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Peri´c3 1 Abdus Salam School of Mathematical Sciences, GC University, Lahore 54000, Pakistan 2 Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia 3 Faculty of Foo

Volume 2009, Article ID 128486, 14 pages

doi:10.1155/2009/128486

Research Article

Some New Results Related to Favard’s Inequality

Naveed Latif,1 J Pe ˇcari ´c,1, 2 and I Peri´c3

1 Abdus Salam School of Mathematical Sciences, GC University, Lahore 54000, Pakistan

2 Faculty of Textile Technology, University of Zagreb, 10000 Zagreb, Croatia

3 Faculty of Food Technology and Biotechnology, University of Zagreb, 10000 Zagreb, Croatia

Correspondence should be addressed to Naveed Latif,sincerehumtum@yahoo.com

Received 31 July 2008; Revised 17 January 2009; Accepted 5 February 2009

Recommended by A Laforgia

Log-convexity of Favard’s difference is proved, and Drescher’s and Lyapunov’s type inequalities for this difference are deduced The weighted case is also considered Related Cauchy type means are defined, and some basic properties are given

Copyrightq 2009 Naveed Latif et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction and Preliminaries

Let f and p be two positive measurable real valued functions defined on a, b ⊆ R with

b

a p xdx  1 From theory of convex means cf 1,2, the well-known Jensen’s inequality

gives that for t < 0 or t > 1,

b

a

p xf t xdx ≥

b

a

p xfxdx

t

and reverse inequality holds for 0 < t < 1 In3, Simic considered the difference

D s  D s a, b, f, p 

b

a

p xf s xdx −

b

a

p xfxdx

s

He has given the following

Theorem 1.1 Let f and p be nonnegative and integrable functions on a, b, withb

a p xdx  1,

then for 0 < r < s < t, r, s, t /  1, one has



D s

s s − 1

t −r

≤



D r

r r − 1

t −s

D t

t t − 1

s −r

Remark 1.2 For an extension ofTheorem 1.1see3

Let us write the well-known Favard’s inequality

Theorem 1.3 Let f be a concave nonnegative function on a, b ⊂ R If q > 1, then

2q

q 1

 1

b − a

b

a

f xdx

q

b − a

b

a

If 0 < q < 1, the reverse inequality holds in1.4.

Note that1.4 is a reversion of 1.1 in the case when px  1/b − a.

Let us note that Theorem 1.3 can be obtained from the following result and also obtained by Favardcf 4, page 212

Theorem 1.4 Let f be a nonnegative continuous concave function on a, b, not identically zero, and

let φ be a convex function on 0, 2  f , where



f  1

b − a

b

a

Then

1

2 f

2 f

0

φ ydy ≥ 1

b − a

b

a

φ

Karlin and Studdencf 5, page 412 gave a more general inequality as follows

Theorem 1.5 Let f be a nonnegative continuous concave function on a, b, not identically zero;  f

is defined in1.5, and let φ be a convex function on c, 2  f − c, where c satisfies 0 < c ≤ fmin(where

fminis the minimum of f) Then

1

2 f − 2c

2 f −c

c

φ ydy ≥ 1

b − a

b

a

φ

For φy  y p , p > 1, we can get the following fromTheorem 1.5

Theorem 1.6 Let f be continuous concave function such that 0 < c ≤ fmin;  f is defined in1.5 If

p > 1, then

1



2 f − 2cp  1



2 f − cp1− c p1 ≥ 1

b − a

b

a

If 0 < p < 1, the reverse inequality holds in1.8.

In this paper, we give a related results to1.3 for Favard’s inequality 1.4 and 1.8

We need the following definitions and lemmas

Definition 1.7 It is said that a positive function f is log-convex in the Jensen sense on some

interval I⊆ R if

f sft ≥ f2



s  t

2



1.9

holds for every s, t ∈ I.

We quote here another useful lemma from log-convexity theorycf 3

Lemma 1.8 A positive function f is log-convex in the Jensen sense on an interval I ⊆ R if and only

if the relation

u2f s  2uwf



s  t

2



holds for each real u, w and s, t ∈ I.

Throughout the paper, we will frequently use the following family of convex functions

on0, ∞:

ϕ s x 

⎧

⎪

⎪

⎪

⎪

x s

s s − 1 , s /  0, 1;

− log x, s 0;

x log x, s  1.

1.11

The following lemma is equivalent to the definition of convex functionsee 4, page 2

Lemma 1.9 If φ is convex on an interval I ⊆ R, then

φ

s1

s3− s2



 φs2

s1− s3



 φs3

s2− s1



holds for every s1< s2< s3, s1, s2, s3∈ I.

Now, we will give our main results

2 Favard’s Inequality

In the following theorem, we construct another interesting family of functions satisfying the Lyapunov inequality The proof is motivated by3

Theorem 2.1 Let f be a positive continuous concave function on a, b;  f is defined in1.5, and

Δs f :

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

1

s s − 1



2s

s 1

 1

b − a

b

a

f xdx

s

− 1

b − a

b

a

f s xdx



, s /  0, 1;

1− log 2 − log f 1

b − a

b

a

log 2 f f log  f−1

2f− 1

b − a

b

a

f x log fxdx, s  1.

2.1

ThenΔs f is log-convex for s ≥ 0, and the following inequality holds for 0 ≤ r < s < t < ∞:

Δt −r

s f ≤ Δ t −s

r fΔ s −r

Proof Let us consider the function defined by

φ x  u2ϕ s x  2uwϕ r x  w2ϕ t x, 2.3

where r  s  t/2, ϕ sis defined by1.11, and u, w ∈ R We have

φx  u2x s−2 2uwx r−2 w2x t−2

ux s/2−1 wx t/2−12

Therefore, φx is convex for x > 0 UsingTheorem 1.4,

1

2 f

2 f

0



u2ϕ s y  2uwϕ r y  w2ϕ t ydy

b − a

b

a



u2ϕ s



f x 2uwϕ r



f x w2ϕ t



f xdx,

2.5

or equivalently

u2

 1

2 f

2 f

0

ϕ s ydy ư 1

b ư a

b

a

ϕ s



f xdx



 2uw

 1

2 f

2 f

0

ϕ r ydy ư 1

b ư a

b

a

ϕ r

f xdx



 w2

 1

2 f

2 f

0

ϕ t ydy ư 1

b ư a

b

a

ϕ t



f xdx



≥ 0.

2.6

Since

Δs f  1

2 f

2 f

0

ϕ s ydy ư 1

b ư a

b

a

ϕ s

we have

u2Δs f  2uwΔ r f  w2Δt f ≥ 0. 2.8

ByLemma 1.8, we have

Δs fΔ t f ≥ Δ2

r f  Δ2

that is,Δs f is log-convex in the Jensen sense for s ≥ 0.

Note thatΔs f is continuous for s ≥ 0 since

lim

s→ 0Δs f  Δ0f and lim

s→ 1Δs f  Δ1f. 2.10

This impliesΔs f is continuous; therefore, it is log-convex.

SinceΔs s f is convex, byLemma 1.9for 0≤ r <

s < t < ∞ and taking φs  log Δ s f, we get

logΔt ưr

s f ≤ log Δ t ưs

r f  log Δ s ưr

which is equivalent to2.2

Theorem 2.2 Let f, Δ s f be defined as in Theorem 2.1 , and let t, s, u, v be nonnegative real numbers such that s ≤ u, t ≤ v, s / t, and u / v Then

 Δt f

Δs f

1/tưs

≤Δv f

Δu f

1/vưu

Proof An equivalent form of1.12 is

ϕ

x2

ư ϕx1

x2ư x1 ≤ ϕ



y2

ư ϕy1

y2ư y1

where x1≤ y1, x2≤ y2, x1/  x2, and y1/  y2 Since byTheorem 2.1,Δs f is log-convex, we can

set in2.13: ϕx  log Δ x f, x1 s, x2 t, y1 u, and y2 v We get

logΔt f ư log Δ s f

t ư s ≤

logΔv f ư log Δ u f

from which2.12 trivially follows

The following extensions of Theorems2.1and2.2can be deduced in the same way fromTheorem 1.5

Theorem 2.3 Let f be a continuous concave function on a, b such that 0 < c ≤ fmin;  f is defined

in1.5, and

Δs f :

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

1

s s ư 1

 

2 f ư cs1



2 f ư 2cs  1ư

c s1



2 f ư 2cs  1ư

1

b ư a

b

a

f s xdx



, s /  0, 1;

1

2 f ư 2c



2 f  c log c ư 2c ư2 f ư clog

2 f ư c 1

b ư a

b

a log f xdx, s  0;

1 22 f ư 2c



2 f ư c2

log

2 f ư cư 2 f2 2c  f ư c2log c  2c

ư 1

b ư a

b

a

2.15

Then Δs f is log-convex for s ≥ 0, and the following inequality holds for 0 ≤ r < s < t < ∞:

Δt ưr

s f ≤ Δ t ưs

r f Δ s ưr

Theorem 2.4 Let f, Δ s f be defined as in Theorem 2.3 , and let t, s, u, v be nonnegative real numbers such that s ≤ u, t ≤ v, s / t, and u / v, one has

 Δt f

Δs f

1/tưs

≤ Δv f

Δu f

1/vưu

3 Weighted Favard’s Inequality

The weighted version of Favard’s inequality was obtained by Maligranda et al in6

Theorem 3.1 1 Let f be a positive increasing concave function on a, b Assume that φ is a convex

function on 0, ∞, where



f i b − a

b

a f twtdt

2b

Then

1

b − a

b

a

φ

f tw tdt ≤

1

0

φ

2r  f i

w

a 1 − r  brdr. 3.2

If f is an increasing convex function on a, b and fa  0, then the reverse inequality in 3.2 holds.

2 Let f be a positive decreasing concave function on a, b Assume that φ is a convex

function on 0, ∞, where



f d b − a

b

a f twtdt

2b

Then

1

b − a

b

a

φ

f tw tdt ≤

1

0

φ

2r  f d



w

ar  b1 − rdr. 3.4

If f is a decreasing convex function on a, b and fb  0, then the reverse inequality in 3.4 holds.

Theorem 3.2 1 Let f be a positive increasing concave function on a, b;  f i is defined in3.1, and

Πs f :

1

0

ϕ s



2r  f i



w

a 1 − r  brdr− 1

b − a

b

a

ϕ s



f tw tdt. 3.5

ThenΠs f is log-convex on 0, ∞, and the following inequality holds for 0 ≤ r < s < t < ∞:

Πt −r

s f ≤ Π t −s

r fΠ s −r

2 Let f be an increasing convex function on a, b, fa  0, Πs f : −Π s f Then Πs f

is log-convex on 0, ∞, and the following inequality holds for 0 ≤ r < s < t < ∞:



Πt −r

s f ≤ Πt −s

r f Πs −r

Proof As in the proof ofTheorem 2.1, we useTheorem 3.11 instead ofTheorem 1.4

Theorem 3.3 1 Let f and Π s f be defined as in Theorem 3.2 (1), and let t, s, u, v ≥ 0 be such that

s ≤ u, t ≤ v, s / t, and u / v Then

 Πt f

Πs f

1/tưs

≤Πv f

Πu f

1/vưu

2 Let f and Πs f be defined as in Theorem 3.2 (2), and let t, s, u, v ≥ 0 be such that s ≤ u,

t ≤ v, s / t, and u / v Then,

 Πt f



Πs f

1/tưs

≤ Πv f



Πu f

1/vưu

Proof Similar to the proof ofTheorem 2.2

Theorem 3.4 1 Let f be a positive decreasing concave function on a, b;  f d is defined as in3.3,

and

Γs f :

1

0

ϕ s

2r  f d

w

ar  b1 ư rdrư 1

b ư a

b

a

ϕ s

f tw tdt. 3.10

ThenΓs f is log-convex on 0, ∞, and the following inequality holds for 0 ≤ r < s < t < ∞:

Γt ưr

s f ≤ Γ t ưs

r fΓ s ưr

2 Let f be a decreasing convex function on a, b, fb  0, Γ s f : ưΓ s f Then Γ s is log-convex on 0, ∞, and the following inequality holds for 0 ≤ r < s < t < ∞:

Γt ưr

s f ≤ Γ t ưs

r fΓ s ưr

Proof As in the proof ofTheorem 2.1, we useTheorem 3.12 instead ofTheorem 1.4

Theorem 3.5 1 Let f and Γ s f be defined as in Theorem 3.4 (1), and let t, s, u, v ≥ 0 be such that

s ≤ u, t ≤ v, s / t, and u / v Then

 Γt f

Γs f

1/tưs

≤Γv f

Γu f

1/vưu

2 Let f and Γ s f be defined as in Theorem 3.4 (2), and let t, s, u, v ≥ 0 be such that s ≤ u,

t ≤ v, s / t, and u / v Then

 Γt f

Γs f

1/tưs

≤ Γv f

Γu f

1/vưu

Proof Similar to the proof ofTheorem 2.2.

Remark 3.6 Let w ≡ 1 If f is a positive concave function on a, b, then the decreasing rearrangement f∗is concave ona, b By applyingTheorem 3.4to f∗, we obtain thatΓs f∗ is

log-convex Equimeasurability of f with f∗givesΓs f  Γ s f∗ and we see thatTheorem 3.4

is equivalent toTheorem 2.1

Remark 3.7 Let w t  t α with α >−1 ThenTheorem 3.2gives that if f is a positive increasing

concave function on0, 1, then Π α

sis log-convex, and

Πα

s  1

s s − 1



α  2 s

α  s  1

1

0

f tt α dt

s

−

1

0

f s tt α dt



, s /  0, 1,

Πα

0 

1

0

log ftt α dt− log



α  21

0f tt α dt

α 1 

1

α  12,

Πα

1  log



α  2

1

0

f tt α dt

1

0

f tt α dt−

1

0f tt α dt

α 2 −

1

0

f t log ftt α dt,

3.15

with zero for the function f t  t.

If f is a positive decreasing concave function on 0, 1, thenTheorem 3.4gives thatΓα

s

is log-convex, and

Γα

s  1

s s − 1



α  1 s α  2 s B s  1, α  1

1

0

f tt α dt

s

−

1

0

f s tt α dt



, s /  0, 1,

Γα

0 

1

0

log f tt α dt 1

α 1H α  1 −

1

α 1log



α  1α  2

1

0

f tt α dt



,

Γα

1 1− Hα  2  logα  1α  21

0

f tt α dt



1

0

f tt α dt log

1

0

f tt α dt−

1

0

f t log ftt α dt,

3.16

with zero for the function ft  1 − t, where B·, · is the beta function, and Hα is the harmonic number defined for α > −1 with Hα  ψα  1  γ, where ψ is the digamma function and γ  0.577215 the Euler constant.

4 Cauchy Means

Let us note that2.12, 2.17, 3.8, 3.9, 3.13, and 3.14 have the form of some known inequalities between means e.g., Stolarsky means, Gini means, etc. Here we will prove that expressions on both sides of3.8 are also means The proofs in the remaining cases are analogous

Lemma 4.1 Let h ∈ C2I, I interval in R, be such that his bounded, that is, m ≤ h ≤ M Then

the functions φ1, φ2defined by

φ1t  M

2 t

2− ht, φ2t  ht − m

2t

are convex functions.

Theorem 4.2 Let w be a nonnegative integrable function on a, b withb

a w xdx  1 Let f be a

positive increasing concave function on a, b, h ∈ C20, 2 f i  Then there exists ξ ∈ 0, 2 f i , such

that

1

0

h

2r  f i



w

a 1 − r  brdr− 1

b − a

b

a

h

f tw tdt

 hξ

2

1 0

2r  f i2w

a 1 − r  brdr− 1

b − a

b

a

f2twtdt



.

4.2

Proof Set m minx ∈0,2 f

ihx, M  max x ∈0,2 f

ihx Applying 3.2 for φ1and φ2defined

inLemma 4.1, we have

1

0

φ12r  f i wa 1 − r  brdr ≥ 1

b − a

b

a

φ1



f tw tdt,

1

0

φ22r  f i wa 1 − r  brdr ≥ 1

b − a

b

a

φ2



f tw tdt,

4.3

that is,

M

2

1 0

2r  f i2w

a 1 − r  brdr− 1

b − a

b

a

f2twtdt



≥

1

0

h 2r  f i wa 1 − r  brdr− 1

b − a

b

a

h

f tw tdt,

4.4

1

0

h

2r  f i



w

a 1 − r  brdr− 1

b − a

b

a

h

f tw tdt

≥ m 2

1

0

2r  f i

2

w

a 1 − r  brdr− 1

b − a

b

a

f2twtdt



.

4.5

By combining4.4 and 4.5, 4.2 follows from continuity of h

Theorem 4.3 Let f be a positive increasing concave nonlinear function on a, b, and let w be a

nonnegative integrable function on a, b withb

a w xdx  1 If h1, h2 ∈ C20, 2 f i , then there

exists ξ ∈ 0, 2 f i  such that

h1ξ

h2ξ 

1

0h1

2r  f i

w

a 1 − r  brdr − 1/b − ab

a h1

f tw tdt

1

0h2

2r  f i

w

a 1 − r  brdr − 1/b − ab

a h2

f tw tdt , 4.6

provided that h2x / 0 for every x ∈ 0, 2 f i .

Proof Define the functional Φ : C2

0, 2  f i

→ R with

Φh 

1

0

h

2r  f i

w

a 1 − r  brdr− 1

b − a

b

a

h

f tw tdt, 4.7

and set h0  Φh2h1− Φh1h2 Obviously,Φh0  0 UsingTheorem 4.2, there exists ξ ∈

0, 2 f i such that

Φh0  h0ξ

2

1 0



2r  f i

2

w

a 1 − r  brdr− 1

b − a

b

a

f2twtdt



We give a proof that the expression in square brackets in4.8 is nonzero actually strictly positive by inequality3.2 for nonlinear function f Suppose that the expression in square

brackets in4.8 is equal to zero, which is by simple rearrangements equivalent to equality

b

a

t − a2w tdt 

b

a

g2twtdt, where gt 

b

a t − awtdt

b

a f twtdt f t. 4.9

Since g is positive concave function, it is easy to see that gt/t − a is decreasing function

ona, b see 6, thus

1 b 1

a t − awtdt

b

a

g twtdt ≤ x 1

a t − awtdt

x

a

g twtdt, x ∈ a, b, 4.10

sox

a t − awtdt ≤x

a g twtdt for every x ∈ a, b Set

F x 

x

a



Obviously, Fx ≤ 0, Fa  Fb  0 By 4.9, obvious estimations and integration by parts,

we have

0

b

a



t − a2− g2tw tdt ≥

b

a 2gtt − a − gtw tdt



b

a 2gtdFt  −

b

a

F td2gt≥ 0.

4.12

This impliesb

a t − a2− g2twtdt b

a 2gtt − a − gtwtdt, which is equivalent to

b

a t − a − gt2w tdt  0 This gives that g is a linear function, which obviously implies that f is a linear function.

Since the function f is nonlinear, the expression in square brackets in4.8 is strictly

positive which implies that h0ξ  0, and this gives 4.6 Notice thatTheorem 4.2for h  h2

implies that the denominator of the right-hand side of4.6 is nonzero

Corollary 4.4 Let w be a nonnegative integrable function withb

a w xdx  1 If f is a positive

increasing concave nonlinear function on a, b, then for 0 < s / t / 1 / s there exists ξ ∈ 0, 2 f i  such

that

ξ t −s s s − 1

t t − 1

1

0



2r  f it

w

a 1 − r  brdr − 1/b − ab

a f t rwrdr

1

02r  f is w

a 1 − r  brdr − 1/b − ab

a f s rwrdr . 4.13

Proof Set h1x  x t and h2x  x s , t /  s / 0, 1 in 4.6, then we get 4.13

Remark 4.5 Since the function ξ t −sis invertible, then from4.13 we have

0 <

⎛

⎝ss − 1

t t − 1

1

0



2r  f it

w

a 1 − r  brdr − 1/b − ab

a f t rwrdr

1

0



2r  f i

s

w

a 1 − r  brdr − 1/b − ab

a f s rwrdr

⎞

⎠

1/t−s

≤ 2 f i

4.14

In fact, similar result can also be given for 4.6 Namely, suppose that h

1/h2 has inverse function Then from4.6, we have

ξ

h

1

h2

−1⎛

⎝

1

0h1

2r  f i

w

a 1 − r  brdr − 1/b − ab

a h1

f tw tdt

1

0h2

2r  f i

w

a 1 − r  brdr − 1/b − ab

a h2

f tw tdt

⎞

⎠ 4.15

So, we have that the expression on the right-hand side of4.15 is also a mean

By the inequality4.14, we can consider

M t,s f; w 

⎛

⎝ss − 1

t t − 1

1

0



2r  f it

w

a 1 − r  brdr − 1/b − ab

a f t rwrdr

1

0



2r  f is

w

a 1 − r  brdr − 1/b − ab

a f s rwrdr

⎞

⎠

1/t−s

4.16