τ_ik(g)− Rb

aτik(g(t)) dµ(t) Rb

a dµ(t)

!

+f⁽ⁱ⁾(β) ν_ik(g)− Rb

aνik(g(t)) dµ(t) Rb

a dµ(t)

!#

and

Φ₄(f) = R_b

af(g(t)) dµ(t) Rb

a dµ(t)

− M−g

M−mf(m)− g−m M−mf(M) (5.4)

−

m−1

X

i=0 m−1−i

X

k=0

m+k−1 k

·h

f⁽ⁱ⁾(α)LR(τik, g, m, M, µ)+f⁽ⁱ⁾(β)LR(νik, g, m, M, µ) i

.

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Theorem 3. Let µ be a regular, real (signed) Borel measure and f : [α, β]→R, f ∈Cⁿ([α, β]), n≥2. Let g: [a, b]→R be integrable with respect to µ such thatg([a, b])⊆[α, β]and g=

Rb

ag(t) dµ(t) Rb

a dµ(t) ∈[α, β]. If for alls∈[α, β]

the reverse inequalities in (2.3), (3.3), (4.6) and (4.11) hold, then there exists ξ ∈[α, β]such that

Φ_i(f) =f⁽ⁿ⁾(ξ)Φ_i(ϕ), (5.5)

where ϕ(x) = ^x_n!ⁿ.

Proof. Let us denote m= minf⁽ⁿ⁾ and M = maxf⁽ⁿ⁾. We first consider the following function φ1(x) = ^{M x}_n!ⁿ −f(x). Then φ⁽ⁿ⁾₁ (x) = M −f⁽ⁿ⁾(x) ≥ 0, x ∈ [α, β], so φ₁ is a n-convex function. Similarly, a function φ₂(x) = f(x)−^mx_n!ⁿ is an-convex function. Now, we use inequalities from (2.4), (3.4), (4.7) and (4.12) for n-convex functions φ1 and φ2. So, we can conclude that there existsξ ∈[α, β] that we are looking for in (5.5).

Corollary 9. Let f, h: [α, β]→Rsuch thatf, h∈Cⁿ([α, β]). If for all s ∈ [α, β] the reverse inequalities in (2.3), (3.3), (4.6) and (4.11) hold, then there exists ξ∈[α, β] such that

Φ_i(f)

Φi(h) = f⁽ⁿ⁾(ξ)

h⁽ⁿ⁾(ξ), i= 1,2,3,4, (5.6)

provided that the denominator of the left-hand side is non-zero.

Proof. We use the following standard technique: Let us define the linear functional L(χ) = Φi(χ), i = 1,2,3,4. Next, we define χ(t) = f(t)L(h)− h(t)L(f). According to Theorems 3, applied to χ, there exists ξ ∈ [α, β] so that

L(χ) =χ⁽ⁿ⁾(ξ)Φi(ϕ), ϕ(x) = xⁿ

n!, i= 1,2,3,4.

From L(χ) = 0, it follows f⁽ⁿ⁾(ξ)L(h) −h⁽ⁿ⁾(ξ)L(f) = 0 and (5.6) is proved.

Now, let us recall some definitions and facts about exponentially convex functions (see [5]):

Definition 3. A function ψ : I → R is n-exponentially convex in the Jensen sense on I if _n

X

i,j=1

ξ_iξ_jψ

x_i+x_j 2

≥0,

holds for all choices ξ1, . . . , ξn∈Rand all choices x1, . . . , xn∈ I.

A function ψ : I → R is n-exponentially convex if it is n-exponentially convex in the Jensen sense and continuous on I.

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Remark 5.It is clear from the definition that 1-exponentially convex functions in the Jensen sense are in fact nonnegative functions. Also, n- exponentially convex functions in the Jensen sense are k-exponentially convex in the Jensen sense for every k∈N, k≤n.

By definition of positive semi-definite matrices and some basic linear al- gebra we have the following proposition:

Proposition 1. If ψisn-exponentially convex in the Jensen sense, then the matrix h

ψ_x

i+xj

2

ik

i,j=1 is positive semi-definite for all k ∈ N, k ≤ n.

Particularly, deth ψ_x

i+xj

2

ik

i,j=1 ≥0 for all k∈N, k≤n.

Definition 4.A function ψ:I →Ris exponentially convex in the Jensen sense on I if it is n-exponentially convex in the Jensen sense for all n∈N.

A functionψ:I →Ris exponentially convex if it is exponentially convex in the Jensen sense and continuous.

Remark 6. It is known (and easy to show) that ψ :I → R is log-convex in the Jensen sense if and only if

α²ψ(x) + 2αβψ

x+y 2

+β²ψ(y)≥0

holds for everyα, β ∈Randx, y∈I. It follows that a function is log-convex in the Jensen sense if and only if it is 2-exponentially convex in the Jensen sense.

A positive function is log-convex if and only if it is 2-exponentially convex.

We use an idea from [13] to give an elegant method of producing n- exponentially convex functions and exponentially convex functions applying the above functionals to a given family with the same property (see also [5]):

Theorem 4. Let Υ = {f_s : s ∈ J}, where J is an interval in R, be a family of functions defined on an interval [α, β] in R, such that the function s7→fs[z0, . . . , z_l]isn-exponentially convex in the Jensen sense on J for every (l+ 1) mutually different points z₀, . . . , z_l ∈[α, β]. Let Φ_i(f), i = 1,2,3,4 be linear functional defined as in (5.1), (5.2) (5.3) and (5.4). Thens7→Φi(fs)is an n-exponentially convex function in the Jensen sense on J. If the function s7→Φ_i(f_s) is continuous on J, then it is n-exponentially convex on J.

Proof. For ξ_i ∈ R, i = 1, . . . , n and s_i ∈ J, i = 1, . . . , n, we define the function

h(z) =

n

X

i,j=1

ξ_iξ_jfsi+sj 2

(z).

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Using the assumption that the functions7→f_s[z₀, . . . , z_l] isn-exponentially convex in the Jensen sense, we have

h[z0, . . . , zl] =

n

X

i,j=1

ξiξjfsi+sj 2

[z0, . . . , zl]≥0,

which in turn implies thathis al-convex function onJ, so it is Φ_k(h)≥0, k= 1,2,3,4, hence,

n

X

i,j=1

ξiξjΦk

fsi+sj

2

≥0.

We conclude that the function s7→ Φ_k(f_s) is n-exponentially convex on J in the Jensen sense.

If the function s 7→ Φ_k(fs) is also continuous on J, then s7→ Φ_k(fs) is n-exponentially convex by definition.

The following corollaries are immediate consequences of the above theorem:

Corollary 10. Let Υ = {f_s : s∈ J}, where J is an interval in R, be a family of functions defined on an interval [α, β]in R, such that the function s 7→ f_s[z₀, . . . , z_l] is exponentially convex in the Jensen sense on J for every (l+ 1) mutually different points z₀, . . . , z_l ∈[α, β]. Let Φ_i(f), i = 1,2,3,4 be linear functional defined as in (5.1), (5.2) (5.3) and (5.4). Then s7→ Φi(fs) is an exponentially convex function in the Jensen sense on J. If the function s7→Φ_i(f_s) is continuous on J, then it is exponentially convex on J.

Corollary 11. Let Υ = {f_s : s∈ J}, where J is an interval in R, be a family of functions defined on an interval [α, β]in R, such that the function s7→fs[z0, . . . , zl]is 2-exponentially convex in the Jensen sense onJ for every (l+ 1) mutually different points z0, . . . , z_l ∈[α, β]. Let Φi(f), i = 1,2,3,4 be linear functional defined as in (5.1), (5.2) (5.3) and (5.4). Then the following statements hold:

(i) If the function s7→Φ_i(f_s) is continuous on J, then it is2-exponentially convex function onJ. Ifs7→Φi(fs) is additionally strictly positive, then it is also log-convex on J. Furthermore, the following inequality holds true:

(5.7) [Φi(fs)]^t−r ≤[Φi(fr)]^t−s[Φi(ft)]^s−r for every choice r, s, t∈J, such thatr < s < t.

(ii) If the functions7→Φ_i(f_s)is strictly positive and differentiable on J, then for everys, q, u, v∈J, such that s≤u andq ≤v, we have

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(5.8) µ_s,q(Φ_i,Υ)≤µ_u,v(Φ_i,Υ), where

(5.9) µ_s,q(Φ_i,Υ) =







Φi(fs) Φi(fq)

_s−q¹

, s6=q, exp

_d

dsΦi(fs) Φi(fq)

, s=q, for f_s, f_q ∈Υ.

Proof. (i) This is an immediate consequence of Theorem 4 and Remark 6.

(ii) Since by (i) the function s 7→ Φi(fs), i = 1,2,3,4 is log-convex on J, that is, the function s7→log Φi(fs) is convex onJ. So, we get

(5.10) log Φi(fs)−log Φi(fq)

s−q ≤ log Φi(fu)−log Φi(fv) u−v

fors≤u, q≤v, s6=q, u6=v, and therefore conclude that µ_s,q(Φ_i,Υ)≤µ_u,v(Φ_i,Υ).

Cases s=q and u=v follow from (5.10) as limit cases.

Remark 7. Note that the results from the above theorem and corollaries still hold when two of the points z₀, . . . , z_l∈[α, β] coincide, say z₁ =z₀, for a family of differentiable functions fs such that the functions7→fs[z0, . . . , zl] is n-exponentially convex in the Jensen sense (exponentially convex in the Jensen sense, log-convex in the Jensen sense), and furthermore, they still hold when all (l+ 1) points coincide for a family of l differentiable functions with the same property. The proofs are obtained by (1.6) and suitable characterization of convexity.

6. APPLICATIONS TO STOLARSKY TYPE MEANS

In this section, we present several families of functions which fulfil the conditions of Theorem 4, Corollary 10, Corollary 11 and Remark 7. This enable us to construct a large family of functions which are exponentially convex. For a discussion related to this problem see [4].

Example 1. Consider a family of functions Ω₁ ={l_s:R→R:s∈R} defined by

l_s(x) = _e^sx

sⁿ, s6= 0,

xⁿ

n!, s= 0.

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We have ^d_dxⁿ^ln^s(x) =e^sx>0 which shows thatl_sisn-convex onRfor every s∈R and s7→ ^d_dxⁿ^l_n^s(x) is exponentially convex by definition. Using analogous arguing as in the proof of Theorem 4 we also have that s 7→ ls[z0, . . . , zn] is exponentially convex (and so exponentially convex in the Jensen sense).

Using Corollary 10 we conclude that s7→Φ_i(l_s), i= 1,2,3,4 are exponentially convex in the Jensen sense. It is easy to verify that this mapping is continuous (although mapping s7→ ls is not continuous fors = 0), so it is exponentially convex.

For this family of functions,µ_s,q(Φ_i,Ω₁), i= 1,2,3,4 from (5.9), becomes

µs,q(Φi,Ω1) =









 _Φ

i(ls) Φi(lq)

_s−q¹

, s6=q,

exp

Φi(id·ls) Φi(ls) − ⁿ_s

, s=q6= 0, exp

1 n+1

Φi(id·l₀) Φi(l0)

, s=q= 0.

Now, using (5.8) it is monotonous function in parameters sand q.

We observe here that _dnls

dnlqdxn dxn

_s−q¹

(lnx) =xso using Corollary 9 it follows that:

Ms,q(Φi,Ω1) = lnµs,q(Φi,Ω1), i= 1,2,3,4 satisfy

α ≤Ms,q(Φi,Ω1)≤β, i= 1,2,3,4.

If we set that the image of the function g is [α, β], we have that α= min

t∈[a,b]{g(t)} ≤Ms,q(Φi,Ω1)≤ max

t∈[a,b]{g(t)}=β, i= 1,2,3,4, which shows that M_s,q(Φ_i,Ω₁) are means of g(t) for i= 1,2,3,4. Because of above inequality, these means are also monotonic.

Example 2. Consider a family of functions Ω2 ={f_s : (0,∞)→R:s∈R} defined by

fs(x) =

( _xs

s(s−1)···(s−n+1), s /∈ {0,1, . . . , n−1},

x^jlnx

(−1)^n−1−jj!(n−1−j)!, s=j∈ {0,1, . . . , n−1}.

Here, ^d_dxⁿ^fn^s(x) = x^s−n = e^{(s−n) ln}^x > 0 which shows that f_s is n-convex forx >0 ands7→ ^d_dxⁿ^fn^s(x) is exponentially convex by definition. Arguing as in Example 1 we get that the mappingss7→Φ_i(f_s), i= 1,2,3,4 are exponentially

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convex. In this case we assume that [α, β]∈R⁺. Functions (5.9) now are equal to:

µs,q(Φi,Ω2) =









 Φi(fs)

Φi(fq)

_s−q¹

, s6=q,

exp

(−1)ⁿ⁻¹(n−1)!^Φ_Φⁱ^(f⁰^f^s⁾

i(fs) +Pn−1 k=0 1

k−s

, s=q /∈ {0,1,. . . ,n−1}, exp

(−1)ⁿ⁻¹(n−1)!^Φ_2Φⁱ^(f⁰^f^s⁾

i(fs) +Pn−1 k=0k6=s

1 k−s

, s=q∈ {0,1,. . . ,n−1}.

We observe that _dnfs

dxn dnfq dxn

_s−q¹

(x) = x, so if Φi(i = 1,2,3,4) are positive, then Corollary 9 yields that there exist someξ_i ∈[α, β], i= 1,2,3,4 such that

ξ^s−q_i = Φi(fs)

Φ_i(f_q), i= 1,2,3,4.

Since the functionξ 7→ξ^s−q is invertible for s6=q, we then have

(6.1) α≤

Φi(fs) Φi(fq)

_s−q¹

≤β, i= 1,2,3,4.

As in the previous example, if we set that the image of the function g is [α, β], in that case we have that

(6.2) α= min

t∈[a,b]g(t)≤

Φi(fs) Φ_i(f_q)

_s−q¹

≤ max

t∈[a,b]g(t) =β, i= 1,2,3,4, which shows thatµs,q(Φi,Ω2), i= 1,2,3,4 are means.

Now, we impose one additional parameter r. For r 6= 0 by substituting g7→g^r, s7→ _r^s and q7→ ^q_r in (6.2), we get the following:

(6.3) min

t∈[a,b](g(t))^r≤

Φ_i(g^r,·, f_s) Φi(g^r,·, f_q)

_s−q^r

≤ max

t∈[a,b](g(t))^r, i= 1,2,3,4.

We define new generalized mean as follows:

(6.4) µ_s,q;r(Φ_i,Ω₂) =







µ^s

r,^q_r(g^r,Φ_i,Ω₂)¹_r

, r6= 0, µ_s,q(lng,Φ_i,Ω₂), r= 0.

These new generalized means are also monotonic. If s, q, u, v∈ R, r 6= 0 such that s≤u, q≤v, then we have

µ_s,q;r(Φ_i,Ω₂)≤µ_u,v;r(Φ_i,Ω₂), i= 1,2,3,4.

The above result follows from the following inequality:

µ^s

r,_r^q(g^r,Φi,Ω2) =

Φi(g^r,·, f_s) Φi(g^r,·, f_q)

_s−q^r

≤

Φi(g^r,·, f_s) Φi(g^r,·, f_q)

_u−v^r

=µ^u

r,^v_r(g^r,Φi,Ω2),

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fors, q, u, v ∈R, r 6= 0, such that ^s_r ≤ ^u_r,^q_r ≤ ^v_r, and the fact that µ_s,q(Φ_i,Ω₂) for i = 1,2,3,4 are monotone in both parameters. For r = 0, we obtain the required result by taking the limitr →0.

Example 3. Consider a family of functions

Ω₃ ={h_s: (0,∞)→R:s∈(0,∞)}

defined by

hs(x) =

( s^−x

(−lns)ⁿ, s6= 1

xⁿ

n!, s= 1.

Since ^d_dxⁿ^hn^s(x) =s^−x is the Laplace transform of a non-negative function (see [15]) it is exponentially convex. Obviously hs are n-convex functions for everys >0. For this family of functions,µ_s,q(Φ_i,Ω₃), i= 1,2,3,4, in this case for [α, β]∈R⁺, from (5.9) becomes

µ_s,q(Φ_i,Ω₃) =











Φi(hs) Φi(hq)

_s−q¹

, s6=q,

exp

−^Φ_sΦⁱ^(id·h^s⁾

i(hs) −_s_lnⁿ_s

, s=q 6= 1, exp

−_n+1¹ ^Φ_Φⁱ^(id·h¹⁾

i(h1)

, s=q = 1.

These are monotone functions in parameters sand q by (5.8).

Example 4. Consider a family of functions

Ω₄={k_s: (0,∞)→R:s∈(0,∞)}

defined by

ks(x) = e^−x

√s

(−√ s)ⁿ. Since ^d_dxⁿ^kn^s(x) =e^−x

√sis the Laplace transform of a non-negative function (see [15]) it is exponentially convex. Obviously ks are n-convex functions for everys >0. For this family of functions,µs,q(Φi,Ω4), i= 1,2,3,4, in this case for [α, β]∈R⁺, from (5.9) becomes

µ_s,q(Φ_i,Ω₄) =







Φi(ks) Φi(kq)

_s−q¹

, s6=q,

exp

− ^Φⁱ^(id·k^s⁾

2√

sΦi(ks)−_2sⁿ

, s=q.

These are monotone functions in parameters sand q by (5.8).

Acknowledgments. The research of the authors was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grant 117-1170889- 0888.

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Received 1 March 2013 University of Zagreb, Faculty of Architecture, Fra Andrije Kacica Miosica 26,

10000 Zagreb, Croatia gorana.aras-gazic@arhitekt.hr

University of Zagreb, Faculty of Civil Engineering, Fra Andrije Kacica Miosica 26,

10000 Zagreb, Croatia vera@master.grad.hr University of Zagreb, Faculty of Textile Technology,

Prilaz baruna Filipovica 28a, 10000 Zagreb, Croatia

pecaric@element.hr University of Zagreb,

Faculty of Food Technology and Biotechnology, Pierottijeva 6,

10000 Zagreb, Croatia avukelic@pbf.hr