A Concept of Synchronicity Associated with Convex Functions in Linear Spaces and Applications

A Concept of Synchronicity Associated with Convex Functions in Linear Spaces and Applications

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CONVEX FUNCTIONS IN LINEAR SPACES AND APPLICATIONS

S.S. DRAGOMIR

Abstract. A concept of synchronicity associated with convex functions in linear spaces and a µCebyšev type inequality are given. Applications for norms, semi-inner products and for convex functions of several real variables are also given.

1. Introduction

The Jensen inequality for convex functions plays a crucial role in the Theory of Inequalities due to the fact that other inequalities such as that arithmetic mean- geometric mean inequality, Hölder and Minkowski inequalities, Ky Fan’s inequality etc. can be obtained as particular cases of it.

LetC be a convex subset of the linear spaceX andf a convex function onC:If p= (p1; : : : ; pn)is a probability sequence andx= (x1; : : : ; xn)2Cⁿ;then

(1.1) f

Xn i=1

p_ix_i

! _n X

i=1

p_if(x_i); is well known in the literature as Jensen’s inequality.

For re…nements of the Jesen inequality and applications related to Ky Fan’s inequality, the arithmetic mean-geometric mean inequality, the generalised triangle inequality, thef-divergence measures etc. see [1]-[7].

Assume thatf :X !Ris aconvex function on the real linear space X. Since for any vectors x; y2X the functiongx;y :R!R; gx;y(t) :=f(x+ty)is convex it follows that the following limits exist

r+( )f(x) (y) := lim

t!0+( )

f(x+ty) f(x) t

and they are called theright(left) Gâteaux derivatives of the functionf in the point xover the directiony:

It is obvious that for anyt >0> swe have (1.2) f(x+ty) f(x)

t r+f(x) (y) = inf

t>0

f(x+ty) f(x) t

sup

s<0

f(x+sy) f(x)

s =r f(x) (y) f(x+sy) f(x) s

Date: 01 July, 2009.

Key words and phrases. Convex functions, Gâteaux derivatives, Norms, Semi-inner products, Synchronous sequences, µCebyšev inequality.

1

for anyx; y2X and, in particular,

(1.3) r f(u) (u v) f(u) f(v) r⁺f(v) (u v)

for any u; v2X:We call this the gradient inequality for the convex functionf: It will be used frequently in the sequel in order to obtain various results related to Jensen’s inequality.

The following properties are also of importance:

(1.4) r⁺f(x) ( y) = r f(x) (y);

and

(1.5) r+( )f(x) ( y) = r+( )f(x) (y) for anyx; y2X and 0:

The right Gâteaux derivative issubadditive while the left one is superadditive, i.e.,

(1.6) r+f(x) (y+z) r+f(x) (y) +r+f(x) (z) and

(1.7) r f(x) (y+z) r f(x) (y) +r f(x) (z) for anyx; y; z2X .

Some natural examples can be provided by the use of normed spaces.

Assume that(X;k k)is a real normed linear space. The function f :X !R, f(x) := ¹₂kxk² is a convex function which generates the superior and the inferior semi-inner products

hy; xis(i):= lim

t!0+( )

kx+tyk² kxk²

t :

For a comprehensive study of the properties of these mappings in the Geometry of Banach Spaces see the monograph [6].

For the convex functionfp:X!R,fp(x) :=kxk^p withp >1;we have r+( )f_p(x) (y) =

8<

:

pkxk^{p 2}hy; xis(i) ifx6= 0

0 ifx= 0

for anyy2X:

Ifp= 1;then we have

r+( )f₁(x) (y) = 8<

:

kxk ¹hy; xis(i) ifx6= 0 + ( )kyk ifx= 0 for anyy2X:

This class of functions will be used to illustrate the inequalities obtained in the general case of convex functions de…ned on an entire linear space.

In the recent paper [9] the following re…nement and reverse of the Jensen in- equality in terms of the gradient have been obtained:

Theorem 1. Letf :X !Rbe a convex function de…ned on a linear spaceX:Then for any n-tuple of vectors x = (x₁; :::; x_n) 2 Xⁿ and any probability distribution p= (p₁; :::; p_n)2Pⁿ we have the inequality

(1.8) Xn k=1

pkr f(xk) (xk) Xn k=1

pkr f(xk) Xn i=1

pixi

!

Xn i=1

p_if(x_i) f Xn i=1

p_ix_i

!

Xn k=1

p_kr+f Xn i=1

p_ix_i

!

(x_k) r+f Xn i=1

p_ix_i

! _n X

i=1

p_ix_i

! 0:

A particular case of interest is forf(x) =kxk^pwhere(X;k k)is a normed linear space. Then for any p 1; for any n-tuple of vectorsx = (x1; :::; xn) 2Xⁿ and any probability distributionp= (p1; :::; pn)2Pⁿ with Pn

i=1pixi 6= 0we have the inequality

(1.9) Xn i=1

pikxik^p Xn i=1

pixi p

p Xn i=1

pixi p 22

4 Xn k=1

pk

* xk;

Xn j=1

pjxj

+

s

Xn i=1

pixi 23

5 0:

Ifp 2the inequality holds for anyn-tuple of vectors and probability distribution.

Also, for anyp 1;for anyn-tuple of vectors x= (x₁; :::; x_n)2Xⁿn f(0; :::;0)g and any probability distributionp= (p1; :::; pn)2Pⁿ we have the inequality (1.10) p

" _n X

k=1

p_kkx_kk^p Xn k=1

p_kkx_kk^{p 2}

* _n X

i=1

p_ix_i; x_k +

i

#

Xn i=1

pikxik^p Xn i=1

pixi p

: Motivated by the above results we introduce in this paper a class of sequences associated with convex functions in linear spaces and establish a µCebyšev type inequality and some new inequalities for convex functions. Applications for norms, semi-inner products and for convex functions of several real variables are also given.

2. rf Synchronicity

Considerf :X !Ra convex function on the linear spaceX:We also assume that u= (u₁; : : : ; u_n)and v= (v₁; : : : ; v_n) are twon tuples of vectors withu_i; v_i 2X;

i2 f1; : : : ; ng:

De…nition 1. We say thatv isrf synchronous with uif (2.1) r f(uk) (vk vj) r⁺f(uj) (vk vj)

for any k; j 2 f1; : : : ; ng: If the inequality is reversed in (2.1) for each k; j 2 f1; : : : ; ng; then we say thatv isrf asynchronous withu:

We notice that in general, ifv isrf asynchronous withu;this does not imply thatuisrf synchronous withv:

As general examples of such convex functions we can consider f(x) = kxk^p; p 1 where(X;k k)is a normed linear space. Since (see Introduction)

r f(x) (y) =pkxk^{p 2}hy; xii forx; y2X withx6= 0;

r f(0) (y) = 8<

:

0 if p >1 kyk if p= 1

; fory2X;

r⁺f(x) (y) =pkxk^{p 2}hy; xis forx; y2X withx6= 0;

r⁺f(0) (y) = 8<

:

0 if p >1 kyk if p= 1

; fory2X;

whereh; isis thesuperior semi-inner product andh; iiis theinferior semi-inner product, then we can de…ne the following concepts of synchronicity for the two n tuples of vectorsu= (u1; : : : ; un)andv= (v1; : : : ; vn):

Letp 1andu; v 2Xⁿ be as above. We say thatvis p r synchronous with uif

(2.2) kukk^{p 2}hvk vj; ukii kujk^{p 2}hvk vj; ujis

for anyk; j2 f1; : : : ; ng:

We observe that forp2[1;2)we should assume thatuk 6= 0fork2 f1; : : : ; ng: Forp= 2;the equation (2.2) reduces to

(2.3) hvk vj; ukii hvk vj; ujis for anyk; j2 f1; : : : ; ng:

If(X;k k)is a smooth normed space, meaning that the norm is Gâteaux di¤er- entiable on anyx2X; x6= 0 and if we denote by[; ]the semi-inner product gen- erating the normk k(see [6, pp. 19-20]), then the fact thatvisp r synchronous withumeans that

(2.4) kukk^{p 2}[vk vj; uk] kujk^{p 2}[vk vj; uj] for anyk; j2 f1; : : : ; ng:Forp= 2;we have

(2.5) [vk vj; uk] [vk vj; uj] for anyk; j2 f1; : : : ; ng:

Moreover, if the norm k k is generated by an inner product h; i; then v is p r synchronous withumeans that

(2.6) D

vk vj;kukk^{p 2}uk kujk^{p 2}vj

E

0 for anyk; j2 f1; : : : ; ng while forp= 2;it reduces to

(2.7) hv_k v_j; u_k u_ji 0 for anyk; j2 f1; : : : ; ng;

which is the concept of synchronous sequences in inner product spaces that has been introduced in [13]. For some inequalities for synchronous sequences in inner product spaces, see [13] and [14].

As some natural examples of synchronous sequences in inner product spaces, we can consider the sequences fx_igi2N and fAx_igi2N whereA :X !X is a positive linear operator onX;i.e., hAx; xi 0 for anyx2X:

For a convex functionf :X !Rwe de…ner~f( ) ( )as (2.8) r~f(x) (y) :=1

2[r f(x) (y) +r⁺f(x) (y)]; wherex; y2X:

We observe that forf as above, we havethe homogeneity property: (2.9) r~f(x) ( y) = r~f(x) (y) for anyx; y2X;

and any 2R.

The following inequality forr f synchronous sequences holds.

Theorem 2. Assume thatv isr f synchronous withuandp= (p₁; : : : ; p_n)is a probability distribution. Then

(2.10)

Xn i=1

pir~f(ui) (vi) Xn i;j=1

pipjr~f(ui) (vj):

Proof. Sincer⁺( ) ( )is subadditive in the second variable, then we have (2.11) r+f(u_i) (v_i v_j) r+f(u_i) (v_i) r+f(u_i) (v_j) for anyi; j2 f1; : : : ; ng:

Also, by the fact thatr ( ) ( )is superadditive in the second variable, we have that

(2.12) r f(ui) (vi) r f(ui) (vj) r f(ui) (vi vj) for alli; j2 f1; : : : ; ng:

Now, by (2.11), (2.12) and by the de…nition ofr f synchronicity, we deduce that

r f(ui) (vi) r f(ui) (vj) r⁺f(ui) (vi) r⁺f(ui) (vj); which is equivalent with

(2.13) r f(ui) (vi) +r⁺f(ui) (vj) r⁺f(ui) (vi) +r f(ui) (vj) for alli; j2 f1; : : : ; ng:

Therefore, by multiplying (2.13) withpipj 0and summing overi andj from 1to n;we get

(2.14) Xn i=1

pir f(ui) (vi) + Xn j=1

pjr⁺f(ui) (vj) Xn

i;j=1

pipjr⁺f(ui) (vi) + Xn i;j=1

pipjr f(ui) (vj): Now, observe that

Xn j=1

p_jr+f(u_j) (v_j) = Xn i=1

p_ir+f(u_i) (v_i)

and Xn

i;j=1

p_ip_jr+f(u_j) (v_i) = Xn i;j=1

p_ip_jr+f(u_i) (v_j); which, by (2.14) divided by 2, provides the desired result (2.10).

Corollary 1. With the assumptions of Theorem 2 and, if in addition r~f(u_i) ( ) is additive for any i2 f1; : : : ; ng; then we have

(2.15)

Xn i=1

pir~f(ui) (vi) Xn i;j=1

pipjr~f(ui) 0

@ Xn j=1

pjuj

1 A:

Remark 1. If f is Gâteaux di¤ erentiable at the points ui; i 2 f1; : : : ; ng; then r~f(ui) ( ) = rf(ui) ( ) and is therefore linear on X: With this assumption, the inequality (2.15) holds withrinstead ofr~:Moreover, there are examples of convex functions de…ned on linear spaces for whichr~f(x) ( )is linear for anyx6= 0without the function f being Gâteaux di¤ erentiable at that point (see[6, pp. 44-45]).

Following [15], we consider theg semi-inner product h; ig:X X !Rde…ned by

hy; xig:= 1

2[hy; xii+hy; xis]; x; y2X:

Utilising this notation, we have the following conditional inequality for semi-inner products and norms in normed linear spaces.

Proposition 1. Let (X;k k) be a normed linear space, u = (u₁; : : : ; u_n), v = (v₁; : : : ; v_n)2Xⁿ andp 1: If

(2.16) ku_kk^{p 2}hv_k v_j; u_ki_i ku_jk^{p 2}hv_k v_j; u_ji_s for any k; j2 f1; : : : ; ng;then

(2.17)

Xn k=1

pkkukk^{p 2}hvk; ukig

Xn k;j=1

pkpjkukk^{p 2}hvj; ukig

for any p a probability distribution. If p 2; then we should have in (2.16) all u_k6= 0: Ifp= 2 and

(2.18) hv_k v_j; u_kii hv_k v_j; u_jis

for any k; j2 f1; : : : ; ng;then (2.19)

Xn k=1

p_khv_k; u_kig

Xn k;j=1

p_kp_jhv_j; u_kig;

for any pa probability distribution.

As a particular case of interest, we state the following result that holds in inner product spaces.

Corollary 2. Let (X;h; i) be a real inner product space, u= (u1; : : : ; un), v = (v₁; : : : ; v_n)2Xⁿ andp 1: If

(2.20) D

vk vj;kukk^{p 2}uk kujk^{p 2}vj

E 0 for any k; j2 f1; : : : ; ng;then

(2.21)

Xn k=1

pkkukk^{p 2}hvk; uki

* _n X

j=1

pjuj; Xn k=1

pkkukk^{p 2}uk

+

for any pa probability distribution.

Remark 2. We observe that if then tuplesuandv above are synchronous, i.e., (2.22) hvk vj; uk uji 0 for anyj; k2 f1; : : : ; ng;

then we have the following µCebyšev type inequality (2.23)

Xn k=1

pkhvk; uki

* _n X

k=1

pkvk; Xn k=1

pkuk

+ :

This result was …rst obtained in [13].

3. Inequalities for Convex Functions The following result for convex functions may be stated:

Theorem 3. Let f : X ! R be a convex function on the linear space X and x; y2Xⁿ:Letpbe a probability distribution so thatPn

i=1pixi =Pn

i=1piyi:Ifx y is r~ f synchronous with y and r~f(y_i) ( ) is additive for each i 2 f1; : : : ; ng; then we have the inequality:

(3.1)

Xn i=1

pif(xi) Xn i=1

pif(yi): Proof. Sincef is convex, then for anyx; y2X we have

(3.2) f(x) f(y) r⁺f(y) (x y) r~f(y) (x y): Then from (3.2) we have the inequality:

(3.3) f(x_i) f(y_i) r~f(y_i) (x_i y_i) for eachi2 f1; : : : ; ng:

Now, if we multiply (3.3) withp_i 0and then sum overifrom 1 ton;we get (3.4)

Xn i=1

pif(xi) Xn i=1

pif(yi) Xn i=1

pir~f(yi) (xi yi):

Now, if we use Corollary 1 forui =yi and vi=xi yi; i2 f1; : : : ; ng;we deduce the inequality

Xn i=1

pir~f(yi) (xi yi) Xn i=1

pir~f(yi) Xn i=1

pi(xi yi)

! (3.5)

= Xn i=1

p_ir~f(y_i) (0) = 0:

Combining (3.4) with (3.5), we deduce the desired inequality (3.1).

Remark 3. It is clear that if f is Gâteaux di¤ erentiable at all the points y_i; i 2 f1; : : : ; ng; thenr~f(yi) ( ) =rf(yi) ( ); i2 f1; : : : ; ng; which are linear onX:

In the case of Gâteaux di¤erentiable functions, we can state the following result as well.

Theorem 4. Let f : X !R be a convex and Gâteaux di¤ erentiable function on the linear space X: Assume that x; y2 Xⁿ andp is a probability distribution. If x y isr~ f synchronous with y and

Xn i=1

pixi

Xn i=1

piyi2

\n i=1

ker (rf(yi) ( )); then

(3.6)

Xn i=1

pif(xi) Xn i=1

pif(yi):

The proof is as in that of Theorem 3 when in (3.5) we take into account that rf(y_i)

Xn i=1

p_ix_i Xn i=1

p_iy_i

!

= 0 for alli2 f1; : : : ; ng since

Xn i=1

pixi

Xn i=1

piyi2

\n i=1

ker (rf(yi) ( )):

The following result in smooth normed linear spaces may be stated.

Proposition 2. Let (X;k k) be a smooth normed linear space and let [; ] be the semi-inner product that generates its norm k k: If x; y2 Xⁿ andp 1 are such that

(3.7) kykk^{p 2}[xk yk xj+yj; yk] kyjk^{p 2}[xk yk xj+yj; yj]

for anyk; j 2 f1; : : : ; ng; then for any probability distribution pwith the property that

(3.8)

Xn j=1

pjxj = Xn j=1

pjyj

we have the inequality (3.9)

Xn k=1

p_kkx_kk^p Xn k=1

p_kky_kk^p: If p2[1;2)we shall assume thatyk6= 0 fork2 f1; : : : ; ng:

If p= 2 and

(3.10) [x_k y_k x_j+y_j; y_k] [x_k y_k x_j+y_j; y_j]

for any k; j 2 f1; : : : ; ng; then for any probability distribution p satisfying (3.8), we have

(3.11)

Xn k=1

pkkxkk² Xn k=1

pkkykk²:

The case of inner product spaces is incorporated in:

Corollary 3. Let(X;h; i)be an inner product space. Ifx; y2Xⁿ andp 1 are such that

(3.12) D

xk xj;kykk^{p 2}yk kyjk^{p 2}yj

E D

yk yj;kykk^{p 2}yk kyjk^{p 2}yj

E

for any k; j 2 f1; : : : ; ng; then for any p satisfying (3.8), we have the inequality (3.9).

If p2[1;2);then we shall assume that y_k 6= 0,k2 f1; : : : ; ng: If p= 2 and

(3.13) hxk xj; yk yji kyk yjk² for any k; j2 f1; : : : ; ng then for any psatisfying (3.8), we have the inequality (3.11).

4. Applications for Convex Functions onR^m

Now, consider an open and convex setCin the real linear spaceR^m; m 1:For a convex and di¤erentiable functionf :C!R, we have

(4.1) rf(x) (y) = @f(x)

@x ; y ; x2C; y2R^m; where

@f(x)

@x = @f(x)

@x¹ ; : : : ;@f(x)

@x^m ; x= x¹; : : : ; x^m 2C and h; iis the usual inner product in R^m; i.e., hu; vi =Pm

k=1uⁱ vⁱ; where u= u¹; : : : ; u^m andv= v¹; : : : ; v^m 2R^m:

Now, ifv:= (v1; : : : ; vn)2R^m andu:= (u1; : : : ; un)2C^m;then we say that v isr f synchronous withuif

(4.2) @f(uk)

@x

@f(uj)

@x ; vk vj 0 for anyk; j2 f1; : : : ; ng: The following result may be stated.

Proposition 3. Let f : C ! R be a di¤ erentiable convex function on the open and convex set C R^m: If v := (v1; : : : ; vn) 2 R^m and u := (u1; : : : ; un) 2 C^m are such thatv isr f synchronous withu, then for any probability distribution p= (p1; : : : ; pn);we have the inequality

(4.3)

Xn i=1

p_i @f(u_i)

@x ; v_i

* _n X

i=1

p_i@f(u_i)

@x ; Xn i=1

p_iv_i +

: The proof is obvious by Corollary 1.

Now, ifuk = u¹_k; : : : ; u^m_k ; k2 f1; : : : ; ngandvk = v¹_k; : : : ; v^m_k ;then (4.4) @f(uk)

@x

@f(uj)

@x ; vk vj = Xm

`=1

@f(uk)

@x

@f(uj)

@x v^{`</sup><sub>k</sub> v<sup>`}_j :

Remark 4. The above relation (4.4) shows that a su¢ cient condition forv to be r f synchronous withuis that all the sequencesn

@f(uk)

@x`

o

k=1;:::;nand v<sup>`</sup><sub>k k=1;:::;n</sub> are synchronous, where`2 f1; : : : ; mg;i.e.,

(4.5) @f(uk)

@x

@f(uj)

@x v_k^{`</sup> v<sub>j</sub><sup>`} 0 for anyk; j2 f1; : : : ; ng and for all`2 f1; : : : ; mg:

The following result is an obvious consequence of Theorem 4.

Proposition 4. Letf :C!Rbe a di¤ erentiable convex function on the open and convex set C R^m:If x= (x₁; : : : ; x_n)2R^m andy= (y₁; : : : ; y_n)2C^m are such that

(4.6) @f(y_k)

@x

@f(y_j)

@x ; x_k x_j @f(y_k)

@x

@f(y_j)

@x ; y_k y_j ;

for each k; j 2 f1; : : : ; ng; then for any probability distribution p = (p1; : : : ; pn) with

(4.7)

Xn i=1

pixi= Xn i=1

piyi

we have the inequality (4.8)

Xn i=1

pif(xi) Xn i=1

pif(yi):

Remark 5. As above, a su¢ cient condition for (4.6) to hold is that the sequences n_@f(y

k)

@x`

o

k=1;:::;n and x^{`</sup><sub>k</sub> y<sup>`}_{k k=1;:::;n} are synchronous for each`2 f1; : : : ; mg: References

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Mathematics, School of Engineering and Science, Victoria University, PO Box 14428, Melbourne City, VIC, Australia. 8001

E-mail address: sever.dragomir@vu.edu.au

URL:http://www.staff.vu.edu.au/rgmia/dragomir/