International Journal of Analysis and Applications

Volume 16, Number 3 (2018), 427-436

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

DOI: 10.28924/2291-8639-16-2018-427

SOME PROPERTIES OF GENERALIZED STRONGLY HARMONIC CONVEX

FUNCTIONS

MUHAMMAD ASLAM NOOR∗, KHALIDA INAYAT NOOR, SABAH IFTIKHAR AND

FARHAT SAFDAR

Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan.

∗Corresponding author: noormaslam@gmail.com

Abstract. In this paper, we introduce a new class of harmonic convex functions with respect to an arbitrary

trifunction F(·, ·, ·) : K×K×[0, 1] → R, which is called generalized strongly harmonic convex functions. We

study some basic properties of strongly harmonic convex functions. We also discuss the sufficient conditions

of optimality for unconstrained and inequality constrained programming under the generalized harmonic

convexity. Several special cases are discussed as applications of our results. Ideas and techniques of this

paper may motivate further research in different fields.

1. Introduction

The concept of convexity and generalized convexity in the study of optimality to solve mathematical

programming, have been extended using innovative ideas and techniques. For example, in earlier paper-

s, Bector and Singh [3] introduced a class of b-vex functions. Chao et al. [4] considered new generalized

sub-b-convex functions and sub-b-convex sets. They proved the sufficient conditions of optimality for both

unconstrained and inequality constrained sub-b-convex programming. Anderson et. al. [1] and Iscan [5]

have investigated various properties of harmonic convex functions. Noor and Noor [9] have shown that the

minimum of the differentiable harmonic convex functions on the harmonic convex set can be characterized

by a class of variational inequalities, which is called harmonic variational inequality. To the best of our

Received 2018-01-19; accepted 2018-03-14; published 2018-05-02.

2010 Mathematics Subject Classification. 26D15; 26D10; 90C23.

Key words and phrases. harmonic convex functions; F-harmonic convex functions; optimality conditions.

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.

427

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-427


Int. J. Anal. Appl. 16 (3) (2018) 428

knowledge, this field is new one and has not developed as yet. A significant class of convex functions is that

of strongly harmonic convex functions introduced by Noor et. al. [11]. For recent applications, generaliza-

tions and other aspects of convex and harmonic convex functions, see [2–4,6,10,12–19] and references therein.

Inspired by the research works [2–4, 6], we introduce a new class of harmonic convex functions with

respect to an arbitrary function F(., ., .), which is called strongly generalized harmonic convex function.

One can show that the generalized strongly harmonic convex functions is quite general and unified one.

Several new and old classes of convex and harmonic convex functions can be obtained from these general

harmonic convex functions. We consider the sufficient conditions of optimality for both unconstrained and

inequality constrained programming. Some properties of generalized strongly harmonic convex functions and

generalized strongly harmonic convex sets are discussed. Results obtained in this paper may be considered

as significant improvement of the known results.

2. Preliminaries

In this Section, we recall some basic concepts and results. We also introduce some new concepts and

discuss some special cases.

Definition 2.1. [1]. A set K = [a,b] ⊂ Rn \{0} is said to be a harmonic convex set, if

xy

tx + (1 − t)y
∈ K, ∀x,y ∈ K, t ∈ [0, 1].

We now consider the concept of generalized strongly harmonic convex functions with respect to an arbi-

trary trifunction F(·, ·, ·) : K ×K × [0, 1] → R.

Definition 2.2. Let K is a nonempty harmonic convex set in Rn \{0}. A function f : K → R is said to

be generalized strongly harmonic convex function on K with respect to map F(·, ·, ·) : K ×K × [0, 1] → R, if

and only if,

f

(
xy

tx + (1 − t)y

)
≤ (1 − t)f(x) + tf(y) + F(x,y,t), ∀x,y ∈ K,t ∈ [0, 1]. (2.1)

(I). If F(x,y,t) = 0 in Definition 2.2, then it reduces to harmonic convex function.

Definition 2.3. [5]. A function f : K → R, where K is a nonempty harmonic convex set Rn \{0}, is said

to be a harmonic convex function on K, if and only if,

f

(
xy

tx + (1 − t)y

)
≤ (1 − t)f(x) + tf(y), ∀x,y ∈ K, t ∈ [0, 1].

This shows that every harmonic convex function f is harmonic sub-F-convex function with respect to the

map F(x,y,t) = 0, but the converse may not be true. See also [8–10, 17].



Int. J. Anal. Appl. 16 (3) (2018) 429

(II). If F(x,y,t) = t(1−t)F( xy
x−y ) in Definition 2.2, then it reduces to F-strongly harmonic convex function

Definition 2.4. [18]. A function f : K ⊆ R \{0}→ R is said to be F-strongly harmonic convex function,

if there exists a non-negative function F : X \{0}→ [0,∞), such that

f

(
xy

tx + (1 − t)y

)
≤ (1 − t)f(x) + tf(y) − t(1 − t)F

(
xy

x−y

)
, ∀x,y ∈ K,t ∈ (0, 1). (2.2)

If t = 1
2

, then (2.2) reduces to

f

(
2xy

x + y

)
≤
f(x) + f(y)

2
−

1

4
F

(
xy

x−y

)
and the functionf is called F-strongly harmonic mid-convex(harmonic Jensen-convex) function.

(III). If F(x,y,t) = −ct(1 − t) ‖ x−y
xy
‖ in Definition 2.2, then it reduces to strongly harmonic convex

function with modulus c > 0.

Definition 2.5. [11]. A function f : K → R, where K is a nonempty harmonic convex set in Rn \{0}, is

said to be strongly harmonic convex function on K with modulus c > 0, if and only if,

f

(
xy

tx + (1 − t)y

)
≤ (1 − t)f(x) + tf(y) − ct(1 − t)

∥∥x−y
xy

∥∥2, ∀x,y ∈ I,t ∈ (0, 1).
Theorem 2.1. [17]. Let K be a nonempty harmonic convex set in Rn \ {0} and let f : K → R be

differentiable on K. Then f is harmonic quasi convex, if and only if,

f(x) ≤ f(y) ⇒〈f′(y),
xy

y −x
〉≤ 0 ∀ x,y ∈ K.

Definition 2.6. Let K is a nonempty harmonic convex set in Rn \{0}. A function f : K → R is said to be

generalized strongly harmonic quasi convex function on K with respect to map F(·, ·, ·) : K×K× [0, 1] → R,

if and only if,

f

(
xy

tx + (1 − t)y

)
≤ max{f(x),f(y)} + F(x,y,t), ∀x,y ∈ K,t ∈ [0, 1]. (2.3)

(I). If F(x,y,t) = 0 in Definition 2.6, then it reduces to harmonic quasi convex function.

Definition 2.7. [20]. A function f : K → R, where K is a nonempty harmonic convex set Rn\{0}, is said

to be a harmonic quasi convex function on K, if and only if,

f

(
xy

tx + (1 − t)y

)
≤ max{f(x),f(y)}, ∀x,y ∈ K.

We now define a new class of generalized strongly harmonic log-convex functions.

Definition 2.8. Let K is a nonempty harmonic convex set in Rn \{0}. A function f : K → R is said to be

generalized strongly harmonic log-convex function on K with respect to map F(·, ·, ·) : K ×K × [0, 1] → R,

if and only if,

f

(
xy

tx + (1 − t)y

)
≤ [f(x)]1−t[f(y)]t + F(x,y,t), ∀ x,y ∈ K, t ∈ [0, 1]. (2.4)



Int. J. Anal. Appl. 16 (3) (2018) 430

From (2.4), it follows that

f

(
xy

tx + (1 − t)y

)
≤ [f(x)]1−t[f(y)]t + F(x,y,t)

≤ (1 − t)f(x) + tf(y) + F(x,y,t)

≤ max{f(x),f(y)} + F(x,y,t), ∀ x,y ∈ K, t ∈ [0, 1],

This shows that every generalized strongly harmonic log-convex function is generalized strongly harmonic

convex and every generalized strongly harmonic convex function is generalized strongly harmonic quasi

convex, but the converse is not true.

3. Main Results

In this section, we introduce the concept of generalized harmonic convex set and harmonic convex functions

with respect to an arbitrary map F(·, ·, ·) and discuss its properties.

Theorem 3.1. If fi : K → R, (i = 1, 2, 3, ...,m) are generalized strongly harmonic convex functions with

respect to map Fi : K ×K × [0, 1] → R, respectively. Then the function

f =

m∑
i=1

aifi, a1 ≥ 0, i = 1, 2, 3, ...,m,

is generalized strongly harmonic convex with respect to F =
∑m

i=1 aibi.

Proof. For all x,y ∈ K and t ∈ [0, 1], we have

f

(
xy

tx + (1 − t)y

)
=

m∑
i=1

aifi

(
xy

tx + (1 − t)y

)

≤
m∑
i=1

ai[(1 − t)fi(x) + tfi(y) + Fi(x,y,t)]

= (1 − t)
m∑
i=1

aifi(x) + t

m∑
i=1

aifi(y) +

m∑
i=1

aiFi(x,y,t)

= (1 − t)f(x) + tf(y) + F(x,y,t).

�

Theorem 3.2. If the function fi : K → R are generalized strongly harmonic convex functions with respect

to Fi(x,y,t) respectively. Then f = max{fi, i = 1, 2, 3, ...,m} is also generalized strongly harmonic convex

with respect to F = max{Fi}.



Int. J. Anal. Appl. 16 (3) (2018) 431

Proof. Consider,

f

(
xy

tx + (1 − t)y

)
= max

{
fi

(
xy

tx + (1 − t)y

)
, i = 1, 2, 3, ...,m

}
≤ (1 − t) max{fi(x)} + t max{fi(y)} + max{Fi(x,y,t)

= (1 − t)f(x) + tf(y) + F(x,y,t).

So, f(x) is harmonic sub-F convex function with respect to

F(x,y,t) = max{fi, i = 1, 2, 3, ...,m}. �

Theorem 3.3. If the function f : K → R is a generalized strongly harmonic convex function with respect to

F(x,y,t) and g : R → R is a linear function, then f ◦g is generalized strongly harmonic convex with respect

to F ′ = g ◦F.

Proof. Since f is generalized strongly harmonic convex function with respect to F(x,y,t) and g is a an

increasing function, it follows that

(g ◦f)
(

xy

tx + (1 − t)y

)
= g

(
f

(
xy

tx + (1 − t)y

))
≤ g((1 − t)f(x) + tf(y) + F(x,y,t)

= (1 − t)g(f(x)) + tg(f(y)) + g(F(x,y,t))

= (1 − t)(g ◦f) + t(g ◦f) + (g ◦F)(x,y,t).

That is, g◦f is generalized strongly harmonic convex function with respect to F ′ = g◦F and this completes

the proof. �

The following theorem gives a necessary and sufficient characterization of a differentiable generalized

strongly harmonic convex function with respect to a map F(·, ·, ·).

Theorem 3.4. Let K be a harmonic convex set. If f : K → R is a differentiable generalized strongly

harmonic convex function on the harmonic convex set K with respect to the map F(x,y,t), then

(1) 〈f′(x), xy
x−y〉≤ f(y) −f(x) + limt→0+

F (x,y,t)
t

, ∀x,y ∈ K.

(2) 〈f′(x) −f′(y), xy
x−y〉≤ limt→0+

F (x,y,t)
t

+ limt→0+
F (y,x,t)

t
, ∀x,y ∈ K.

Proof. (1). Let f be a generalized strongly harmonic convex function. Then

f

(
xy

tx + (1 − t)y

)
≤ (1 − t)f(x) + tf(y) + F(x,y,t),

which can be written as

f(y) −f(x) +
F(x,y,t)

t
≥
f
(

xy
y+t(x−y)

)
−f(x)

t
.



Int. J. Anal. Appl. 16 (3) (2018) 432

Since f is differentiable function, so taking the limit in the above inequality, as t → 0, we have

f(y) −f(x) + lim
t→0+

F(x,y,t)

t
≥〈f′(x),

xy

x−y
〉, (3.1)

which is (1).

Changing the role of x and y, in (3.1), we obtain

f(x) −f(y) + lim
t→0+

F(y,x,t)

t
≥〈f′(y),

xy

y −x
〉. (3.2)

Adding (3.1) and (3.2), we have

〈f′(x) −f′(y),
xy

x−y
〉≤ lim

t→0+
F(x,y,t)

t
+ lim

t→0+
F(y,x,t)

t
,

which is the required (2). This completes the proof. �

Definition 3.1. [17]. A function f : K → R, where K is a nonempty harmonic convex set Rn \{0} is said

to be a harmonic quasi convex function, if, for each x,y ∈ K with f(x) ≤ f(y), we have 〈f′(y), xy
y−x〉≤ 0; or

equivalently, if 〈f′(y), xy
y−x〉 > 0, then f(x) > f(y).

Definition 3.2. [17]. A function f : K → R, where K is a nonempty harmonic convex set Rn \{0} is said

to be a harmonic pseudo-convex function, if, for each x,y ∈ K with 〈f′(y), xy
y−x〉 ≥ 0, we have f(x) ≥ f(y);

or equivalently, if f(x) < f(y), then 〈f′(y), xy
y−x〉 < 0.

Theorem 3.5. Let K be a harmonic convex set and f : K → R be a differentiable generalized strongly har-

monic convex function on the harmonic convex set K with respect to the map F(x,y,t). If limt→0
F (x,y,t)

t
≤

|f(x) −f(y)|, ∀ x,y ∈ K, then f is harmonic quasi-convex. Furthermore, if limt→0
F (x,y,t)

t
< |f(x) −f(y)|,

∀ x,y ∈ K, then f is harmonic pseudo-convex.

Proof. Suppose that f(x) ≤ f(y), for any x,y ∈ K and t ∈ (0, 1). Then from Theorem 3.4, we have

〈f′(y),
xy

y −x
〉≤ f(x) −f(y) + lim

t→0+
F(y,x,t)

t
.

If limt→0
F (y,x,t)

t
≤ |f(x) − f(y)|, then f(x) − f(y) + limt→0

F (x,y,t)
t

≤ 0. So, 〈f′(y), xy
y−x〉 ≤ 0. Therefore,

from Definition 3.1, we have f is harmonic quasi-convex function.

Similarly, if f(x) < f(y), we also have 〈f′(x), xy
x−y〉 < 0. So, from the Definition 3.2, we have f is harmonic

pseudo-convex function. �

Now, we are going to introduce a new concept of generalized harmonic convex set.

Definition 3.3. Let Ks ⊂ Rn+1\{0} be a nonempty set. A set Ks is said to be generalized harmonic convex

with respect to F(x,y,t) : Rn ×Rn × [0, 1] → R, if(
xy

tx + (1 − t)y
, (1 − t)α + tβ + F(x,y,t)

)
∈ Ks, ∀(x,α), (y,β) ∈ Ks,



Int. J. Anal. Appl. 16 (3) (2018) 433

for all x,y ∈ Rn, and t ∈ [0, 1].

We now investigate some characterizations of generalized strongly harmonic convex function f : K → R

in term of their epigraph E(f).

Definition 3.4. [2]. Let K be a nonempty in Rn and let f : K → R be a function. Then epigraph of f,

denoted by E(f) is defined by

E(f) = {(x,α) : x ∈ K,α ∈ R,f(x) ≤ α}.

Theorem 3.6. A function f : K → R is generalized strongly harmonic convex with respect to F(x,y,t) :

Rn×Rn×[0, 1] → R, if and only if, epigraph of f is generalized harmonic convex set with respect to F(·, ·, ·).

Proof. Suppose that f is harmonic sub-F-convex function with respect to F(x,y,t). Let (x,α) and (y,β) ∈

E(f). Then x,y ∈ K, f(x) ≤ α and f(y) ≤ β, we have

f

(
xy

tx + (1 − t)y

)
≤ (1 − t)f(x) + tf(y) + F(x,y,t)

≤ (1 − t)α + tβ + F(x,y,t).

Hence from Definition, one has(
xy

tx + (1 − t)y
, (1 − t)α + tβ + F(x,y,t)

)
∈ E(f)

Thus E(f) is generalized harmonic convex set with respect to F.

Conversely, assume that E(f) is generalized harmonic convex set with respect to F. Let x,y ∈ K, then

(x,f(x)) and (y,f(y)) belong to E(f). Thus for t ∈ [0, 1],(
xy

tx + (1 − t)y
, (1 − t)f(x) + tf(y) + F(x,y,t)

)
∈ E(f).

This further follows that,

f

(
xy

tx + (1 − t)y

)
≤ (1 − t)f(x) + tf(y) + F(x,y,t)

That is f is a generalized strongly harmonic convex function with respect to F(., ., .). �

Theorem 3.7. If Ksi is a family of generalized harmonic convex set with respect to a map F(x,y,t), then

∩i∈IKsi is a generalized harmonic convex set with respect to F(x,y,t).

Proof. Let (x,α), (y,β) ∈ ∩i∈IKsi , t ∈ [0, 1]. Then for each i ∈ I, (x,α), (y,β) ∈ Ii. Since Ksi is a

generalized harmonic convex set with respect to F(., ., .), it follows that(
xy

tx + (1 − t)y
, (1 − t)f(x) + tf(y) + F(x,y,t)

)
∈ Ksi,∀i ∈ I.



Int. J. Anal. Appl. 16 (3) (2018) 434

Thus (
xy

tx + (1 − t)y
, (1 − t)f(x) + tf(y) + F(x,y,t)

)
∈∩i∈IKsi.

Hence, ∩i∈IKsi is generalized harmonic convex set with respect to F(x,y,t). �

We apply the above results to the nonlinear programming problem. First, we consider the unconstraint

problem.

Theorem 3.8. Let f : K → R be differentiable and generalized strongly harmonic convex function with

respect to map F(·, ·, ·). Consider the optimal problem min{f(x)|x ∈ K}. If x̄ ∈ K and the relation

〈f′(x),
x̄x

x̄−x
〉− lim

t→0+
F(x̄,x,t)

t
≥ 0, (3.3)

holds for each x ∈ K, then x̄ is the optimal solution of f on K.

Proof. For any x ∈ k, from Theorem 3.4, we have

〈f′(x),
x̄x

x̄−x
〉− lim

t→0+
F(x̄,x,t)

t
≤ f(x) −f(x̄). (3.4)

From (3.3) and (3.4), we have f(x) −f(x̄) ≥ 0. So, x̄ is an optimal solution of f on K. �

Next, we apply the above results to the nonlinear programming problem with inequality constraints:

min f(x)

(Pg) s.t gi(x) ≤ 0, i ∈ I = {1, 2, 3, ...,m}

x ∈ Rn

Denote the feasible set of (Pg) by Sg = {x ∈ Rn|gi(x) ≤ 0, i ∈ I}.

Definition 3.5. Let K be a nonempty harmonic convex set in Rn. The function f : K → R is said to

be generalized strongly harmonic pseudo convex on K with respect to F : K × K × [0, 1] → R, if for each

x,y ∈ K and t ∈ (0, 1), from 〈f′(y), xy
y−x〉 + limt→0

F (y,x,t)
t
≥ 0 one can have f(x) ≥ f(y).

Theorem 3.9. (Karush-Kuhn-Tucker Sufficient Conditions) The function f is differentiable generalized

strongly harmonic pseudo convex with respect to F(., ., .) : K×K×[0, 1] → R, gi(x) (i ∈ I) are differentiable

and generalized strongly harmonic convex with respect to F : K ×K × [0, 1] → R. Assume that x̄ ∈ Sg is a

KKT point of (Pg), that is, there exist multiplier λi ≥ 0 (i ∈ I) such that

〈f′(x̄),
x̄x

x̄−x
〉 +

∑
i∈I(x̄)

λi〈g′(x̄),
x̄x

x̄−x
〉 = 0, λi〈g′(x̄),

x̄x

x̄−x
〉 = 0. (3.5)



Int. J. Anal. Appl. 16 (3) (2018) 435

If

lim
t→0

F(x,x̄,t)

t
≥
∑
i∈I

λi lim
t→0

F(x,x̄,t)

t
, ∀x ∈ Sg. (3.6)

Then x̄ is an optimal solution of the problem (Pg).

Proof. For any x ∈ Sg, we have gi(x) ≤ 0, gi(x̄) = 0, i ∈ I(x̄) = {i ∈ I | gi(x̄) = 0}. Therefore, from the

generalized strongly harmonic convexity of gi(x) and Theorem 3.4, we obtain

〈g′(x̄),
x̄x

x̄−x
〉− lim

t→0+
F(x̄,x,t)

t
≤ 0, for i ∈ I(x̄). (3.7)

From (3.5), one has

〈f′(x̄),
x̄x

x̄−x
〉 = −

∑
i∈I(x̄)

λi〈g′(x̄),
x̄x

x̄−x
〉.

In view of (3.6) and from (3.7), we have

〈f′(x̄),
x̄x

x̄−x
〉 + lim

t→0+
F(x̄,x,t)

t
≥ −

∑
i∈I(x̄)

λi〈g′(x̄),
x̄x

x̄−x
〉 +

∑
i∈I(x̄)

λi lim
t→0+

F(x̄,x,t)

t

= −
∑

i∈I(x̄)

λi

[
〈g′(x̄),

x̄x

x̄−x
〉− lim

t→0+
F(x̄,x,t)

t

]
≥ 0.

So,

〈f′(x̄),
x̄x

x̄−x
〉 + lim

t→0+
F(x̄,x,t)

t
≥ 0.

From the generalized strongly harmonic pseudo convexity of f(x), we have f(x) ≥ f(x̄). Therefore x̄ is an

optimal solution of the problem (Pg). �

Acknowledgements

The authors would like to thank the Rector, COMSATS Institute of Information Technology, Pakistan,

for providing excellent research and academic environments.

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	1. Introduction
	2. Preliminaries
	3. Main Results
	Acknowledgements
	References