EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 4, 2020, 814-829
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

New refinement of Niezgoda’s inequality with
applications to Ky Fan inequality

Sadia Chanan1,∗, Asif R. Khan1

1 Department of Mathematics, University of Karachi, University Road, Karachi-75270,
Pakistan

Abstract. The aim of this article is to give the refinement of Niezgoda’s inequality with its
applications to Ky Fan inequality and cyclic mixed symmetric means.

2020 Mathematics Subject Classifications: 26A51, 39B62, 26D15, 26D20, 26D99
Key Words and Phrases: Convex functions, Niezgoda’s inequality, Ky Fan Inequality, cyclic
mixed symmetric means

1. Introduction and Preliminaries

Jensen’s inequality for convex functions is one of the most celebrated inequality in
Mathematics and Statistics. Due to its high importance there are given numerous variants,
generalizations and refinements of Jensen’s inequalities (for reference see [8, 9, 12, 13, 29]).
We also adduce to [25] and [28] for detailed discussion on Jensen’s inequality and for some
remarks on literature and history of the topic.

A variant of Jensen’s inequality named as Jensen-Mercer inequality was established by
Mercer [24] given as follows:

Theorem 1. Let x1 ≤ x2 ≤ · · · ≤ xn and let w1, w2, . . . , wn be nonnegative real numbers

such that

n∑
i=1

wi = 1. If φ is a convex function defined on an interval containing all xi’s

for 1 ≤ i ≤ n. Then

φ

(
x1 + xn −

n∑
i=1

wixi

)
≤ φ (x1) + φ (xn)−

n∑
i=1

wiφ (xi) . (1)

Now, we recall a prerequisite concept of majorization from [23].
Let x = (x1, . . . , xm) and y = (y1, . . . , ym) denote two m-tuples and x[1] ≥ · · · ≥

x[m], y[1] ≥ · · · ≥ y[m] be their ordered components.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i4.3776

Email addresses: sadiachanankhan@yahoo.com (S. Chanan), asifrk@uok.edu.pk (A. R. Khan)

https://www.ejpam.com 814 c© 2020 EJPAM All rights reserved.



S. Chanan, A. R. Khan / Eur. J. Pure Appl. Math, 13 (4) (2020), 814-829 815

Definition 1. For x, y ∈ Rm,

x ≺ y if


k∑
i=1

x[i] ≤
k∑
i=1

y[i] , k ∈ {1, . . . ,m− 1},

m∑
i=1

x[i] =
m∑
i=1

y[i]

when x ≺ y, x is said to be majorized by y or y majorizes x.

In the book [10] we find a very power result namely majorization theorem (see also
[23]).

Theorem 2. Let x, y ∈ Rn then following inequality is true for all continuous convex
functions φ : R→ R,

n∑
i=1

φ (xi) ≤
n∑
i=1

φ (yi)

if and only if x ≺ y.

We now define an extension of Jensen-Mercer inequality which is referred as Niezgoda’s
inequality by Niezgoda [26]. For recent work on Niezgoda inequality we refer the reader
[1, 15–17, 27].

Theorem 3. Suppose that a be an m-tuple such that ai ∈ J and a n ×m matrix X =
(xj) = (xij) with xij ∈ J for all i ∈ {1, . . . , n} and j ∈ {1, . . . ,m}.

If a majorizes each row of X, that is,

xi. = (xi1, . . . , xim) ≺ (a1, . . . , am) = a for each i ∈ {1, . . . , n},

then for a continuous convex function φ on J following inequality holds.

φ

 m∑
j=1

aj −
m−1∑
j=1

n∑
i=1

wixij

 ≤ m∑
j=1

φ(aj)−
m−1∑
j=1

n∑
i=1

wiφ(xij), (2)

with wi ≥ 0 such that
∑n

i=1wi = 1.

Especially, the inequality stated below is also valid for wi = 1
n , i ∈ {1, . . . , n}

φ

 m∑
j=1

aj −
1

n

m−1∑
j=1

n∑
i=1

xij

 ≤ m∑
j=1

φ(aj)−
1

n

m−1∑
j=1

n∑
i=1

φ(xij). (3)

The cyclic refinement of the Jensen’s inequality in paper [3] is given as follows:



S. Chanan, A. R. Khan / Eur. J. Pure Appl. Math, 13 (4) (2020), 814-829 816

Theorem 4. Let φ : I → R be a convex function and I be an interval in R,x =
(x1, . . . , xn) ∈ In and λ = (λ1, . . . , λn) be a nonnegative n-tuple such that

∑k
i=1 λi = 1 for

some k, 2 ≤ k ≤ n. Then

φ

(
1

n

n∑
i=1

xi

)
≤ 1

n

n∑
i=1

φ

k−1∑
j=0

λj+1xi+j

 ≤ 1

n

n∑
i=1

φ (xi) , (4)

where i+ j means i+ j − n in case of i+ j > n.

The cyclic refinement of Jensen-Mercer inequality in paper [5] is defined as follows:

Theorem 5. Let I ⊂ R be an interval, x = (x1, . . . , xn) ∈ In such that
(
c+ d−

∑k−1
j=0 λj+1xi+j

)
∈

I and λ = (λ1, . . . , λn) be a positive n-tuple such that
∑k

i=1 λi = 1 for some k, 2 ≤ k ≤ n,
then for convex function φ : I → R, [c, d] ⊂ I, following inequalities hold:

φ

(
c+ d−

n∑
i=1

wixi

)

≤
n∑
i=1

wiφ

c+ d−
k−1∑
j=0

λj+1xi+j


≤ φ (c) + φ (d)−

n∑
i=1

wiφ (xi) , (5)

where i+ j means i+ j − n in case of i+ j > n.

In this article we are going to use some of the following assumptions:

• (C1): Let φ : J → R be a convex function.

• (C2): Let a be a m-tuple such that aj ∈ Jn and a n×m matrix X = (xij) ∈ Jn,∀i ∈
{1, . . . , n} and ∀j ∈ {1, . . . ,m} such that (xi1+k, . . . , xim+k) = (xi1, . . . , xim),∀i ∈
{1, . . . , n}) and λ : (λ1, . . . , λn) ba a n-tuple such that

∑l
k=1 λk = 1, l ∈ {2, . . . , n}.

Moreover, wi’s are non-negative real weights for 1 ≤ i ≤ n such that
∑n

i=1wi = 1

• (C3): Let φ, ψ : J → R be continuous and strictly monotone functions.

Under the assumptions stated above it should be noted that m∑
j=1

aj −
m−1∑
j=1

l−1∑
k=0

λk+1xij+k

 ∈ J.
The aim of this paper is to present new refinement of Theorem 3. In main result

section, we will give refinement for weighted version of Niezgoda’s inequality, then we will
define its special case for equal weights. In application section, with the help of main
results we will give refinements of Ky Fan and arithmetic-geometric means inequalities
and their related results. We also define cyclic mixed symmetric means, power mean and
generalized quasi-arithmetic means and study their properties. We follow the techniques
given in [3]. The final section gives suggestions for further work and future ideas.



S. Chanan, A. R. Khan / Eur. J. Pure Appl. Math, 13 (4) (2020), 814-829 817

2. Main Result

Theorem 6. Let the assumptions stated in Theorem 3 be true. In addition we suppose
that the assumptions given in (C1) and (C2) are also valid. Then we have

φ

 m∑
j=1

aj −
m−1∑
j=1

n∑
i=1

wixij


≤

n∑
i=1

wiφ

 m∑
j=1

aj −
m−1∑
j=1

l−1∑
k=0

λk+1xij+k


≤

m∑
j=1

φ(aj)−
m−1∑
j=1

n∑
i=1

wiφ (xij) . (6)

Proof. To prove first inequality of (6), since φ is a convex function and m∑
j=1

aj −
m−1∑
j=1

l−1∑
k=0

λk+1xij+k

 ∈ J,
therefore by Jensen’s Inequality,

n∑
i=1

wiφ

 m∑
j=1

aj −
m−1∑
j=1

l−1∑
k=0

λk+1xij+k


≥ φ

 n∑
i=1

wi

m∑
j=1

aj −
n∑
i=1

m−1∑
j=1

l−1∑
k=0

wiλk+1xij+k


= φ

 m∑
j=1

aj −

(
l∑

k=1

λk

)
m−1∑
j=1

n∑
i=1

wixij


= φ

 m∑
j=1

aj −
m−1∑
j=1

n∑
i=1

wixij

 .

On the other hand, to prove second inequality of (6), we consider following expression

φ

 m∑
j=1

aj −
m−1∑
j=1

l−1∑
k=0

λk+1xij+k


for fixed i ∈ {1, 2, . . . , n} and proceed as follows:

φ

 m∑
j=1

aj −
m−1∑
j=1

l−1∑
k=0

λk+1xij+k





S. Chanan, A. R. Khan / Eur. J. Pure Appl. Math, 13 (4) (2020), 814-829 818

= φ

 m∑
j=1

aj −
m−1∑
j=1

l−1∑
k=0

λk+1xij


= φ

 m∑
j=1

aj −
m−1∑
j=1

l∑
k=1

λkxij


= φ

 m∑
j=1

aj −

(
l∑

k=1

λk

)
m−1∑
j=1

xij


= φ

 m∑
j=1

aj −
m−1∑
j=1

xij


Using majorization property we have

φ

 m∑
j=1

aj −
m−1∑
j=1

xij

 = φ (xim) ≤
m∑
j=1

φ (aj)−
m−1∑
j=1

φ (xij)

Or we can write,

φ

 m∑
j=1

aj −
m−1∑
j=1

l−1∑
k=0

λk+1xij+k

 ≤ m∑
j=1

φ (aj)−
m−1∑
j=1

φ (xij) . (7)

Now multiplying inequality (7) with wi and summing over i from 1 to n we get our required
result.

Corollary 1. Under the assumptions of Theorem 6 and for wi = 1
n , i ∈ {1, . . . , n}, we

have

φ

 m∑
j=1

aj −
1

n

m−1∑
j=1

n∑
i=1

xij


≤ 1

n

n∑
i=1

φ

 m∑
j=1

aj −
m−1∑
j=1

l−1∑
k=0

λk+1xij+k


≤

m∑
j=1

φ(aj)−
1

n

m−1∑
j=1

n∑
i=1

φ (xij) . (8)

Remark 1. If we set m = 2, a1 = c, a2 = d and xi1 = xi for i ∈ {1, . . . , n}, then
Theorem 5 will become special case of Theorem 6.



S. Chanan, A. R. Khan / Eur. J. Pure Appl. Math, 13 (4) (2020), 814-829 819

3. Refinement of the Ky Fan inequality

Throughout this section, let the assumptions stated in Theorem 6 be valid with 0 <
c < d. We define generalized (or modified) arithmetic, geometric and harmonic mean
respectively as follow (for general discussion on mean and related inequalities we refer
[4]):

Ân =
m∑
j=1

aj −
m−1∑
j=1

n∑
i=1

wixij ,

Ĝn =

m∏
j=1

aj

m−1∏
j=1

n∏
i=1

(xij)
wi

,

Ĥn =

 m∑
j=1

(aj)
−1 −

m−1∑
j=1

n∑
i=1

wi(xij)
−1

−1 .
Also for xij ∈ (0, 12 ], we define arithmetic, geometric and harmonic means as follows:

Â′n =

m∑
j=1

(1− aj)−
m−1∑
j=1

n∑
i=1

wi (1− xij) ,

Ĝ′n =

m∏
j=1

(1− aj)

m−1∏
j=1

n∏
i=1

((1− xij)wi

,

Ĥ ′n =

 m∑
j=1

(1− aj)−1 −
m−1∑
j=1

n∑
i=1

wi (1− xij)−1
−1 .

We also define new notations Â(λ; x) and Ĝ(λ; x) as under:

Â(λ; x) =
m∑
j=1

aj −
m−1∑
j=1

l−1∑
k=0

λk+1xij+k,

Ĝ(λ; x) =

m∏
j=1

aj

m−1∏
j=1

l−1∏
k=0

(xij+k)
λk+1

.



S. Chanan, A. R. Khan / Eur. J. Pure Appl. Math, 13 (4) (2020), 814-829 820

Also for xij ∈ (0, 12 ], we define

Â′(λ; x) =
m∑
j=1

(1− aj)−
m−1∑
j=1

l−1∑
k=0

λk+1(1− xij+k),

Ĝ′(λ; x) =

m∏
j=1

(1− aj)

m−1∏
j=1

l−1∏
k=0

(1− xij+k)λk+1

.

Now, we present the refinement of the Ky Fan type inequality. For Ky Fan inequality
and related results, see [2], [6] and [5] and references given therein.

Theorem 7. Let assumptions stated in Theorem 6 be true. Then following inequality
holds:

Ân

Â′n
≤

n∏
i=1

(
Â(λ,x)

Â′(λ,x)

)wi

≤ Ĝn

Ĝ′n
.

Proof. By applying the convex function φ(x) = ln
(

x
1−x

)
for all x ∈ (0, 12 ], to the

inequality (6), we get,

ln

( ∑m
j=1 aj −

∑m−1
j=1

∑n
i=1wixij

1−
∑m

j=1 aj +
∑m−1

j=1

∑n
i=1wixij

)

≤
n∑
i=1

wi ln

( ∑m
j=1 aj −

∑m−1
j=1

∑l−1
k=0 λk+1xij+k

1−
∑m

j=1 aj +
∑m−1

j=1

∑l−1
k=0 λk+1xij+k

)

≤
m∑
j=1

ln

(
ai

1− aj

)
−
m−1∑
j=1

n∑
i=1

wi ln

(
xij

1− xij

)
consequently,

ln

(
Ân

Â′n

)
≤ ln

n∏
i=1

( ∑m
j=1 aj −

∑m−1
j=1

∑l−1
k=0 λk+1xij+k

1−
∑m

j=1 aj +
∑m−1

j=1

∑l−1
k=0 λk+1xij+k

)wi

≤ ln

(
Ĝn

Ĝ′n

)

Finally, we obtain (
Ân

Â′n

)
≤

n∏
i=1

(
Â(λ,x)

Â′(λ,x)

)wi

≤

(
Ĝn

Ĝ′n

)
,

which completes the proof.



S. Chanan, A. R. Khan / Eur. J. Pure Appl. Math, 13 (4) (2020), 814-829 821

Remark 2. For wi = 1
n , we obtain the special case of Theorem 7 as follows:

An
A′n
≤

n∏
i=1

(
A(λ,x)

A′(λ,x)

) 1
n

≤ Gn
G′n

,

where

An =
m∑
j=1

aj −
1

n

m−1∑
j=1

n∑
i=1

xij ,

Gn =

∏m
j=1 ajm−1∏

j=1

n∏
i=1

xij

 1
n

,

and

A′n =
m∑
j=1

(1− aj)−
1

n

m−1∑
j=1

n∑
i=1

(1− xij) ,

G′n =

∏m
j=1(1− aj)m−1∏

j=1

n∏
i=1

(1− xij)

 1
n

.

Now, we present refinement of arithmetic-geometric mean type inequality as follows:

Corollary 2. Let the assumptions stated in Theorem 6 be true. Then following inequality
holds:

Ân ≥
n∏
i=1

(
Â(λ,x)

)wi

≥ Ĝ′n.

Proof. By applying the convex function φ (x) = − ln (x) , x ∈ (0, 12 ] to Theorem 6 we
obtain required result.

Now, we present refinement of harmonic and geometric means inequality as follows:

Corollary 3. Let the assumptions stated in Theorem 6 be true. Then following inequalities
hold: (

Ĝ′n

)−1
≤

n∑
i=1

wi

(
Ĝ′(λ,x)

)−1
≤
(
Ĥ ′n

)−1
.



S. Chanan, A. R. Khan / Eur. J. Pure Appl. Math, 13 (4) (2020), 814-829 822

Proof. By applying the convex function φ (x) = exp (x) , x ∈ (0, 12 ] to Theorem 6 and

by replacing aj and xij by ln
(

1
1−aj

)
and ln

(
1

1−xij

)
respectively, we get

exp

 m∑
j=1

ln

(
1

1− aj

)
−
m−1∑
j=1

n∑
i=1

wi ln

(
1

1− xij

)
≤

n∑
i=1

wi exp

 m∑
j=1

ln

(
1

1− aj

)
−
m−1∑
j=1

l−1∑
k=0

λk+1 ln

(
1

1− xij+k

)
≤

m∑
j=1

exp

(
ln

(
1

1− aj

))
−
m−1∑
j=1

n∑
i=0

wi exp

(
ln

(
1

1− xij

))
,

Consequently,

exp

(
− ln

( ∏m
j=1(1− aj)∏n

i=1

∏m−1
j=1 (1− xij)wi

))

≤
n∑
i=1

wi exp

− ln

 ∏m
j=1(1− aj)∏m−1

j=1

∏l−1
k=0

(
1

1−xij+k

)λj+1




≤

 m∑
j=1

(
1

(1− aj)

)
−
m−1∑
j=1

n∑
i=1

wi

(
1

1− xij

) ,

which is equivalent to

(
Ĝ′n

)−1
≤

n∑
i=1

wi

(
Ĝ′ (λ,x)

)−1
≤

 m∑
j=1

(
1

(1− aj)

)
−
m−1∑
j=1

n∑
i=1

wi

(
1

1− xij

) ,

finally, we obtain (
Ĝ′n

)−1
≤

n∑
i=1

wi

(
Ĝ′ (λ,x)

)−1
≤
(
Ĥ ′n

)−1
,

which completes the proof.

We would also establish refinements related to arithmetic-harmonic means inequalities
as follows:



S. Chanan, A. R. Khan / Eur. J. Pure Appl. Math, 13 (4) (2020), 814-829 823

Corollary 4. Let the assumptions stated in Theorem 6 be true. Then following inequalities
hold:

1

Ân
≤

n∑
i=1

wi

(
1

Â(λ,x)

)
≤ 1

Ĥn

, (9)

1

Â′n
≤

n∑
i=1

wi

(
1

Â′(λ,x)

)
≤ 1

Ĥ ′n
. (10)

Proof. By applying convex function f(x) = 1
x , x ∈ (0, 12 ] to Theorem 6 we get inequality

(9). Similarly, by using convex function f(x) = 1
1−x , x ∈ (0, 12 ] to Theorem 6 we get

inequality (10).

We establish a refinement of the difference of the arithmetic and harmonic means.

Corollary 5. Let the assumptions stated in Theorem 6 be true. Then following inequalities
hold:

1

Ân
− 1

Â′n
≤

n∑
i=1

wi

(
1

Â(λ,x)
− 1

Â′(λ,x)

)
≤ 1

Ĥn

− 1

Ĥ ′n
.

Proof. By applying convex function f(x) = 1
x −

1
1−x , x ∈ (0, 12 ] to Theorem 6 we obtain

required result.

4. Cyclic mixed symmetric means

The Jensen’s inequality and Jensen-Mercer inequality are much fertile to study about
mixed means (see [11] and [21]). Let the assumptions stated in Theorem 6 be true. Then
we define power mean of the order r ∈ R, for positive n-tuple x as follows:

M̂r (xij , . . . , xij+l−1;λ1, . . . , λl)

=



∑m
j=1 a

r
j −

m−1∑
j=1

l−1∑
k=0

λk+1x
r
ij+k

 1
r

, r 6= 0,

m∏
j=1

aj

m−1∏
j=1

l−1∏
k=0

(xij+k)
λk+1

, r = 0,



S. Chanan, A. R. Khan / Eur. J. Pure Appl. Math, 13 (4) (2020), 814-829 824

and cyclic mixed symmetric means corresponding to (6) is given as:

M̂r,s(x, λ) =



(
n∑
i=1

wiM̂
s
r (xij , · · · , xij+l−1, λ1, · · · , λl)

) 1
s

, s 6= 0,

(
n∏
i=1

M̂r (xij , · · · , xij+l−1, λ1, · · · , λl)

)wi

, s = 0.

The standard power mean of order r ∈ R for the positive n-tuple x are defined as
follows:

M̂r (x) =



∑m
j=1 a

r
j −

m−1∑
j=1

n∑
i=1

wix
r
ij

 1
r

, r 6= 0,∏m
j=1 aj

m−1∏
j=1

n∏
i=1

(xij)
wi

, r = 0.

Corollary 6. For r ≤ 1 and by considering the assumptions stated in (C1) for positive
m-tuple a and x, the following inequality hold.

M̂r (x) ≤ M̂r (x, λ) ≤ Ân. (11)

For r ≥ 0, the inequality (11) is reversed.

Proof. For r ≤ 1, r 6= 0, by applying the convex function φ(x) = x
1
r to Theorem 6 and

replacing aj and xij with arj and xrij respectively and for r = 0 applying convex function
φ(x) = exp(x) to the Theorem 6, replacing ajand xij with ln aj and lnxij respectively, we
obtain 11.

If r ≥ 1, then the function φ(x) = x
1
r is concave, so the inequalities in (11) is reversed.

Now, we define the bounds for power mean and cyclic mixed symmetric means as
follows:

Corollary 7. Let r, s ∈ R such that r ≤ s and considering the assumption stated in (C1)
for positive n-tuple x, following inequalities hold.

M̂r (x) ≤ M̂r,s (x, λ) ≤ M̂s (x) .

(12)

Proof. Let r, s 6= 0. By applying Theorem 6 for convex function φ (x) = x
s
r , x > 0 and

by replacing aj and positive n-tuple x by arj and (xr) respectively, and then raising the

power 1
s we get,

M̂r (x) ≤ M̂r,s (x, λ) ≤ M̂s (x) .



S. Chanan, A. R. Khan / Eur. J. Pure Appl. Math, 13 (4) (2020), 814-829 825

For s = 0 or r = 0, we obtain the required result by applying appropriate limits.

Let φ : [a, b] → R be a continuous and strictly monotone function then cyclic quasi-
arithmetic means are defined as

M̂φ (x) := φ−1

 m∑
j=1

φ (aj)−
m−1∑
j=1

n∑
i=1

wiφ (xij)

 (13)

.

Let φ, ψ : [a, b] → R be strictly monotonic and continuous functions and under the
assumptions stated in (A1) and (A2), we define generalized means with respect to (6) as
follows:

Mφ,ψ (x, λ)

= φ−1

φoψ−1
 m∑
j=1

aj

− n∑
i=1

wi
(
φoψ−1

)m−1∑
j=1

l−1∑
k=0

λk+1ψ (xij+k)

 . (14)

Now, we establish the relation among generalized means and quasi-arithmetic means as
follows:

Corollary 8. Let assumptions (C1) and (C3) be true. Then

M̂ψ (x) ≤ M̂φ,ψ (x, λ) ≤ M̂φ (x) ,

if either φoψ−1 is convex and ψ is strictly increasing or φoψ−1 is concave and ψ is strictly
decreasing.

Proof. By applying Theorem 6 to the convex function φoψ−1 and replacing aj by ψ (aj)
and n-tuples x by ψ (x), we get

φoψ−1

 m∑
j=1

ψ(aj)−
m−1∑
j=1

n∑
i=1

wiψ(xij)


≤

n∑
i=1

wiφoψ
−1

 m∑
j=1

ψ(aj)−
l−1∑
k=0

m−1∑
j=1

λk+1ψ(xij+k)


≤

m∑
j=1

φoψ−1 (ψ(aj))−
m−1∑
j=1

n∑
i=1

wiφoψ
−1 (ψ(xij)) ,

consequently,

φoψ−1

 m∑
j=1

ψ(aj)−
m−1∑
j=1

n∑
i=1

wiψ(xij)





S. Chanan, A. R. Khan / Eur. J. Pure Appl. Math, 13 (4) (2020), 814-829 826

≤
n∑
i=1

wiφoψ
−1

 m∑
j=1

ψ(aj)−
l−1∑
k=0

m−1∑
j=1

λk+1ψ(xij+k)


≤

m∑
j=1

φ(aj)−
m−1∑
j=1

n∑
i=1

wiφ(xij),

by applying φ−1, we get

φ−1φoψ−1

 m∑
j=1

ψ(aj)−
m−1∑
j=1

n∑
i=1

wiψ(xij)


≤ φ−1

 n∑
i=1

wiφoψ
−1

 m∑
j=1

ψ(aj)−
l−1∑
k=0

m−1∑
j=1

λk+1ψ(xij+k)


≤ φ−1

 m∑
j=1

φ(aj)−
m−1∑
j=1

n∑
i=1

wiφ(xij)

 ,

and after some simplification we obtained required result.

5. Further Results and Future Work

Under the assumptions of Theorem 6, we define two positive linear functionals as

ϕ1 (x, λ, φ) =

m∑
j=1

φ (aj)−
m−1∑
j=1

n∑
i=1

wiφ (xij)

−
n∑
i=1

wiφ

 m∑
j=1

aj −
l−1∑
k=0

m−1∑
j=1

λk+1xij+k


ϕ2 (x, λ, φ) =

n∑
i=1

wiφ

 m∑
j=1

aj −
l−1∑
k=0

m−1∑
j=1

λk+1xij+k


−

 m∑
j=1

φ(aj)−
m−1∑
j=1

n∑
i=1

wiφ (xij)


We can state different results for these two functionals defined above which may be listed
as follows:

(i) We can state Lagrange type and Cauchy type mean value theorems and results
related to n−exponential and logarithmic convexity by using similar techniques as
stated in [3] and [14].



REFERENCES 827

(ii) We can also state number of applications by using method of article [22].

(iii) We can state further results using technique of index set function with series of refine-
ments and plenty of applications including Rado and Popovicu series of inequality
by using method of [18] and [19].

(iv) We can also prove all inequalities in reverse direction by considering concave function
instead of convex function by using simple relation: f is concave iff and −f is convex.

Here we state some future ideas for interested readers:

(i) One can also work on similar results as stated in this article for generalized convex
functions including functions with nondecreasing increments and functions with non-
decreasing increments of convex type see for example [1], [7] and [20].

(ii) One can also state similar results as stated in this article for arbitrary real numbers
(not only non-negative real numbers) for example by working with assumptions of
Jensen-Steffensen inequality.

(iii) One can also try its integral version as well.

References

[1] M. Maqsood Ali, Asif R. Kha, Inam Ullah Khan, and Sumayyah Saadi. Improve-
ment of Jensen and Levinson Type Inequalities for Functions with Nondecreasing
Increments. Global J. Pure Appl. Math., 6(15):945–970, 2019.

[2] H. Alzer. The inequality of ky fan’s and related results. Acta App. Math., 38:305–354,
1995.

[3] I. Brentic, K. A. Khan, and J. Pečarić. Refinements of Jensen’s Inequality with
applications to cyclic mixed symmetric means and Cauchy means. J. Math. Inequal,
4(9):1309–1321, 2015.

[4] P. S. Bullen, D. S. Mitrinović, and P. M. Vasić. Means and Their Inequalities. Reidel,
Dordrech, 1988.

[5] S. Chanan and Asif R. Khan. On Some refinements of Jensen-Mercer inequality with
applications. submitted.

[6] S. Chanan, Asif R. Khan, S. Ahmed, and N. Raisat. Generalizations of Ky Fan
inequality and related results. J. Inequal. and Special Functions, 10:123–142, 2019.

[7] S. Chanan, Asif R. Khan, and Inam Ullah Khan. Gabler inequality for functions with
nondecreasing increments of convex type. Adv. Inequal. Appl., 10(3), 2020.

[8] M. Klariči´ c Bakula and J. Pečarić. On the Jensens inequality for convex functions on
the co-ordinates in a rectangle from the plane. Taiwanese J. Math., 10(5):1271–1292,
2006.



REFERENCES 828

[9] S. S. Dragomir. A new refinement of Jensen’s inequality in linear spaces with appli-
cations. Mathematical and Computer Modelling, 52:1497–1505, 2010.

[10] G. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities. Cambridge University
Press, Cambridge,, 1978.

[11] L. Horv́ath, K. A. Khan, and J. Pečarić. Further Refinement of Results about Mixed
Symmetric Means and Cauchy Means. Advances in Inequalities and Applications,
1:12–32, 2012.

[12] S. Hussain and J. Pečarić. An improvement of Jensens inequality with some applica-
tions. Asian-European J. Math., 2:85–94, 2009.

[13] Asif R. Khan, Josip Pečarić, and Mirna Rodić Lipanović. nExponential Convexity
for Jensen-Type Inequalities. J. Math. Inequal., 7:313–335, 2013.

[14] Asif R. Khan, Josip Pečarić, and Marjan Praljak. Popoviciu type inequalities for
n-convex functions via extension of Montgomery identity. An. Şt. Univ. Ovidius
Constanţa, 24:161–188, 2016.

[15] Asif R. Khan, Josip Pečarić, and Marjan Praljak. A Note on Generalized Mercer’s
Inequality. Bull. Malays. Math. Sci. Soc., 2017:1–11, 2017.

[16] Asif R. Khan and Inam Ullah Khan. Some remarks on Niezgoda’s extension of Jensen-
Mercer Inequality. Adv. Inequal. Appl., 12:1–11, 2016.

[17] Asif R. Khan and Inam Ullah Khan. An Extension of Jensen-Mercer Inequality for
Functions with Nondecreasing Increments. J. Inequal. Special Funct, 10:1–15, 2019.

[18] Asif R. Khan, Inam Ullah Khan, and Shahid Sultan Ali Ramji. Generalization and
Refinements of Jensen-Mercer Inequality with Applications. J. Math. Inequal., to
appear.

[19] Asif R. Khan and Sumayyah Saadi. Generalized and Refinements of Generalized
Niezgoda-type Inequality for Similary Separable Vectors with Applications. FILO-
MAT.

[20] Asif R. Khan and Sumayyah Saadi. Generalized Jensen-Mercer Inequality for Func-
tions with Nondecreasing Increments. Abs. and Appl. Anal., 2016:12, 2016.

[21] K. A. Khan, J. Pečarić, and I. Peric. Differences of weighted mixed symmetric means
and related results. Journal of Inequalities and Applications, 2010, 2010.

[22] M. Adil Khan and J. Peˇ carić Asif R. Khan. On the refinements of Jensen-Mercer’s
inequality. Rev. Anal. Numer. Theor. Approx., 41:62–81, 2012.

[23] A. W. Marshall, I. Olkin, and B. C. Arnold. Inequalities: Theory of majorization and
its applications (Second Edition). Springer Series in Statistics, New York, 2011.



REFERENCES 829

[24] A. Mcd. Mercer. A variant of Jensen’s inequality. J. Ineq. Pure and Appl. Math., 4,
2013.

[25] D. S. Mitrinović, J. E. Pečarić, and A. M. Fink. Classical and new inequalities in
analysis. Kluwer Academic Publishers Group, Dordrecht, Dordrecht, 1993.

[26] M. Niezgoda. A generalization of Mercer’s result on convex functions. Nonlinear
Anal., 71:2771–2779, 2009.

[27] M. Niezgoda. A generalization of Mercer’s result on convex functions, II. Math.
Inequal. Appl., 18:1013–1023, 2015.

[28] J. Pečarić, F. Proschan, and Y. L. Tong. Convex functions, Partial Orderings and
Statistical Applications. Academic Press, New York, Academic Press, New York,
1992.

[29] J. Rooin. Some refinements of discrete Jensens inequality and some of its applications.
Nonlinear Functional Anal. Appl., 1:107–118, 2007.