International Journal of Analysis and Applications

Volume 16, Number 2 (2018), 178-192

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-178

ON GIACCARDI’S INEQUALITY AND ASSOCIATED FUNCTIONAL IN THE PLANE

ATIQ UR REHMAN1,∗, M. HASSAAN AKBAR2 AND G. FARID1

1COMSATS Institute of Information Technology, Attock, Pakistan

2Government Higher Secondary School, Khunda, Tehsil Jand, District Attock, Pakistan

∗Corresponding author: atiq@mathcity.org

Abstract. In this paper the authors extend Giaccardi’s inequality to coordinates in the plane. The au-

thors consider the nonnegative associated functional due to Giaccardi’s inequality in plane and discuss its

properties for certain class of parametrized functions. Also the authors proved related mean value theorems.

1. Introduction

Let I be a real interval. A function f : I → R is said to be convex on I if

f(λx + (1 −λ)y) ≤ λf(x) + (1 −λ)f(y)

for all x,y ∈ R and λ ∈ [0,1].

In [2], Dragomir gave the definition of convex functions on coordinates as follows.

Definition 1.1. Let ∆ = [a,b] × [c,d] ⊆ R2 and f : ∆ → R be a mapping. Define partial mappings

fy : [a,b] → R by fy(u) = f(u,y) (1.1)

and

fx : [c,d] → R by fx(v) = f(x,v). (1.2)

Received 2017-08-21; accepted 2017-10-19; published 2018-03-07.

2010 Mathematics Subject Classification. 26A51; 26D15; 35B05.

Key words and phrases. convex functions; log-convexity; convex functions on coordinates; Giaccardi’s inequality.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

178

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-178


Int. J. Anal. Appl. 16 (2) (2018) 179

Then f is said to be convex on coordinates (or coordinated convex) in ∆ if fy and fx are convex on [a,b]

and [c,d] respectively for all x ∈ [a,b] and y ∈ [c,d]. A mapping f is said to be strictly convex on coordinates

(or strictly coordinated convex) in ∆ if fy and fx are strictly convex on [a,b] and [c,d] respectively for all

x ∈ [a,b] and y ∈ [c,d].

Now we define another important subclass of convex functions i.e. log convex functions.

Definition 1.2. A function f : I → R+ is called log convex on I if

f(αx + βy) ≤ fα(x)fβ(y)

where α + β = 1 α,β ≥ 0 and x,y ∈ I.

Log-convex functions have excellent closure properties. The sum and product of two log-convex functions

is convex. If f is convex function and g is log-convex function then the functional composition g ◦f is also

log-convex. Many authors consider this function e.g. see [12], in which some of the properties of log convex

functions has been discussed (also see [6, 7, 9] and references therein). In the following defintion, we define

log convex function on coordinates.

Definition 1.3. A function f : ∆ → R+ is called log convex on coordinates in ∆ if partial mappings defined

in (1.1) and (1.2) are log convex on [a,b] and [c,d] respectively for all x ∈ [a,b] and y ∈ [c,d].

Remark 1.1. Every log convex function is log convex on coordinates but the converse is not true in general.

For example, f : [0, 1]2 → [0,∞) defined by f(x,y) = exy is convex on coordinates but not convex.

Giaccardi’s inequality is stated as follows (see [8, page 153, 155] or [10]).

Theorem 1.1. Let [0,a) ⊂ R, (x1, ...,xn) ∈ [0,a)n and (p1, ...,pn) be nonnegative n−tuples such that

(xi −x0) (x̃n −xi) > 0 for i = 1, . . . ,n and x̃n 6= x0, where x0 ∈ [0,a) and x̃n =
∑n
k=1 pkxk.

If f is convex, then the inequality

n∑
k=1

pkf(xk) 6 Af (x̃n) + B

(
n∑
k=1

pk − 1

)
f(x0) (1.3)

is valid, where

A =

∑n
k=1 pk(xk −x0)
x̃n −x0

B =
x̃n

x̃n −x0
.

Remark 1.2. Condition that f is convex, can be replaced with
f(x)−f(x0)

x−x0
is an increasing function, then

inequality (1.3) is also valid.



Int. J. Anal. Appl. 16 (2) (2018) 180

Remark 1.3. If f is strictly convex, then strict inequality holds in (1.3) unless x1 = ... = xn and
∑n
i=1 pi =

1.

Remark 1.4. For pi = 1 (i = 1, ...,n), the above inequality becomes

n∑
i=1

f(xi) 6 f

(
n∑
i=1

xi

)
+ (n− 1) f(x0). (1.4)

Remark 1.5. If we put x0 = 0 in above inequality, we get Petrović’s inequality for convex functions on real

line.

In this paper we extend Giaccardi’s inequality to coordinates in the plane. We consider functionals due to

Giaccardi’s inequality in plane and discuss its properties for certain class of coordinated log-convex functions.

Also we proved related mean value theorems.

2. Main results

In the following theorem we give our first result that is Giaccardi’s inequality for coordinated convex

functions.

Theorem 2.1. Let ∆ = [0,a)× [0,b) ⊂ R2, (x1, ...,xn) ∈ [0,a)n, (y1, ...,yn) ∈ [0,b)n, (p1, ...,pn), (q1, ...,qn)

be non-negative n−tuples and
∑n
i=1 pixi = x̃n,

∑n
j=1 qjyj = ỹn such that

(xi −x0) (x̃n −xi) > 0 for i = 1, ...,n, x̃n 6= x0 and
n∑
i=1

pi ≥ 1, (2.1)

and

(yj −yo) (ỹn −yj) > 0 for j = 1, ...,n and ỹn 6= yo. (2.2)

If f is coordinated convex function, then

n∑
i,j=1

piqjf(xi,yj) 6 A1


A2f (x̃n, ỹn) + B2


 n∑
j=1

qj − 1


f (x̃n,yo)


 +

B1

(
n∑
i=1

pi − 1

)A2f (x0, ỹn) + B2

 n∑
j=1

qj − 1


f(x0,yo)


 ,

(2.3)

holds where

A1 =

∑n
i=1 pi(xi −x0)
x̃n −x0

, B1 =
x̃n

x̃n −x0
. (2.4)

and

A2 =

∑n
j=1 qj(yj −y0)
ỹn −y0

, B2 =
ỹn

ỹn −y0
(2.5)



Int. J. Anal. Appl. 16 (2) (2018) 181

Proof. Let fx : [0,b) → R and fy : [0,a) → R be mappings such that fx(v) = f(x,v) and fy(u) = f(u,y).

Since f is coordinated convex on ∆, therefore fy is convex on [0,a). By Theorem 1.1, one has

n∑
i=1

pify(xi) 6 A1fy (x̃n) + B1

(
n∑
i=1

pi − 1

)
fy(x0),

where A1 and B1 are defined in (2.4). We write

n∑
i=1

pif(xi,y) 6 A1f (x̃n,y) + B1

(
n∑
i=1

pi − 1

)
f(x0,y).

By setting y = yj, we have

n∑
i=1

pif(xi,yj) 6 A1f (x̃n,yj) + B1

(
n∑
i=1

pi − 1

)
f(x0,yj),

this gives
n∑
i=1

n∑
j=1

piqjf(xi,yj) 6 A1

n∑
j=1

qjf (x̃n,yj) + B1

(
n∑
i=1

pi − 1

)
n∑
j=1

qjf(x0,yj). (2.6)

Again using Theorem 1.1 on terms of right hand side for second coordinates, we have

n∑
j=1

qjf (x̃n,yj) 6 A2f (x̃n, ỹn) + B2


 n∑
j=1

qj − 1


f (x̃n,y0)

and (
n∑
i=1

pi − 1

)
n∑
j=1

qjf(x0,yj) 6 A2

(
n∑
i=1

pi − 1

)f (x0, ỹn) + B2

 n∑
j=1

qj − 1


f(x0,y0)


 .

where A2 and B2 are defined in (2.5) Using above inequalities in (2.6), we get

n∑
i,j=1

piqjf(xi,yj) 6 A1


A2f (x̃n, ỹn) + B2


 n∑
j=1

qj − 1


f (x̃n,yo)


 +

B1

(
n∑
i=1

pi − 1

)A2f (x0, ỹn) + B2

 n∑
j=1

qj − 1


f(x0,yo)


 ,

which is the required result. �

Remark 2.1. If f is strictly coordinated convex then above inequality is strict unless all xi’s and yi’s are

not equal or
n∑
i=1

pi 6= 1 and
n∑
j=1

qj 6= 1.

Remark 2.2. If we take yj = 0 and qj = 1, (i,j = 1, ...,n) with f(xi, 0) 7→ f(xi), then we get inequality

(1.3).

The following corollary is particular case of Theorem 2.1, which is stated in [11, Theorem 2].



Int. J. Anal. Appl. 16 (2) (2018) 182

Corollary 2.1. Let ∆ = [0,a)×[0,b) ⊂ R2, (x1, ...,xn) ∈ [0,a)n, (y1, ...,yn) ∈ [0,b)n, (p1, ...,pn), (q1, ...,qn)

be non-negative n−tuples and
∑n
i=1 pixi = x̃n,

∑n
j=1 qjyj = ỹn such that x̃n ≥ xj and ỹn ≥ yj for j = 1, ...,n.

Also let that x̃n ∈ [0,a),
∑n
i=1 pi ≥ 1 and ỹn ∈ [0,b). If f : ∆ → R is coordinated convex function, then

n∑
i,j=1

piqjf(xi,yj) 6 f (x̃n, ỹn) +


 n∑
j=1

qj − 1


f (x̃n, 0) +

(
n∑
i=1

pi − 1

)f (0, ỹn) +

 n∑
j=1

qj − 1


f(0, 0)


 ,

(2.7)

holds.

Proof. If we put x0 = y0 = 0 in Theorem 2, conditions (2.4) and (2.5) becomes A1 = A2 = B1 = B2 = 1, so

(2.3) takes the form

n∑
i,j=1

piqjf(xi,yj) 6 f (x̃n, ỹn) +


 n∑
j=1

qj − 1


f (x̃n, 0) +

(
n∑
i=1

pi − 1

)f (0, ỹn) +

 n∑
j=1

qj − 1


f(0, 0)


 ,

as required. �

Let I ⊆ R be an interval and f : I → R be a function. Then for distinct points ui ∈ I,i = 0, 1, 2. The

divided differences of first and second order are defined as follows.

[ui,ui+1,f] =
f(ui+1) −f(ui)

ui+1 −ui
, (i = 0, 1) (2.8)

[u0,u1,u2,f] =
[u1,u2,f] − [u0,u1,f]

u2 −u0
. (2.9)

The values of the divided differences are independent of the order of the points u0,u1,u2 and may be extended

to include the cases when some or all points are equal, that is

[u0,u0,f] = lim
u1→u0

[u0,u1,f] = f
′(u0) (2.10)

provided that f′ exists. Now passing the limit u1 → u0 and replacing u2 by u in second order divided

difference, we have

[u0,u0,u,f] = lim
u1→u0

[u0,u1,u,f] =
f(u) −f(u0) − (u−u0)f′(u0)

(u−u0)2
,u 6= u0 (2.11)

provided that f′ exists. Also passing to the limit ui → u (i = 0, 1, 2) in second order divided difference, we

have

[u,u,u,f] = lim
ui→u

[u0,u1,u2,f] =
f′′(u)

2
(2.12)

provided that f′′ exists.



Int. J. Anal. Appl. 16 (2) (2018) 183

One can note that, if for all u0,u1 ∈ I, [u0,u1,f] ≥ 0, then f is increasing on I and if for all u0,u1,u2 ∈ I,

[u0,u1,u2,f] ≥ 0, then f is convex on I.

Now we define some families of parametric functions which we use in sequal.

Let I = [0,a) and J = [0,b) be intervals and let for t ∈ (c,d) ⊆ R, ft : I ×J → R be a mapping. Then

we define functions

ft,y : I → R by ft,y(u) = ft(u,y)

and

ft,x : J → R by ft,x(v) = ft(x,v),

where x ∈ I and y ∈ J.

Suppose M1 denotes the class of functions ft : I ×J → R for t ∈ (c,d) such that

t 7→ [u0,u1,u2,ft,y] ∀ u0,u1,u2 ∈ I

and

t 7→ [v0,v1,v2,ft,x] ∀ v0,v1,v2 ∈ J

are log convex functions in Jensen sense on (c,d) for all x ∈ I and y ∈ J.

Under the assumptions of Theorem 2.1 we define linear functional G(f; x0,y0) as a non negative difference

of inequality (2.3)

G(f; x0,y0) = A1


A2f (x̃n, ỹn) + B2


 n∑
j=1

qj − 1


f (x̃n,y0)


 +

B1

(
n∑
i=1

pi − 1

)A2f (x0, ỹn) + B2

 n∑
j=1

qj − 1


f(x0,y0)


− n∑

i,j=1

piqjf(xi,yj)

(2.13)

where A1,B1 and A2,B2 are defined in (2.4) and (2.5) respectively.

Remark 2.3. Under the assumptions of Theorem 2.1, if f is coordinated convex in ∆, then G(f; x0,y0) ≥ 0.

Remark 2.4. As a special case, if we put x0 = y0 = 0, in (2.13), then we get

Υ(f) = f (x̃n, ỹn) +

(
n∑
i=1

qj − 1

)
f (x̃n, 0) +

(
n∑
i=1

pi − 1

)
[
f(0, ỹn) +

(
n∑
i=1

qj − 1

)
f(0, 0)

]
−

n∑
i,j=1

pi,qjf(xi,yj),

(2.14)

that is G(f; 0, 0) = Υ(f).



Int. J. Anal. Appl. 16 (2) (2018) 184

Remark 2.5. If we put yj = 1 for j = 1, ...,n in (2.13) then we get functional

P(f) = f (x̃n) −
n∑
i=1

pif(xi) −

(
1 −

n∑
i=1

pi

)
f(0) (2.15)

defined in [1].

The following lemmas are given in [9].

Lemma 2.1. Let I ⊆ R be an interval. A function f : I → (0,∞) is log-convex in Jensen sense on I, that

is, for each r,t ∈ I

f(r)f(t) ≥ f2
(
t + r

2

)
if and only if the relation

m2f(t) + 2mnf

(
t + r

2

)
+ n2f(r) ≥ 0

holds for each m,n ∈ R and r,t ∈ I.

Lemma 2.2. If f is convex function on interval I then for all x1,x2,x3 ∈ I for which x1 < x2 < x3, the

following inequality is valid:

(x3 −x2)f(x1) + (x1 −x3)f(x2) + (x2 −x1)f(x3) ≥ 0.

In [11], authors has given some important properties related to the functional defined for Petrović’s

inequality on coordinates. Our next result comprises similar properties of functional defined in (2.13).

Theorem 2.2. Suppose ft ∈ M1 and G be a functional defined in (2.13). Then G(ft,x0,y0) is log-convex

function in Jensen sense for all t ∈ (c,d).

Proof. Let

h(u,v) = m2ft(u,v) + 2mnft+r
2

(u,v) + n2fr(u,v),

where m,n ∈ R and t,r ∈ (c,d). We can assume that

hy(u) = m
2ft,y(u) + 2mnft+r

2
,y(u) + n

2fr,y(u)

and

hx(v) = m
2ft,x(v) + 2mnft+r

2
,x(v) + n

2fr,x(v).

Since divided differences satisfy the linearity property, therefore we can have

[u0,u1,u2,hy] = m
2[u0,u1,u2,ft,y] + 2mn[u0,u1,u2,ft+r

2
,y] + n

2[u0,u1,u2,fr,y].



Int. J. Anal. Appl. 16 (2) (2018) 185

Since we have given that [u0,u1,u2; hy] is log-convex in Jensen sense, therefore using ft = [u0,u1,u2; hy] in

Lemma 2.1, we get that

[u0,u1,u2,hy] = m
2[u0,u1,u2,ft,y] + 2mn[u0,u1,u2,ft+r

2
,y] + n

2[u0,u1,u2,fr,y] ≥ 0

which is equivalent to write

[u0,u1,u2; hy] ≥ 0.

This shows that hy is convex on interval I. In the similar way, one can prove that hx is convex on J. This

concludes that h is coordinated convex on ∆. By Remark 2.3, we have

G(h,x0,y0) ≥ 0,

that is,

m2G(ft,x0,y0) + 2mnG(ft+r
2
,x0,y0) + n

2G(fr,x0,y0) ≥ 0.

Thus by Lemma 2.1 we have that G(ft,x0,y0) is log-convex in Jensen sense on (c,d). �

Corollary 2.2. Let the functional Υ defined in (2.14) and ft ∈ M1. Then the function t 7→ Υ(ft) is log

convex in Jensen sense on (c,d)

Proof. On putting x0 = y0 = 0 in above theorem, we get G(ft; 0, 0) = Υ(ft), hence the required result

follows. �

Theorem 2.3. Suppose ft is from class M1 and G be a functional defined in (2.13), If G(ft,x0,y0) is

continuous for all t ∈ (c,d), then G(ft,x0,y0) is log convex for all t ∈ (c,d).

Proof. Since we know that if a function is log convex in Jensen sense and continuous, then it is log convex.

From Theorem 2.2, if ft ∈M1, then G(ft,x0,y0) is log convex in Jensen sense and we have given that it is

continuous, hence G(ft,x0,y0) is log convex for all t ∈ (c,d). �

Corollary 2.3. Let the functional Υ defined in (2.14) and ft ∈M1. If the function t 7→ Υ(ft) is continuous

on (c,d), then it is log convex on (c,d)

Proof. On putting x0 = y0 = 0 in above theorem, we get G(ft; 0, 0) = Υ(ft), hence the required result

follows. �

Theorem 2.4. Suppose ft ∈ M1 and G be a functional defined in (2.13). If G(ft; x0,y0) is positive, then

for r,s,t ∈ (c,d) such that r < s < t, one has

[G(fs; x0,y0)]
t−r ≤ [G(fr; x0,y0)]

t−s
[G(ft; x0,y0)]

s−r
. (2.16)



Int. J. Anal. Appl. 16 (2) (2018) 186

Proof. By taking f = logG(ft,x0,y0) in Lemma 2.2, we have for t 6= r, u 6= v,

(t−s) logG(fr; x0,y0) + (r − t) logG(fs; x0,y0) + (s−r) logG(ft; x0,y0) ≥ 0,

which is equivalent to

[G(fs; x0,y0)]
t−r ≤ [G(fr; x0,y0)]

t−s
[G(ft; x0,y0)]

s−r
(2.17)

that is our required result. �

Corollary 2.4. Let the functional Υ defined in (2.14) and ft ∈ M1. If Υ(ft) is positive, then for some

r < s < t, where r,s,t ∈ (c,d), one has

[Υ(fs)]
t−r ≤ [Υ(fr)]

t−s
[Υ(ft)]

s−r
. (2.18)

Proof. On putting x0 = y0 = 0 in above theorem, we get G(ft; 0, 0) = Υ(ft), hence the required result

follows. �

The following Lemma is equivalent to the definition of convex function (see [5, Page 2]).

Lemma 2.3. Let I be an interval in R. A function f : I → R is convex if and only if for all t,r,u,v ∈ I

such that t ≤ u,r ≤ v,t 6= r,u 6= v, one has

f(t) −f(r)
t−r

≤
f(u) −f(v)

u−v
.

Theorem 2.5. Let G(ft; x0,y0) be the linear functional defined in (2.13), where ft ∈ M1. If the function

t 7→ G(ft; x0,y0) is derivable on (c,d), then for t,r,u,v ∈ (c,d) such that t 6 u,r 6 v, we have

C1(t,r) 6 C1(u,v),

where

C1(t,r) =




(
G(ft;x0,y0)
G(fr;x0,y0)

) 1
t−r

, t 6= r,

exp
(

d
dt

(G(ft;x0,y0))
G(ft;x0,y0)

)
, t = r.

(2.19)

Proof. By taking f = G(ft,x0,y0) in Lemma 2.3, we have for t 6= r,u 6= v,

logG(ft; x0,y0) − logG(fr; x0,y0)
t−r

6
logG(fu; x0,y0) − logG(fv; x0,y0)

u−v
.

This gives

C1(t,r) 6 C1(u,v), t 6= r,u 6= v.

For t = r,u = v, we consider limiting cases in above inequality, when r → t and v → u.

�



Int. J. Anal. Appl. 16 (2) (2018) 187

The following corollaries that are stated in [11], are special cases of Theorem 2.5.

Corollary 2.5. Under the assumptions of Theorem (2.5), let Υ(ft) be the linear functional defined in (2.14)

then E(t,r,ft) 6 E(u,v,ft), where

E(t,r,ft) =




(
Υ(ft)
Υ(fr)

) 1
t−r

, t 6= r,

exp
(

d
dt

(Υ(ft))

Υ(ft)

)
, t = r.

(2.20)

Proof. On putting x0 = y0 = 0 in Theorem (2.5), we get G(ft; x0,y0) = Υ(ft), hence the required result

follows. �

Corollary 2.6. Under the assumptions of Theorem (2.5), let P(ft) be the linear functional defined in (2.15)

then T (t,r,ft) 6 T (u,v,ft), where

T (t,r,ft) =




(
P(ft)
P(fr)

) 1
t−r

, t 6= r,

exp
(

d
dt

(P(ft))
P(ft)

)
, t = r.

(2.21)

Proof. On putting, yj = 1 for j = 1, ...,n in Corollary 2.5, we get our required result. �

Example 2.1. Let t ∈ (0,∞) and ϕt : [0,∞)2 → R be a function defined as

ϕt(u,v) =




utvt

t(t−1), t 6= 1,

uv(log u + log v), t = 1.
(2.22)

Define partial mappings

ϕt,v : [0,∞) → R by ϕt,v(u) = ϕt(u,v)

and

ϕt,u : [0,∞) → R by ϕt,u(v) = ϕt(u,v).

As we have

[u,u,u,ϕt,v] =
∂2ϕt,v
∂u2

= ut−2vt ≥ 0 ∀ t ∈ (0,∞).

This gives t 7→ [u0,u0,u0,ϕt,v] is log convex in Jensen sense. Similarly one can deduce that t 7→ [v0,v0,v0,ϕt,u]

is also log-convex in Jensen sense. If we choose ft = ϕt in Theorem 2.2, we get log convexity of the functional

G(γt).

In special case, if we choose ϕt(u,v) = ϕt(u, 1), then we get [1, Example 3].

Example 2.2. Let t ∈ [0,∞) and δt : [0,∞)2 → R be a function defined as

δt(u,v) =



uveuvt

t
, t 6= 0,

u2v2, t = 0.
(2.23)



Int. J. Anal. Appl. 16 (2) (2018) 188

Define partial mappings

δt,v : [0,∞) → R by δt,v(u) = δt(u,v)

and

δt,u : [0,∞) → R by δt,u(v) = δt(u,v)

for all u,v ∈ [0,∞).

As we have

[u,u,u,δt,v] =
∂2δt,v
δu2

= euvt(2v2 + uv2) ≥ 0 ∀ t ∈ (0,∞).

This gives t 7→ [u0,u0,u0,δt,v] is log convex in Jensen sense. Similarly one can deduce that t 7→ [v0,v0,v0,δt,u]

is also log-convex in Jensen sense. If we choose ft = δt in Theorem 2.2, we get log convexity of the functional

G(δt).

In special case, if we choose δt(u,v) = δt(u, 1), then we get [1, Example 8].

Example 2.3. Let t ∈ [0,∞) and γt : [0,∞)2 → R be a function defined as

γt(u,v) =



euvt

t
, t 6= 0,

uv, t = 0.
(2.24)

Define partial mappings

γt,v : [0,∞) → R by γt,v(u) = γt(u,v)

and

γt,u : [0,∞) → R by γt,u(v) = γt(u,v).

As we have

[u,u,u,γt,v] =
∂2γt,v
∂u2

= tv2euvt ≥ 0 ∀ t ∈ (0,∞).

This gives t 7→ [u0,u0,u0,γt,v] is log convex in Jensen sense. Similarly one can deduce that t 7→ [v0,v0,v0,γt,u]

is also log-convex in Jensen sense. If we choose ft = γt in Theorem 2.2, we get log convexity of the functional

G(γt).

In special case, if we choose γt(u,v) = γt(u, 1), then we get [1, Example 9].

Example 2.4. Let t ∈ [0,∞) and λt : [0,∞)2 → R be a function defined as

λt(u,v) =
ueu
√
t

√
t

(2.25)

Define partial mappings

λt,v : [0,∞) → R by λt,v(u) = λt(u,v)

and

λt,u : [0,∞) → R by λt,u(v) = λt(u,v).



Int. J. Anal. Appl. 16 (2) (2018) 189

As we have

[u,u,u,λt,v] =
∂2λt,v
∂u2

= v2euv
√
t
(

2 + uv
√
t
)
≥ 0 ∀ t ∈ (0,∞).

This gives t 7→ [u0,u0,u0,λt,v] is log convex in Jensen sense. Similarly one can deduce that t 7→ [v0,v0,v0,λt,u]

is also log-convex in Jensen sense. If we choose ft = λt in Theorem 2.2, we get log convexity of the functional

G(λt).

In special case, if we choose λt(u,v) = λt(u,−1), then we get [1, Example 6].

3. Mean value theorems

If a function is twice differentiable on an interval I, then it is convex on I if and only if its second order

derivative is nonnegative. If a function f(X) := f(x,y) has continuous second order partial derivatives on

interior of ∆ then it is convex on ∆ if the Hessian matrix

Hf (X) :=


∂2f(X)∂x2 ∂2f(X)∂y∂x
∂2f(X)
∂x∂y

∂2f(X)
∂y2




is nonnegative definite, that is, vHf (X)v
t is nonnegative for all real nonnegative vector v. It is easy to see

that f : ∆ → R is coordinated covex on ∆ iff

f′′x (y) =
∂2f(x,y)

∂y2
and f′′y (x) =

∂2f(x,y)

∂x2

are nonnegative for all interior points (x,y) in ∆.

Lemma 3.1. Let f : ∆ → R be a function such that

M1 ≤
∂2f(x,y)

∂x2
≤M1

and

m2 ≤
∂2f(x,y)

∂y2
≤ M2

for all interior points (x,y) in ∆2. Consider the function ψ1,ψ2 : ∆ → R defined as

ψ1 =
1

2
max{M1,M2}(x2 + y2) −f(x,y)

ψ2 = f(x,y) −
1

2
min{M1,m2}(x2 + y2),

then ψ1,ψ2 are convex on coordinates in ∆.

Proof. Since

∂2ψ1(x,y)

∂x2
= max{M1,M2}−

∂2f(x,y)

∂x2
≥ 0

and

∂2ψ1(x,y)

∂y2
= max{M1,M2}−

∂2f(x,y)

∂y2
≥ 0



Int. J. Anal. Appl. 16 (2) (2018) 190

for all interior points (x,y) in ∆, ψ1 is convex on coordinates in ∆. Similarly one can prove that ψ2 is also

convex on coordinates in ∆. �

In [3] and [4], we have given mean value theorems of Lagrange type and Cauchy type for certain functional.

Here we give a theorem similar to those but for functional introduced in (2.13).

Theorem 3.1. Let ∆̃ = [0,a1] × [0,b1] ⊂ ∆ and f : ∆̃ → R which has continuous partial derivatives of

second order in ∆̃ and ϕ(x,y) := x2 + y2. Then there exist (β1,γ1) and (β2,γ2) in the interior of ∆̃ such

that

G(f; x0,y0) =
1

2

∂2f(β1,γ1)

∂x2
Υ(ϕ)

and

G(f; x0,y0) =
1

2

∂2f(β2,γ2)

∂y2
Υ(ϕ)

provided that G(ϕ; x0,y0) is non-zero.

Proof. Since f has continuous partial derivatives of second order in ∆̃ and ∆̃ is compact, there exist real

numbers M1,m2,M1 and M2 such that

M1 ≤
∂2f(x,y)

∂x2
≤M1 and m2 ≤

∂2f(x,y)

∂y2
≤ M2,

for all (x,y) ∈ ∆̃.

Now consider functions ψ1 and ψ2 defined in Lemma 3.1. As ψ1 is convex on coordinates in ∆,

G(ψ1; x0,y0) ≥ 0,

that is

G
(

1

2
max{M1,M2}ϕ(x,y) −f(x,y)

)
≥ 0,

this leads us to

2G(f; x0,y0) ≤ max{M1,M2}G(ϕ; x0,y0). (3.1)

On the other hand for function ψ2, one has

min{M1,m2}G(ϕ; x0,y0) ≤ 2G(f; x0,y0). (3.2)

As G(ϕ; x0,y0) 6= 0, combining inequalities (3.1) and (3.2), we get

min{M1,m2}≤
2G(f; x0,y0)
G(ϕ; x0,y0)

≤ max{M1,M2}.

Then there exist (β1,γ1) and (β2,γ2) in the interior of ∆ such that

2G(f; x0,y0)
G(ϕ; x0,y0)

=
∂2f(β1,γ1)

∂x2



Int. J. Anal. Appl. 16 (2) (2018) 191

and
2G(f; x0,y0)
G(ϕ; x0,y0)

=
∂2f(β2,γ2)

∂y2
,

hence the required result follows. �

The following corollary is particular case of Theorem 3.1, which is stated in [11, Theorem 4].

Corollary 3.1. Under the assumptions of above theorem, let Υ(f) be the linear functional defined in (2.14),

then

Υ(f) =
1

2

∂2f(β1,γ1)

∂x2
Υ(ϕ)

and

Υ(f) =
1

2

∂2f(β2,γ2)

∂y2
Υ(ϕ)

provided that Υ(f) is non-zero.

Proof. On putting x0 = y0 = 0 in Theorem 3.1, we get G(f; x0,y0) = Υ(f), hence the required result

follows. �

Theorem 3.2. Let ψ1,ψ2 : ∆̃ → R be mappings which have continuous partial derivatives of second order

in ∆̃. Then there exists (η1,ξ1) and (η2,ξ2) in ∆̃ such that

G(ψ1; x0,y0)
G(ψ2; x0,y0)

=
∂2ψ1(η1,ξ1)

∂x2

∂2ψ2(η1,ξ1)
∂x2

(3.3)

and

G(ψ1; x0,y0)
G(ψ2; x0,y0)

=

∂2ψ1(η2,ξ2)
∂y2

∂2ψ2(η2,ξ2)
∂y2

. (3.4)

Proof. We define the mapping P : ∆̃ → R such that

P = k1ψ1 −k2ψ2,

where k1 = G(ψ2; x0,y0) and k2 = G(ψ1; x0,y0).

Using Theorem 3.1 with f = P, we have

2G(P ; x0,y0) = 0 =
{
k1
∂2ψ1
∂x2

−k2
∂2ψ2
∂x2

}
G(ϕ; x0,y0)

and

2G(P ; x0,y0) = 0 =
{
k1
∂2ψ1
∂y2

−k2
∂2ψ2
∂y2

}
G(ϕ; x0,y0).

Since G(ϕ; x0,y0) 6= 0, we have
k2
k1

=
∂2ψ1(η1,ξ1)

∂x2

∂2ψ2(η1,ξ1)
∂x2

and

k2
k1

=

∂2ψ1(η2,ξ2)
∂y2

∂2ψ2(η2,ξ2)
∂y2

,



Int. J. Anal. Appl. 16 (2) (2018) 192

which are equivalent to required results. �

Corollary 3.2. Under the assumptions of above theorem, let Υ(f) be the linear functional defined in (2.14)

then

Υ(ψ1)

Υ(ψ2)
=

∂2ψ1(η1,ξ1)
∂x2

∂2ψ2(η1,ξ1)
∂x2

(3.5)

and

Υ(ψ1)

Υ(ψ2)
=

∂2ψ1(η2,ξ2)
∂y2

∂2ψ2(η2,ξ2)
∂y2

. (3.6)

Proof. On putting x0 = y0 = 0 in Theorem 3.2, we get G(f; x0,y0) = Υ(f), hence the required result

follows. �

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	1. Introduction
	2. Main results
	3. Mean value theorems
	References