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Wong We discuss some variants of the Hermite-Hadamard inequality for convex functions on time scales.Some improvements and applications are also included.. Based on the known Δ delta and

Volume 2008, Article ID 287947, 24 pages

doi:10.1155/2008/287947

Research Article

Hermite-Hadamard Inequality on Time Scales

Cristian Dinu

Department of Mathematics, University of Craiova, 200585 Craiova, Romania

Correspondence should be addressed to Cristian Dinu,c.dinu@yahoo.com

Received 21 April 2008; Revised 30 June 2008; Accepted 15 August 2008

Recommended by Patricia J Y Wong

We discuss some variants of the Hermite-Hadamard inequality for convex functions on time scales.Some improvements and applications are also included

Copyrightq 2008 Cristian Dinu This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

1 Introduction

Recently, new developments of the theory and applications of dynamic derivatives ontime scales were made The study provides an unification and an extension of traditionaldifferential and difference equations and, in the same time, it is a unification of thediscrete theory with the continuous theory, from the scientific point of view Moreover, it

is a crucial tool in many computational and numerical applications Based on the known Δ delta and ∇ nabla dynamic derivatives, a combined dynamic derivative, so-called α diamond-α dynamic derivative, was introduced as a linear combination of Δ

well-and ∇ dynamic derivatives on time scales The diamond-α dynamic derivative reduces to

the Δ derivative for α  1 and to the ∇ derivative for α  0 On the other hand, it

represents a “weighted dynamic derivative” on any uniformly discrete time scale when

α  1/2 See 1 5 for the basic rules of calculus associated with the diamond-α dynamic

derivatives

The classical Hermite-Hadamard inequality gives us an estimate, from below and fromabove, of the mean value of a convex function The aim of this paper is to establish a fullanalogue of this inequality if we compute the mean value with the help of the delta, nabla,

and diamond-α integral.

The left-hand side of the Hermite-Hadamard inequality is a special case of the Jenseninequality

Recently, it has been proven a variant of diamond-α Jensen’s inequalitysee 6

Theorem 1.1 Let a, b ∈ T and c, d ∈ R If g ∈ Ca, bT, c, d, and f ∈ Cc, d, R is convex,

then

improvements and applications are presented in Section 4, together with an extension

of Hermite-Hadamard inequality for some symmetric functions A special case is that

of diamond-1/2 integral, which enables us to gain a number of consequences of our

Hermite-Hadamard type inequality; we present them inSection 5together with a discussionconcerning the case of convex-concave symmetric functions

We make the convention:

If σt > t, then t is said to be right-scattered, and if ρr < r, then r is said to be

left-scattered The points that are simultaneously right-scattered and left-scattered are called isolated If σ t  t, then t is said to be right dense, and if ρr  r, then r is said to be left dense The points that are simultaneously right-dense and left-dense are called dense.

The mappings μ, ν : T → 0, ∞ defined by

μ t : σt − t,

are called, respectively, the forward and backward graininess functions.

IfT has a right-scattered minimum m, then define T κ  T − {m}; otherwise T κ T If

T has a left-scattered maximum M, then define T κ  T − {M}; otherwise T κ T Finally, put

Tκ

κ Tκ∩ Tκ

Definition 2.1 For f : T → R and t ∈ T κ , one defines the delta derivative of f in t, to be the number denoted by fΔt when it exists, with the property that, for any ε > 0, there is a neighborhood U of t such that

f

ρ t− fs − f∇tρ t − s < ερ t − s, 2.5

for all s ∈ V

We say that f is delta di fferentiable on T κ , provided that fΔt exists for all t ∈ T κand

that f is nabla di fferentiable on T κ , provided that f∇t exists for all t ∈ T κ

is the backward difference operator

For a function f : T → R, we define f σ :T → R by f σ t  fσt, for all t ∈ T, i.e.,

f σ  f ◦ σ We also define f ρ:T → R by f ρ t  fρt, for all t ∈ T, i.e., f ρ  f ◦ ρ For all t∈ Tκ, we have the following properties

i If f is delta differentiable at t, then f is continuous at t.

ii If f is left continuous at t and t is right-scattered, then f is delta differentiable at t with fΔt  f σ t − ft/μt.

iii If t is right-dense, then f is delta differentiable at t, if and only if, the limit

lims →t ft − fs/t − s exists as a finite number In this case, fΔt 

lims →t ft − fs/t − s.

iv If f is delta differentiable at t, then f σ t  ft μtfΔt.

In the same manner, for all t∈ Tκwe have the following properties

i If f is nabla differentiable at t, then f is continuous at t.

ii If f is right continuous at t and t is left-scattered, then f is nabla differentiable at t with f∇t  ft − f ρ t/νt.

iii If t is left-dense, then f is nabla differentiable at t, if and only if, the limit

lims →t ft − fs/t − s exists as a finite number In this case, f∇t 

lims →t ft − fs/t − s.

iv If f is nabla differentiable at t, then f ρ t  ft − νtf∇t.

Definition 2.2 A function f : T → R is called rd-continuous, if it is continuous at all dense points inT and its left-sided limits are finite at all left-dense points in T One denotes

right-by Crdthe set of all rd-continuous functions

A function f :T → R is called ld-continuous, if it is continuous at all left-dense points

inT and its right-sided limits are finite at all right-dense points in T One denotes by Cldtheset of all ld-continuous functions

It is easy to remark that the set of continuous functions onT contains both Crdand Cld

f t, for all t ∈ T κ Then, one defines the delta integral byt

UsingTheorem 2.5,viii we get

i if ft ≤ gt for all t, thenb

A similar theorem works for the nabla antiderivativefor f, g ∈ Cld

Now, we give a brief introduction of the diamond-α dynamic derivative and of the diamond-α integral.

number denoted by fαt when it exists, with the property that, for any ε > 0, there is a neighborhood U of t such that for all s ∈ U

α

f

σ t− fs νts 1 − αf

ρ t− fs μts− fαtμtsνts< εμtsνts. 2.13

A function is called diamond-α di fferentiable on T κ if fαt exists for all t ∈ T κ If f :T →

R is differentiable on T in the sense of Δ and ∇, then f is diamond-α differentiable at t ∈ T κ,

and the diamond-α derivative fαt is given by

fαt  αfΔt 1 − αf∇t, 0 ≤ α ≤ 1. 2.14

As it was proved in5, Theorem 3.9, if f is diamond-α differentiable for 0 < α < 1

then f is both Δ and ∇ differentiable It is obvious that for α  1 the diamond-α derivative

reduces to the standardΔ derivative and for α  0 the diamond-α derivative reduces to the

standard∇ derivative For α ∈ 0, 1, it represents a “weighted dynamic derivative.”

We present here some operations with the diamond-α derivative For that, let f, g :

T → R be diamond-α differentiable at t ∈ T Then,

i f g : T → R is diamond-α differentiable at t ∈ T and

Let a, b ∈ T and f : T → R The diamond-α integral of f from a to b is defined by

b a

provided that f has a delta and a nabla integral on a, bT Obviously, each continuous

function has a diamond-α integral The combined derivativeαis not a dynamic derivative,since we do not have aαantiderivative See6, Example 2.1 In general,

t a

f s α s

α

/

but we still have some of the “classical” properties, as one can easily be deduced from

Theorem 2.7 If a, b, c ∈ T, β ∈ R, and f, g are continuous functions, then

3 The Hermite-Hadamard inequality

In this section, we present an extension of the Hermite-Hadamard inequality, for time scales.For that, we need to find the conditions fulfilled by the functions defined on a time scale Wewant to evaluateb

a t Δt andb

a t ∇t on such sets, because they provide us with a useful tool for

the proof of Hermite-Hadamard inequality We start with a few technical lemmas

Lemma 3.1 Let f : T → R be a continuous function and a, b ∈ T.

i If f is nondecreasing on T, then

time scaleT, remains between the values ofb

a f tΔt for T  {a, b} and for T  a, b.

all t∈ T First, we will show that by adding a point or an interval, the corresponding integralincreases

Let us suppose that we add a point c to T, where a < c < b If T1 T ∪ {c}, and c /∈ T

is an isolated point ofT1withb

a f tΔ1t the corresponding integral, then

Using the same methods, we show that if we “extract” an isolated point or an intervalfrom an initial times scale, the corresponding integral decreases And so, the value ofb

a f tΔt

is between its minimum value corresponding to T  {a, b} and its maximum value

corresponding to T  a, b, that is

f t dt  α T

b a

f tΔt 1− α T

b a

that is

b a

f t α T t b

a

Remark 3.2 The above proof covers the case of adding or extracting a set of the form

{l1, l1, , l n , , l }, where n ∈ N and l nn∈Nis a sequence of real numbers such that limn→∞

way for nonincreasing sequences, while the case of nonmonotone sequences can be split intwo subcases with monotone sequences Let ε > 0 Since lnn∈N is convergent, we have

N1 ∈ N such that |l − l n | < ε, for all n ≥ N1 Since f is rd-continuous and l is left dense, the

limit limn→∞f l n  exists and it is finite Denoting by b this limit, we have N2 ∈ N such that

|b − fl n | < ε, for all n ≥ N2and so f l n  ∈ b − ε, b ε, for all n ≥ N2 UsingTheorem 2.5iv,

we have, for N  max{N1, N2},

Taking the delta integral in the following inequality b − ε < fl n  < b ε and using

b − εl − l N



<

l l

f tΔt < b εl − l N



Taking the modulus in the last inequality and using|l − l N | < ε, we get

If ε goes to 0 and N goes to∞, then limN→∞b

a N f tΔt  0 Passing to the limit as

Now we will prove that if f : T → R is a linear function, i.e., ft  ut v then

b

a f tΔt andb

a f t∇t are symmetric with respect tob

ut v is the corresponding linear function, defined on the interval a, b.

f : a, b → R be the corresponding linear

the conclusion is clear IfT  a, b \ c, d, then

b a

and, obvious, if we choose C  d − c2/2 the conclusion is clear.

By repeating the same arguments several times, we can “extract” any number ofintervals froma, b and get the same conclusion.

If we “extract” an interval, but we “add” an isolated pointi.e., T  a, b \ c, e ∪

e, d  a, c ∪ {e} ∪ d, b, then

b a

and thus, for C  e − c2/2 d − e2/2, we get the conclusion.

For a general linear function, f t  ut v, we have

between a and b to be the function G :T × T → R by

It is clear that the two sums are equal, noticing that

and so Ga, b is finite.

In other words, the function G measures the square of distances between all scattered points between a and b and it depends on the “geometry” of the time scaleT

a t Δt andb

function In fact, we have

b a

a

t dt Ga, b,

b a

t Δt ≤ a b

b a

|t − s| α s t − a2 b − t2

where G is the function introduced in Definition 3.5

Proof UsingRemark 3.6, we have

b a

Now, we are able to give the Hermite-Hadamard inequality for the time scales

Theorem 3.9 Hermite-Hadamard inequality Let T be a time scale and a, b ∈ T Let f : a, b →

R be a continuous convex function Then,

f t α t ≤ fab − a f b − fa

b a

f t α t≤ b − x α

b − a f a

x α − a

and so we have proved the right-hand side

For the left-hand side, we useTheorem 1.1, by taking g : T → T, gs  s for all s ∈ T.

0≤ α ≤ λ, including for the nabla integral, if fb ≤ fa and for all λ ≤ α ≤ 1, including for the delta integral, if fb ≥ fa, where x λ is the λ-center of the time-scaled interval a, bT

Indeed, let us suppose that fb ≥ fa Then, by taking the diamond-α integral side

by side to the inequality ft ≤ fa fb − fa/b − at − a, we get

b a

f t α t ≤ fab − a f b − fa

b a

0 ≤ α ≤ λ, including the nabla integral, if f is nonincreasing and for all λ ≤ α ≤ 1, including the delta integral, if f is nondecreasing.

Indeed, let us suppose that f is nonincreasing Then, usingTheorem 1.1, let g :T → T,

g s  s for all s ∈ T We have

The same arguments are used to prove the case of f nondecreasing function.

Using the last remarks, we can give a more general Hermite-Hadamard inequality fortime scales

Theorem 3.12 a general version of Hermite-Hadamard inequality Let T be a time scale, α, λ ∈

0, 1 and a, b ∈ T Let f : a, b → R be a continuous convex function Then,

i if f is nondecreasing on a, bT, then, for all α ∈ 0, λ one has

f t α t≤ b − x λ

b − a f a

x λ − a

ii If f is nonincreasing on a, bT, then, for all α ∈ 0, λ one has the above inequality 3.47

and for all α ∈ 0, λ one has the above inequality 3.46.

Remark 3.13 In the above inequalities3.46 and 3.47, we have equalities if f is a constant function and α, λ ∈ 0, 1 or if f is a linear function and α  λ.

Theorem 3.14 a weighted version of Hermite-Hadamard inequality Let T be a time scale and

a, b ∈ T Let f : a, b → R be a continuous convex function and let w : T → R be a continuous

function such that w t ≥ 0 for all t ∈ T andb

f twt α t≤ b − x w,α

b − a f a

x w,α − a

and so we have proved the right-hand side

For the left-hand side, we useTheorem 1.2, by taking g : T → T, gs  s for all s ∈ T and h : T → R, ht  wt We have

Remark 3.15 If we consider concave functions instead of convex functions, the above

Hermite-Hadamard inequalities3.36, 3.46, 3.47, and 3.48 are reversed

4 The Hermite-Hadamard inequality forw, α-symmetric functions

In7, Florea and Niculescu proved the following theorem

Theorem 4.1 see 7, Theorem 3 Suppose that f : I → R verifies a symmetry condition (i.e.,

f x f2m − x  2fm for all x ∈ I ∩ −∞, m and is convex over the interval I ∩ −∞, m

and concave over the interval I ∩ m, ∞.

If a b/2 ≥ m and μ is a Hermite-Hadamard measure on each of the intervals a, 2m − a

and 2m − a, b, and is invariant with respect to the map Tx  2m − x on a, 2m − a, then

f xdμ ≥ b − x μ

b − a f a

x μ − a

If a b/2 ≤ m, then the inequalities GHH work in a reverse way, provided μ is a

Hermite-Hadamard measure on each of the intervals a, 2m − b and 2m − b, b, and is invariant with respect

to the map T x  2m − x on 2m − b, b.

We will give an extension of this theorem, for time scales, using functions notnecessarily symmetric in the usual sense For that, we need the following definition

0, 1 One says that a function f : a, b → R is w, α-symmetric on a, bTif the followingconditions are satisfied:

Notice that the function f should be continuous only on a, bT not on a, b An

example of such a function is the following

Example 4.3 Let T  {1} ∪ 3, 4, w : {1} ∪ 3, 4 → R , w1  1, wt  2 for all t ∈ 3, 4 and

We can provide also a continuous function on1, 4, such as

while conditionii can be restated as

4 1

f twt 1/2 t  5f

145

and it is easy to check that both are fulfilled

Now, we can state our theorem, that is a generalization ofTheorem 4.1

Theorem 4.4 Let T be a time scale, a ≤ c ≤ b ∈ T, w : T → R be a positive weight and α ∈ 0, 1.

f twt α t≤ b − x w,α

b − a f a

x w,α − a

If f is concave on p, b then the inequalities in 4.7 are reversed.

ii If the function f : a, b → R is w, α-symmetric on c, bTand concave on a, q then

one has4.7.

If f is convex on a, q, then the inequalities in 4.7 are reversed.

the left-hand side inequality in4.7 For that, we notice that

b a

We can provide also a continuous function on< i>1, 4, such as

while conditionii can be restated... 7

Lemma 3.1 Let f : T → R be a continuous function and a, b ∈ T.

i If f is nondecreasing on T, then

time scaleT,...

In this section, we present an extension of the Hermite-Hadamard inequality, for time scales.For that, we need to find the conditions fulfilled by the functions defined on a time scale Wewant