A Note on Generalized Strongly p-Convex Functions of Higher Order


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(2) (2022), Pages 152-157 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: July 16, 2021  Reviewed: October 12, 2021 Accepted: October 18, 2021 
DOI: http://dx.doi.org/10.18860/ca.v7i1.12938  

A Note on Generalized Strongly p-Convex Functions of Higher 
Order 

Corina Karim*, Ekadion Maulana 

Mathematics Department, Faculty of Mathematics and Natural Sciences,  
Brawijaya University, Malang, Indonesia 

*Corresponding Author 
Email: co_mathub@ub.ac.id* 

rm.ekadion.m@gmail.com  

ABSTRACT 

Generalized strongly 𝑝-convex functions of higher order is a new concept of convex functions 
which introduced by Saleem et al. in 2020. The Schur type inequality for generalized strongly 𝑝-
convex functions of higher order also studied by them. This paper aims to revise Schur type 
inequality for generalized strongly 𝑝-convex functions of higher order in their paper. In order to 
revise it, we show that the contradiction was true. This paper showed that  Schur type inequality 
for generalized strongly 𝑝-convex functions of higher order previously is not valid and we give 
the correct Schur type inequality for generalized strongly 𝑝-convex functions of higher order. 

Keywords: Schur type inequality; 𝑝-Convex functions; Strongly convex of higher order 

INTRODUCTION 

Convexity is a basic notion in many branches of applied mathematics. Convexity 
is an important thing on Functional analysis, Geometry, Mathematical programming, 
Probability, and Statistics. On Functional analysis, convexity has intended to ensure 
existence and uniqueness of solutions of problems of Calculus of variations and optimal 
control. On Mathematical programming, convexity has intended to ensure convergence 
of optimization algorithms [1]. 

Convexity appears in ancient Greek Geometry. Archimedes (ca. 250 BC) used 
convexity on study of the area and arch length. Archimedes has been the first person 
who gave a definition of convexity, similar to the geometric definition which used till 
today, a set is said to be convex if it contains all line segments between each of its points 
[1]. 

Some geometric properties of convex sets and functions have studied before 
1960 by great mathematicians Hermann Minkowski and Werner Fenchel. In 1891, 
Minkowski proved that, in Euclidean space ℝ𝑛 , every compact convex set with center at 
the origin and volume greater than 2𝑛  contains at least one point with integer 
coordinates different from the origin [2]. Afterwards, in 1951, Werner Fenchel’s 
monograph stimulated the development of convexity theory. 

Several researchers have been considered for classical convexity such that some 
of these new concepts are based on an extension of the domain of convex functions or 

http://dx.doi.org/10.18860/ca.v7i1.12938
mailto:co_mathub@ub.ac.id
mailto:rm.ekadion.m@gmail.com


A Note on Generalized Strongly p-Convex Functions of  Higher Order 

Corina Karim 153 

sets to a generalized form[3-9]. Some examples of these new concepts are quasi-
convexity [10], exponential convexity [11, 12], logarithmical convexity [13], ℎ-convexity 
[14, 15], and 𝑝-convexity [3, 4, 16, 17]. 𝑝-Convex funtions and their properties was 
introduced by Zhang and Wan [16] in 2007. In 2018 Maden et al. [17] discussed strongly 
𝑝-convex funtions and Hermite-Hadamard inequality for it. In 2020, Saleem et al. [3, 4] 
discussed about generalized 𝑝-convex funtions and generalized strongly 𝑝-convex 
funtions of higher order. 

In [3] discussed definition and properties of generalized strongly 𝑝-convex 
functions of higher order also some of type inequalities which are Hermite-Hadamard, 
Fejér, and Schur. In Schur type inequality for generalized strongly 𝑝-convex funtions of 

higher order in [3] indirectly mentioned that 𝜙(𝑥3
𝑝

− 𝑥2
𝑝

) ≤ (𝑥3
𝑝

− 𝑥1
𝑝

)𝜙 (
𝑥3

𝑝
−𝑥2

𝑝

𝑥3
𝑝

−𝑥1
𝑝) for any 

𝑥3
𝑝

− 𝑥2
𝑝

, 𝑥3
𝑝

− 𝑥1
𝑝

∈ (0,1) and 𝑥3
𝑝

− 𝑥2
𝑝

< 𝑥3
𝑝

− 𝑥1
𝑝

 with 𝑞 > 0 and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞 . 
So, this paper revises the correction of Schur type inequality in [3]. 

 

METHODS 

 In this section, we discussed about some definitions which are implemented with 
generalized strongly 𝑝-convex functions of higher order. We also give examples of 
strongly 𝑝-convex funtions of higher order and its generalized.  
 
Definition 1. (See [1]) (Convex Set) 

Let 𝑋 ⊂ ℝ𝑛. 𝑋 is called to be convex if  
𝑡𝑥 + (1 − 𝑡)𝑦 ∈ 𝑋, (1) 

For any 𝑥, 𝑦 ∈ 𝑋 and 𝑡 ∈ [0,1]. 
 
Definition 2. (See [3]) ( 𝒑-Convex Set) 

Let 𝑋 ⊂ ℝ𝑛. 𝑋 is called to be 𝑝-convex if  

[𝑡𝑥𝑝 + (1 − 𝑡)𝑦𝑝]
1

𝑝 ∈ 𝑋, (2) 

 
for any 𝑥, 𝑦 ∈ 𝑋 and 𝑡 ∈ [0,1]. 
 
Definition 3. (See [18]) (q-Convex Uniform) 

Let 𝑋 be a Banach space and real number 𝑞 ≥ 2. Defined 𝛿𝑋 (𝜖) = inf {1 − ‖
𝑓+𝑔

2
‖ ; 𝑓, 𝑔 ∈

𝑋, ‖𝑓‖ ≤ 1, ‖𝑔‖ ≤ 1, ‖𝑓 − 𝑔‖ ≥ 𝜖}. 𝑋 is called to be 𝑝-convex uniform if there exists a 

constant 𝑐 > 0   such that 
𝛿𝑋 (𝜖) ≥ 𝑐𝜖

𝑞 , 
for 0 < 𝜖 ≤ 2.  
 
Lemma 1. (See [19]) 

Let 𝑋 be a 𝑞-convex uniform with 𝑞 ≥ 2, then there exists a constant 𝜇 > 0 such that 
‖𝑡𝑥 + (1 − 𝑡)𝑦‖𝑞 ≤ 𝑡‖𝑥‖𝑞 + (1 − 𝑡)‖𝑦‖𝑞 − 𝜇𝜙(𝑡)‖𝑥 − 𝑦‖𝑞 

for every 𝑥, 𝑦 ∈ 𝑋 and 𝑡 ∈ (0,1), where 
𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞 . 

Lemma 1 used to prove an example of strongly 𝑝-convex funtions of higher order and 
generalized strongly 𝑝-convex funtions of higer order. It has been proved in [19]. 
 



A Note on Generalized Strongly p-Convex Functions of  Higher Order 

Corina Karim 154 

Definition 4. (See [3]) (Strongly 𝒑-Convex Functions of Higher Order) 

Let 𝑋 be a 𝑝-convex set. A function 𝑓: 𝑋 → ℝ is called to be strongly 𝑝-convex functions of 
higher order if for any 𝑥, 𝑦 ∈ 𝑋 and 𝑡 ∈ [0,1], then 

𝑓 ([𝑡𝑥𝑝 + (1 − 𝑡)𝑦𝑝]
1

𝑝) ≤ 𝑡𝑓(𝑥) + (1 − 𝑡)𝑓(𝑦) − 𝜇𝜙(𝑡)‖𝑥𝑝 − 𝑦𝑝‖𝑞 , 

 
(3) 

with 𝜇 ≥ 0, 𝑞 > 0, and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞.   
 
Example: 
Let 𝑞 ≥ 2 and 𝑝 > 1. Defined 𝑋 = {𝑥 ∈ ℝ; |𝑥|𝑝 ∈ 𝑙𝑞 } and 𝜓: 𝑋 → ℝ, where 𝜓(𝑥) =
‖|𝑥|𝑝‖𝑞

𝑞
, then 𝜓 is strongly 𝑝–convex functions of higher order. 

 
Definition 5. (See [3]) (Generalized Strongly 𝒑-Convex Funtions of Higher Order) 

Let 𝑋 be a 𝑝-convex set. A function 𝑓: 𝑋 → ℝ is called to be strongly 𝑝-convex functions of 
higher order with respect to 𝜂: 𝐴 × 𝐴 → 𝐵, with 𝐴, 𝐵 ⊆ ℝ if for any 𝑥, 𝑦 ∈ 𝑋 and 𝑡 ∈ [0,1], 
then 

𝑓 ([𝑡𝑥𝑝 + (1 − 𝑡)𝑦𝑝]
1

𝑝) ≤ 𝑓(𝑦) + 𝜂(𝑓(𝑥), 𝑓(𝑦)) − 𝜇𝜙(𝑡)‖𝑥𝑝 − 𝑦𝑝‖𝑞 , (4) 

with 𝜇 ≥ 0, 𝑞 > 0, and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞. 
 
Example: 
Function 𝜓 in example of strongly 𝑝-convex functions of higher order with respect to 
𝜂(𝑥, 𝑦) = 𝑥 − 𝑦 is generalized strongly 𝑝-convex functions of higher order with rescpect 
to 𝜂. 

RESULTS AND DISCUSSION  

In this section, we discussed about revised Schur type inequality for generalized 
strongly 𝑝-convex functions of higher order. We showed that 𝜙(𝑥3

𝑝
− 𝑥2

𝑝
) ≤ (𝑥3

𝑝
−

𝑥1
𝑝

)𝜙 (
𝑥3

𝑝
−𝑥2

𝑝

𝑥3
𝑝

−𝑥1
𝑝) is not valid for any 𝑥3

𝑝
− 𝑥2

𝑝
, 𝑥3

𝑝
− 𝑥1

𝑝
∈ (0,1) and 𝑥3

𝑝
− 𝑥2

𝑝
< 𝑥3

𝑝
− 𝑥1

𝑝
 with 𝑞 > 0 and 

𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞 . 
 

Theorem 1.  (Schur type inequality) 

Let 𝑋 is 𝑝-convex set and 𝐴, 𝐵 ⊂ ℝ. Defined a generalized strongly 𝑝-convex function of 
higher order with respect to 𝜂(∙,∙): 𝐴 × 𝐴 → 𝐵 with 𝜇 ≥ 0 and 𝑝 > 0, then for every 
𝑥1, 𝑥2, 𝑥3 ∈ 𝑋 such that 𝑥1 < 𝑥2 < 𝑥3 and 𝑥3

𝑝
− 𝑥1

𝑝
, 𝑥3

𝑝
− 𝑥2

𝑝
, 𝑥2

𝑝
− 𝑥1

𝑝
∈ (0,1), then the 

following inequality hold 

𝑓(𝑥3)(𝑥3
𝑝

− 𝑥1
𝑝

) − 𝑓(𝑥2)(𝑥3
𝑝

− 𝑥1
𝑝

) + (𝑥3
𝑝

− 𝑥2
𝑝

)𝜂(𝑓(𝑥1), 𝑓(𝑥3))

− (𝑥3
𝑝

− 𝑥1
𝑝

)𝜇𝜙 (
𝑥3

𝑝
− 𝑥2

𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝) ‖𝑥3

𝑝
− 𝑥1

𝑝
‖

𝑞
≥ 0. 

 

(5) 

Proof: 
Let 𝑥1, 𝑥2, 𝑥3 ∈ 𝑋 with some criterion which are 𝑥1 < 𝑥2 < 𝑥3 and 𝑥3

𝑝
− 𝑥1

𝑝
, 𝑥3

𝑝
− 𝑥2

𝑝
, 𝑥2

𝑝
−

𝑥1
𝑝

∈ (0,1). Based on those criteria, then we have 

0 < 𝑥3
𝑝

− 𝑥2
𝑝

< 𝑥3
𝑝

− 𝑥1
𝑝

 and 0 < 𝑥2
𝑝

− 𝑥1
𝑝

< 𝑥3
𝑝

− 𝑥1
𝑝

. (6) 
Based on (6), then we have 



A Note on Generalized Strongly p-Convex Functions of  Higher Order 

Corina Karim 155 

𝑥3
𝑝

− 𝑥2
𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝 ,

𝑥2
𝑝

− 𝑥1
𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝 ∈ (0,1). (7) 

It’s clear that 
𝑥3

𝑝
− 𝑥1

𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝 = 1 ⟺

𝑥3
𝑝

− 𝑥1
𝑝

− 𝑥2
𝑝

+ 𝑥2
𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝 = 1 ⇔

(𝑥3
𝑝

− 𝑥2
𝑝

) + (𝑥2
𝑝

− 𝑥1
𝑝

)

𝑥
3
𝑝

− 𝑥
1
𝑝 = 1

⇔
𝑥3

𝑝
− 𝑥2

𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝 +

𝑥2
𝑝

− 𝑥1
𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝 = 1. 

 

(8) 

Choose 𝑡 =
𝑥3

𝑝
−𝑥2

𝑝

𝑥3
𝑝

−𝑥1
𝑝, so from (8) can get 

𝑡 +
𝑥2

𝑝
− 𝑥1

𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝 = 1 ⇔

𝑥2
𝑝

− 𝑥1
𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝 = 1 − 𝑡 ⇔ 𝑥2

𝑝
− 𝑥1

𝑝
= (1 − 𝑡)(𝑥3

𝑝
− 𝑥1

𝑝
) ⇔ 𝑥2

𝑝
− 𝑥1

𝑝

= (1 − 𝑡)𝑥3
𝑝

− (1 − 𝑡)𝑥1
𝑝

⇔ 𝑥2
𝑝

− 𝑥1
𝑝

= (1 − 𝑡)𝑥3
𝑝

− 𝑥1
𝑝

+ 𝑡𝑥1
𝑝

⇔ 𝑥2
𝑝

= 𝑡𝑥1
𝑝

+ (1 − 𝑡)𝑥3
𝑝

⇔ 𝑥2 = (𝑡𝑥1
𝑝

+ (1 − 𝑡)𝑥3
𝑝

)
1

𝑝. 

 

 
 
(9) 

From (9), we can write  

𝑓(𝑥2) = 𝑓 ([𝑡𝑥1
𝑝

+ (1 − 𝑡)𝑥3
𝑝

]
1

𝑝). (10) 

After that, because 𝑓 is generalized strongly 𝑝-convex functions of higher order, then 

from (10) and 𝑡 =
𝑥3

𝑝
−𝑥2

𝑝

𝑥3
𝑝

−𝑥1
𝑝 we can get 

𝑓(𝑥2) ≤ 𝑓(𝑥3) + (
𝑥3

𝑝
− 𝑥2

𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝) 𝜂(𝑓(𝑥1), 𝑓(𝑥3)) − 𝜇𝜙 (

𝑥3
𝑝

− 𝑥2
𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝) ‖𝑥3

𝑝
− 𝑥1

𝑝
‖

𝑞
 

 

(11) 

If all segments on (11) times by 𝑥3
𝑝

− 𝑥1
𝑝

, then we have 

𝑓(𝑥2)(𝑥3
𝑝

− 𝑥1
𝑝

)

≤ 𝑓(𝑥3)(𝑥3
𝑝

− 𝑥1
𝑝

) + (
𝑥3

𝑝
− 𝑥2

𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝) 𝜂(𝑓(𝑥1), 𝑓(𝑥3))(𝑥3

𝑝
− 𝑥1

𝑝
)

− 𝜇𝜙 (
𝑥3

𝑝
− 𝑥2

𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝) ‖𝑥3

𝑝
− 𝑥1

𝑝
‖

𝑞
(𝑥3

𝑝
− 𝑥1

𝑝
) 

⟺ 𝑓(𝑥3)(𝑥3
𝑝

− 𝑥1
𝑝

) − 𝑓(𝑥2)(𝑥3
𝑝

− 𝑥1
𝑝

) + (𝑥3
𝑝

− 𝑥2
𝑝

)𝜂(𝑓(𝑥1), 𝑓(𝑥3))

− (𝑥3
𝑝

− 𝑥1
𝑝

)𝜇𝜙 (
𝑥3

𝑝
− 𝑥2

𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝 ) ‖𝑥3

𝑝
− 𝑥1

𝑝
‖

𝑞
≥ 0, 

this completes the proof of theorem 1. 

 To show that 𝜙(𝑥3
𝑝

− 𝑥2
𝑝

) ≤ (𝑥3
𝑝

− 𝑥1
𝑝

)𝜙 (
𝑥3

𝑝
−𝑥2

𝑝

𝑥3
𝑝

−𝑥1
𝑝) is not valid for any 𝑥3

𝑝
− 𝑥2

𝑝
, 𝑥3

𝑝
− 𝑥1

𝑝
∈

(0,1) and 𝑥3
𝑝

− 𝑥2
𝑝

< 𝑥3
𝑝

− 𝑥1
𝑝

 with 𝑞 > 0 and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞 , first we take 𝑞 = 2, 
then we have 

𝜙(𝑡) = 𝑡2(1 − 𝑡) + 𝑡(1 − 𝑡)2 = 𝑡2 − 𝑡3 + 𝑡(1 − 2𝑡 + 𝑡2) = 𝑡2 − 𝑡3 + 𝑡 − 2𝑡2 + 𝑡3

= 𝑡 − 𝑡2 = 𝑡(1 − 𝑡). 
 

(12) 

After that, we have to take 𝑥 = 𝑥3
𝑝

− 𝑥1
𝑝

=
1

2
 and 𝑦 = 𝑥3

𝑝
− 𝑥2

𝑝
=

1

4
, then its clear that 𝑥3

𝑝
− 𝑥2

𝑝
=

1

4
<

1

2
= 𝑥3

𝑝
− 𝑥1

𝑝
. So, we can get 

𝜙 (
1

4
) =

1

4
(1 −

1

4
) =

3

16
, 



A Note on Generalized Strongly p-Convex Functions of  Higher Order 

Corina Karim 156 

1

2
𝜙 (

1
4⁄

1
2⁄

) =
1

2
𝜙 (

1

2
) =

1

2
[
1

2
(1 −

1

2
)] =

1

8
=

2

16
. 

And 𝜙 (
1

4
) =

3

16
>

2

16
=

1

2
𝜙 (

1
4⁄

1
2⁄
). So, there exists 𝑥3

𝑝
− 𝑥2

𝑝
=

1

4
, 𝑥3

𝑝
− 𝑥1

𝑝
=

1

2
 and 𝑥3

𝑝
− 𝑥2

𝑝
=

1

4
<

𝑥3
𝑝

− 𝑥1
𝑝

=
1

2
 with 𝑞 = 2 and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞  but (𝑥3

𝑝
− 𝑥2

𝑝
) > (𝑥3

𝑝
− 𝑥1

𝑝
)𝜙 (

𝑥3
𝑝

−𝑥2
𝑝

𝑥3
𝑝

−𝑥1
𝑝) . 

In other words, 𝜙(𝑥3
𝑝

− 𝑥2
𝑝

) > (𝑥3
𝑝

− 𝑥1
𝑝

)𝜙 (
𝑥3

𝑝
−𝑥2

𝑝

𝑥3
𝑝

−𝑥1
𝑝) is not valid for 𝑥3

𝑝
− 𝑥2

𝑝
, 𝑥3

𝑝
− 𝑥1

𝑝
∈ (0,1) and 

𝑥3
𝑝

− 𝑥2
𝑝

< 𝑥3
𝑝

− 𝑥1
𝑝

 with 𝑞 > 0 and 𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞 . 

 
Remarks: 

If 𝜙 satisfies 𝑥𝜙(𝑦) ≥ 𝜙(𝑥𝑦) for any 𝑥, 𝑦 ∈ (0,1) and 𝑞 > 0, then (5) can be written as 

𝑓(𝑥3)(𝑥3
𝑝

− 𝑥1
𝑝

) − 𝑓(𝑥2)(𝑥3
𝑝

− 𝑥1
𝑝

) + (𝑥3
𝑝

− 𝑥2
𝑝

)𝜂(𝑓(𝑥1), 𝑓(𝑥3))

− 𝜇𝜙(𝑥3
𝑝

− 𝑥2
𝑝

)‖𝑥3
𝑝

− 𝑥1
𝑝

‖
𝑞

≥ 0. 

 

CONCLUSION 

Schur type inequality for generalized strongly 𝑝-convex functions of higher order on [3] 
has a correction.   

(𝑥3
𝑝

− 𝑥1
𝑝

)𝜇𝜙 (
𝑥3

𝑝
− 𝑥2

𝑝

𝑥
3
𝑝

− 𝑥
1
𝑝) ≥ 𝜙(𝑥3

𝑝
− 𝑥1

𝑝
), 

is not valid for any 𝑥3
𝑝

− 𝑥2
𝑝

, 𝑥3
𝑝

− 𝑥1
𝑝

∈ (0,1) and 𝑥3
𝑝

− 𝑥2
𝑝

< 𝑥3
𝑝

− 𝑥1
𝑝

 with 𝑝, 𝑞 > 0, 𝑡 ∈ [0,1], and 

𝜙(𝑡) = 𝑡𝑞 (1 − 𝑡) + 𝑡(1 − 𝑡)𝑞 . So, (5) is the correct Schur type inequality for generalized 
strongly 𝑝-convex funtions of higher order. 

ACKNOWLEDGMENTS  

This paper supported by the DPP/SPP grant No. 1519/UN10.F09/PN/2021 at Mathematics and 
Natural Sciences Faculty, Universitas Brawijaya. 

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