Rehman et al., Cogent Mathematics (2016), 3: 1227298
http://dx.doi.org/10.1080/23311835.2016.1227298
PURE MATHEMATICS | RESEARCH ARTICLE
Petrović’s inequality on coordinates and related
results
Atiq Ur Rehman1*, Muhammad Mudessir1, Hafiza Tahira Fazal2 and Ghulam Farid1
Received: 19 July 2016
Accepted: 17 August 2016
Published: 13 September 2016
*Corresponding author: Atiq Ur Rehman,
Department of Mathematics, COMSATS
Institute of Information Technology,
Attock, Pakistan
E-mail: atiq@mathcity.org
Reviewing editor:
Lishan Liu, Qufu Normal University,
China
Additional information is available at
the end of the article
Abstract: In this paper, the authors extend Petrović’s inequality to coordinates in
the plane. The authors consider functionals due to Petrović’s inequality in plane and
discuss its properties for certain class of coordinated log-convex functions. Also, the
authors proved related mean value theorems.
Subjects: Foundations & Theorems; Mathematics & Statistics; Science
Keywords: Petrović’s inequality; log-convexity; convex functions on coordinates
2000 Mathematics subject classifications: Primary 26A51; Secondary 26D15
1. Introduction
A function f : [a, b] → ℝ is called mid-convex or convex in Jensen sense if for all x, y ∈ [a, b], the
inequality
f
(x + y )
2
≤
f (x) + f (y)
2
is valid.
In 1905, J. Jensen was the first to define convex functions using above inequality (see, Jensen,
1905; Robert & Varberg, 1974, p. 8) and draw attention to their importance.
ABOUT THE AUTHORS
PUBLIC INTEREST STATEMENT
Atiq Ur Rehman and Ghulam Farid are assistant
professors in the Department of Mathematics at
the COMSATS Institute of Information Technology
(CIIT), Attock, Pakistan. Their primary research
interests include real functions, mathematical
inequalities, and difference equation.
Muhammad Mudessir has successfully
completed his MS degree in mathematics from
CIIT in this year. He is a teacher in Government
Pilot Secondary School, Attock, Pakistan. His
area of research includes convex analysis and
inequalities in mathematics.
Hafiza Tahira Fazal received her master of
philosophy degree from National College of Business
Administration and Economics, Lahore, Pakistan.
She is working as a lecturer in the Department of
Mathematics at the University of Lahore, Sargodha,
Pakistan from last two years. Her area of research
includes inequalities in mathematics.
A real-valued function defined on an interval is
called convex if the line segment between any two
points on the graph of the function lies above or on
the graph. Convex functions play an important role
in many areas of mathematics. They are especially
important in the study of optimization problems
where they are distinguished by a number of
convenient properties. One of the important
subclass of convex functions is log-convex
functions. Apparently, it would seem that logconvex functions would be unremarkable because
they are simply related to convex functions. But
they have some surprising properties. Recently, the
concept of convex functions has been generalized
by many mathematicians and different functions
related or close to convex functions are defined.
In this work, the variant of Petrovic’s inequality
for convex functions on coordinates is given. Few
generalization of the results related to it are given.
© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution
(CC-BY) 4.0 license.
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Definition 1 A function f : [a, b] → ℝ is said to be convex if
(1.1)
f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y)
holds, for all x, y ∈ [a, b] and t ∈ [0, 1]. A function f is said to be strictly convex if strict inequality holds
in (1.1).
A mapping f : Δ → ℝ is said to be convex in Δ if
f (tx + (1 − t)z, ty + (1 − t)w) ≤ tf (x, y) + (1 − t)f (z, w)
2
for all (x, y), (z, w) ∈ Δ, where Δ: = [a, b] × [c, d] ⊂ ℝ and t ∈ [0,1].
In Dragomir (2001) gave the definition of convex functions on coordinates as follows.
Definition 2 Let Δ = [a, b] × [c, d] ⊆ ℝ2 and f : Δ → ℝ be a mapping. Define partial mappings
fy : [a, b] → ℝ by fy (u) = f (u, y)
(1.2)
and
fx : [c, d] → ℝ by fx (v) = f (x, v).
(1.3)
Then f is said to be convex on coordinates (or coordinated convex) in Δ 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 Δ if fy and fx are strictly convex on [a, b] and [c, d]
respectively for all x ∈ [a, b] and y ∈ [c, d].
One of the important subclass of convex functions is log-convex functions. Apparently, it would
seem that log-convex functions would be unremarkable because they are simply related to convex
functions. But they have some surprising properties. The Laplace transform of a non-negative function is a log-convex. The product of log-convex functions is log-convex. Due to their interesting properties, the log-convex functions appear frequently in many problems of classical analysis and
probability theory, e.g. see (Farid, Marwan, & Rehman, 2015; Niculescu, 2012; Noor, Qi, & Awan, 2013;
Pečarić, & Rehman, 2008a, 2008b; Xi & Qi, 2015; Zhang & Jiang, 2012) and the references therein.
Definition 3 A function f : I → ℝ+ is called log-convex on I if
f (𝛼x + 𝛽y) ≤ f 𝛼 (x)f 𝛽 (y)
where 𝛼, 𝛽 > 0 with 𝛼 + 𝛽 = 1 and x, y ∈ I.
Definition 4 (Alomari & Darus, 2009) A function f : Δ → ℝ+ is called log-convex on coordinates in Δ
if partial mappings defined in (1.2) and (1.3) are log-convex on [a, b] and [c, d], respectively, for all
x ∈ [a, b] and y ∈ [c, d].
Remark 1 Every log-convex function is log convex on coordinates but the converse is not true in gen2
eral. For example, f : [0, 1] → [0, ∞) defined by f (x, y) = exy is log-convex on coodinates but not
log-convex.
In Pečarić, Proschan, and Tong (1992, p. 154), Petrović’s inequality for convex function is stated as
follows.
Theorem 1 Let [0, a) ⊂ ℝ, (x1 , … , xn ) ∈ (0, a]n and (p1 , … , pn ) be nonnegative n-tuples such that
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n
∑
pi xi ⩾ xj for j = 1, 2, 3, … , n and
n
∑
i=1
pi xi ∈ [0, a).
i=1
If f is a convex function on [0, a), then the inequality
n
∑
(
pi f (xi ) ⩽ f
i=1
n
∑
)
pi xi
(
+
i=1
n
∑
)
(1.4)
pi − 1 f (0)
i=1
is valid.
Remark 2 If f is strictly convex, then strict inequality holds in (1.4) unless x1 = ⋯ = xn and
∑n
i=1
pi = 1.
Remark 3 For pi = 1 (i = 1, … , n), the above inequality becomes
n
∑
(
f (xi ) ⩽ f
i=1
)
n
∑
xi
(1.5)
+ (n − 1)f (0).
i=1
This was proved by Petrović in 1932 (see Petrović, 1932).
In this paper, we extend Petrović’s inequality to coordinates in the plane. We consider functionals
due to Petrović’s inequality in plane and discuss its properties for certain class of coordinated logconvex functions. Also we proved related mean value theorems.
2. Main results
In the following theorem, we give our first result that is Petrović’s inequality for coordinated convex
functions.
Theorem 2 Let Δ = [0, a) × [0, b) ⊂ ℝ2, (x1 , … , xn ) ∈ (0, a]n, (y1 , … , yn ) ∈ (0, b]n, (p1 , … , pn ) and
∑n
∑n
(q1 , … , qn ) be non-negative n-tuples such that i=1 pi xi ≥ xj and i=1 qi yi ≥ yj for j = 1, … , n. Also let
∑n
∑n
∑n
i=1 pi xi ∈ [0, a),
i=1 pi ≥ 1 and
i=1 qi yi ∈ [0, b). If f : Δ → ℝ is coordinated convex function, then
n
∑
pi qj f (xi , yj ) ⩽ f
( n
∑
pi x i ,
n
∑
qj yj
+
( n
∑
)
) ( n
∑
pi x i , 0
qj − 1 f
i=1
j=1
j=1
i=1
i,j=1
)
)
]
) ( n
)[ ( n
( n
∑
∑
∑
qj − 1 f (0, 0) .
pi − 1 f 0,
qj yj +
+
i=1
(2.1)
j=1
j=1
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 Δ, therefore fy is convex on [0, a). By Theorem 1, one has
n
∑
(
pi fy (xi ) ⩽ fy
i=1
n
∑
)
pi xi
(
+
i=1
n
∑
)
pi − 1 fy (0).
i=1
By setting y = yj, we have
n
∑
(
pi f (xi , yj ) ⩽ f
i=1
n
∑
)
pi xi , yj
(
+
i=1
n
∑
)
pi − 1 f (0, yj ),
i=1
this gives
n
n
∑
∑
i=1 j=1
pi qj f (xi , yj ) ⩽
n
∑
j=1
(
qj f
n
∑
i=1
)
pi xi , yj
+
( n
∑
i=1
)
pi − 1
n
∑
qj f (0, yj ).
(2.2)
j=1
Again, using Theorem 1 on terms of right-hand side for second coordinates, we have
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n
∑
(
qj f
j=1
n
∑
)
pi xi , yj
(
⩽f
i=1
n
∑
pi xi ,
i=1
n
∑
)
(
qj yj
+
j=1
n
∑
) (
qj − 1 f
n
∑
j=1
)
pi xi , 0
i=1
and
(
n
∑
i=1
)
pi − 1
(
n
∑
qj f (0, yj ) ⩽
n
∑
j=1
)[ (
pi − 1
f 0,
i=1
n
∑
)
qj yj
(
+
j=1
n
∑
)
]
qj − 1 f (0, 0) .
j=1
Using above inequities in (2.2), we get inequality (2.1). ✷
Remark 4 If f is strictly coordinated convex, then above inequality is strict unless all xi’s and yi’s are
∑n
∑n
not equal or i=1 pi ≠ 1 and j=1 qj ≠ 1.
Remark 5 If we take yi = 0 and qj = 1, (i, j = 1,...,n) with f (xi , 0) ↦ f (xi ), then we get inequality (1.4).
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
[u0 , u1 , u2 , f ] =
(2.3)
, (i = 0, 1),
[u1 , u2 , f ] − [u0 , u1 , f ]
.
u2 − u 0
(2.4)
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 [u0 , u1 , f ] = f � (u0 )
(2.5)
u1 →u0
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 [u0 , u1 , u, f ] =
f (u) − f (u0 ) − (u − u0 )f � (u0 )
u1 →u0
(u − u0 )2
, u ≠ u0
(2.6)
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 [u0 , u1 , u2 , f ] =
ui →u
f �� (u)
2
(2.7)
provided that f exists.
′′
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
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ft,x : J → ℝ by ft,x (v) = ft (x, v),
where x ∈ I and y ∈ J.
Suppose  denotes the class of functions ft : I × J → ℝ for t ∈ (c, d) such that
t ↦ [u0 , u1 , u2 , ft,y ] ∀ u0 , u1 , u2 ∈ I
and
t ↦ [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.
We define linear functional 𝛶 (f ) as a non-negative difference of inequality (2.1)
𝛶 (f ) = f
( n
∑
pi x i ,
n
∑
+
qj yj
(
+
n
∑
)[ (
pi − 1
f 0,
n
∑
j=1
i=1
) (
qj − 1 f
j=1
j=1
i=1
( n
∑
)
qj yj
)
(
+
n
∑
i=1
n
∑
)
pi x i , 0
)
]
qj − 1 f (0, 0) −
j=1
n
∑
(2.8)
pi qj f (xi , yj ).
i,j=1
Remark 6 Under the assumptions of Theorem 2, if f is coordinated convex in Δ, then 𝛶 (f ) ≥ 0.
The following lemmas are given in Pečarić and Rehman (2008b).
Lemma 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
)
(
t+r
f (r)f (t) ≥ f 2
2
if and only if the relation
m2 f (t) + 2mnf
(
t+r
2
)
+ n2 f (r) ≥ 0
holds for each m, n ∈ ℝ and r, t ∈ I.
Lemma 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.
Our next result comprises properties of functional defined in (2.8).
Theorem 3 Let the functional 𝛶 defined in (2.8) and ft ∈ . Then the following are valid:
(a) The function t ↦ 𝛶 (ft ) is log-convex in Jensen sense on (c, d).
(b) If the function t ↦ 𝛶 (ft ) is continuous on (c, d), then it is log-convex on (c, d).
(c) If 𝛶 (ft ) is positive, then for some r < s < t, where r, s, t ∈ (c, d), one has
[
]t−r [
]t−s [
]s−r
𝛶 (fs )
≤ 𝛶 (fr )
𝛶 (ft ) .
(2.9)
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Proof (a) Let
h(u, v) = m2 ft (u, v) + 2mnf t+r (u, v) + n2 fr (u, v)
2
where m, n ∈ ℝ and t, r ∈ (c, d). We can consider
hy (u) = m2 ft,y (u) + 2mnf t+r ,y (u) + n2 fr,y (u)
2
and
hx (v) = m2 ft,x (v) + 2mnf t+r ,x (v) + n2 fr,x (v).
2
Now we take
[u0 , u1 , u2 , hy ] = m2 [u0 , u1 , u2 , ft,y ] + 2mn[u0 , u1 , u2 , f t+r ,y ] + n2 [u0 , u1 , u2 , fr,y ].
2
As [u0 , u1 , u2 , ft,y ] is log convex in Jensen sense, so using Lemma 1, the right-hand side of above expression is non-negative, so hy is convex on I. Similarly, one can show that hx is also convex on J,
which concludes h is coordinated convex on Δ. By Remark 6, 𝛶 (h) ≥ 0, that is,
m2 𝛶 (ft ) + 2mn𝛶 (f t+r ) + n2 𝛶 (fr ) ≥ 0,
2
so t ↦ 𝛶 (ft ) is log-convex in Jensen sense on (c, d).
(b) Additionally, we have t ↦ 𝛶 (ft ) is continuous on (c, d), hence we have t ↦ 𝛶 (ft ) is log-convex on
(c, d).
(c) Since t ↦ 𝛶 (ft ) is log-convex on (c, d), therefore for r, s, t ∈ (c, d) with r < s < t and f (t) = log𝛶 (t)
in Lemma 2, we have
(t − s) log 𝛶 (fr ) + (r − t) log 𝛶 (fs ) + (s − r) log 𝛶 (ft ) ≥ 0,
which is equivalent to (2.9).
✷
Example 1 Let t ∈ (0, ∞) and 𝜑t :[0, ∞)2 → ℝ be a function defined as
{
𝜑t (u, v) =
ut v t
,
t(t−1)
t ≠ 1,
uv(log u + log v),
t = 1.
(2.10)
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 ] =
𝜕 2 𝜑t,v
𝜕u2
= ut−2 v t ≥ 0
∀ t ∈ (0, ∞).
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This gives t ↦ [u0 , u0 , u0 , 𝜑t,v ] is log-convex in Jensen sense. Similarly, one can deduce that
t ↦ [v0 , v0 , v0 , 𝜑t,u ] is also log-convex in Jensen sense. If we choose ft = 𝜑t in Theorem 3, we get log
convexity of the functional 𝛶 (𝜑t ).
In special case, if we choose 𝜑t (u, v) = 𝜑t (u, 1), then we get (Butt, Pečarić, & Rehman, A. U.
2011, Example 3).
Example 2 Let t ∈ [0, ∞) and 𝛿t :[0, ∞)2 → ℝ be a function defined as
{
𝛿t (u, v) =
uveuvt
,
t
2 2
t ≠ 0,
u v ,
(2.11)
t = 0.
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
𝜕 2 𝛿t,v
= euvt (2v 2 + uv 2 ) ≥ 0 ∀ t ∈ (0, ∞).
𝛿u2
This gives t ↦ [u0 , u0 , u0 , 𝛿t,v ] is log convex in Jensen sense. Similarly, one can deduce that
t ↦ [v0 , v0 , v0 , 𝛿t,u ] is also log-convex in Jensen sense. If we choose ft = 𝛿t in Theorem 3, we get log
convexity of the functional 𝛶 (𝛿t ).
[u, u, u, 𝛿t,v ] =
In special case, if we choose 𝛿t (u, v) = 𝛿t (u, 1), then we get (Butt et al., 2011, Example 8).
Example 3 Let t ∈ [0, ∞) and 𝛾t :[0, ∞)2 → ℝ be a function defined as
{
𝛾t (u, v) =
euvt
t
uv,
,
t ≠ 0,
t = 0.
(2.12)
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 ] =
𝜕 2 𝛾t,v
𝜕u2
= tv 2 euvt ≥ 0
∀ t ∈ (0, ∞).
This gives t ↦ [u0 , u0 , u0 , 𝛾t,v ] is log-convex in Jensen sense. Similarly one can deduce that
t ↦ [v0 , v0 , v0 , 𝛾t,u ] is also log-convex in Jensen sense. If we choose ft = 𝛾t in Theorem 3, we get log
convexity of the functional 𝛶 (𝛾t ).
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In special case, if we choose 𝛾t (u, v) = 𝛾t (u, 1), then we get (Butt et al., 2011, Example 9).
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 non-negative. If a function f (X): = f (x, y) has continuous second-order partial derivatives on interior of Δ, then it is convex on Δ if the Hessian matrix
⎛
Hf (X) = ⎜
⎜
⎝
𝜕 2 f (X)
𝜕 2 f (X)
𝜕x2
𝜕 2 f (X)
𝜕y𝜕x
𝜕 2 f (X)
𝜕x𝜕y
𝜕y 2
⎞
⎟
⎟
⎠
is non-negative definite, that is, vHf (X)v is non-negative for all real non-negative vector v.
t
It is easy to see that f : Δ → ℝ is coordinated convex on Δ iff
fx�� (y) =
𝜕 2 f (x, y)
𝜕y 2
and fy�� (x) =
𝜕 2 f (x, y)
𝜕x2
2
are non-negative for all interior points (x, y) in Δ .
Lemma 3 Let f :Δ → ℝ be a function such that
m1 ≤
𝜕 2 f (x, y)
𝜕x2
≤ M1
and
m2 ≤
𝜕 2 f (x, y)
𝜕y 2
≤ M2
for all interior points (x, y) in Δ2. Consider the function 𝜓1 , 𝜓2 : Δ → ℝ defined as
𝜓1 =
1
max{M1 , M2 }(x2 + y 2 ) − f (x, y)
2
𝜓2 = f (x, y) −
1
min{m1 , m2 }(x2 + y 2 )
2
then 𝜓1 , 𝜓2 are convex on coordinates in Δ.
Proof Since
𝜕 2 𝜓1 (x, y)
𝜕x2
= max{M1 , M2 } −
𝜕 2 f (x, y)
𝜕x2
≥0
and
𝜕 2 𝜓1 (x, y)
𝜕y
2
= max{M1 , M2 } −
𝜕 2 f (x, y)
𝜕y 2
≥0
for all interior points (x, y) in Δ, 𝜓1 is convex on coordinates in Δ. Similarly, one can prove that 𝜓2 is also
convex on coordinates in Δ. ✷
Theorem 4 Let f : Δ → ℝ be a mapping which has continuous partial derivatives of second order in Δ
and 𝜑(x, y): = x2 + y 2. Then, there exist (𝛽1 , 𝛾1 ) and (𝛽2 , 𝛾2 ) in the interior of Δ such that
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𝛶 (f ) =
2
1 𝜕 f (𝛽1 , 𝛾1 )
𝛶 (𝜑)
2
𝜕x2
and
𝛶 (f ) =
2
1 𝜕 f (𝛽2 , 𝛾2 )
𝛶 (𝜑)
2
𝜕y 2
provided that 𝛶 (𝜑) is non-zero.
Proof Since f has continuous partial derivatives of second order in Δ, there exist real numbers
m1 , m2 , M1 and M2 such that
m1 ≤
𝜕 2 f (x, y)
𝜕x2
≤ M1
and
m2 ≤
𝜕 2 f (x, y)
𝜕y 2
≤ M2 ,
for all (x, y) ∈ Δ.
Now consider functions 𝜓1 and 𝜓2 defined in Lemma 3. As 𝜓1 is convex on coordinates in Δ,
𝛶 (𝜓1 ) ≥ 0,
that is
(
𝛶
)
1
max{M1 , M2 }𝜑(x, y) − f (x, y) ≥ 0,
2
this leads us to
2𝛶 (f ) ≤ max{M1 , M2 }𝛶 (𝜑).
(3.1)
On the other hand, for function 𝜓2, one has
min{m1 , m2 }𝛶 (𝜑) ≤ 2𝛶 (f ).
(3.2)
As 𝛶 (𝜑) ≠ 0, combining inequalities (3.1) and (3.2), we get
min{m1 , m2 } ≤
2𝛶 (f )
≤ max{M1 , M2 }.
𝛶 (𝜑)
Then there exist (𝛽1 , 𝛾1 ) and (𝛽2 , 𝛾2 ) in the interior of Δ such that
𝜕 2 f (𝛽1 , 𝛾1 )
2𝛶 (f )
=
𝛶 (𝜑)
𝜕x2
and
𝜕 2 f (𝛽2 , 𝛾2 )
2𝛶 (f )
,
=
𝛶 (𝜑)
𝜕y 2
hence the required result follows.
✷
Theorem 5 Let 𝜓1 , 𝜓2 : Δ → ℝ be mappings which have continuous partial derivatives of second order in Δ. Then there exists (𝜂1 , 𝜉1 ) and (𝜂2 , 𝜉2 ) in Δ such that
𝛶 (𝜓1 )
=
𝛶 (𝜓2 )
𝜕 2 𝜓1 (𝜂1 ,𝜉1 )
𝜕x2
𝜕 2 𝜓2 (𝜂1 ,𝜉1 )
(3.3)
𝜕x2
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and
𝛶 (𝜓1 )
=
𝛶 (𝜓2 )
𝜕 2 𝜓1 (𝜂2 ,𝜉2 )
𝜕y 2
𝜕 2 𝜓2 (𝜂2 ,𝜉2 )
(3.4)
.
𝜕y 2
Proof We define the mapping P: Δ → ℝ such that
P = k1 𝜓1 − k2 𝜓2 ,
where k1 = 𝛶 (𝜓2 ) and k2 = 𝛶 (𝜓1 ).
Using Theorem 4 with f = P, we have
{
k1
2𝛶 (P) = 0 =
𝜕 2 𝜓1
𝜕x2
− k2
𝜕 2 𝜓2
𝜕x2
}
𝛶 (𝜑)
and
{
k1
2𝛶 (P) = 0 =
𝜕 2 𝜓1
𝜕y 2
− k2
𝜕 2 𝜓2
𝜕y 2
}
𝛶 (𝜑).
Since 𝛶 (𝜑) ≠ 0, we have
k2
k1
𝜕 2 𝜓1 (𝜂1 ,𝜉1 )
=
𝜕x2
𝜕 2 𝜓2 (𝜂1 ,𝜉1 )
𝜕x2
and
k2
k1
𝜕 2 𝜓1 (𝜂2 ,𝜉2 )
=
𝜕y 2
𝜕 2 𝜓2 (𝜂2 ,𝜉2 )
𝜕y
,
2
which are equivalent to required results. ✷
Funding
The authors received no direct funding for this research.
Author details
Atiq Ur Rehman1
E-mail: atiq@mathcity.org
ORCID ID: http://orcid.org/0000-0002-7368-0700
Muhammad Mudessir1
E-mail: mudessir001@gmail.com
Hafiza Tahira Fazal2
E-mail: tahiramalik1230@gmail.com
Ghulam Farid1
E-mails: faridphdsms@hotmail.com, ghlmfarid@ciit-attock.
edu.pk
1
Department of Mathematics, COMSATS Institute of
Information Technology, Attock, Pakistan.
2
Department of Mathematics, University of Lahore, Sargodha
Campus, Pakistan.
Citation information
Cite this article as: Petrović’s inequality on coordinates
and related results, Atiq Ur Rehman, Muhammad
Mudessir, Hafiza Tahira Fazal & Ghulam Farid, Cogent
Mathematics (2016), 3: 1227298.
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