1. Introduction
In many branches of mathematical analysis, having a metric structure is essential for the study of several problems. For instance, the concept of distance between elements of an abstract set allows us to define many topological properties, such as convergence, Cauchy sequences, continuity and others [
1,
2,
3,
4]. One of the important properties of a (standard) distance function
D on an abstract set
M is the triangle inequality, i.e.,
Many generalizations of the concept of a distance function achieved by relaxing the triangle inequality have been introduced in the literature, and examples can be found in [
5,
6,
7,
8,
9,
10]. For instance, in [
5], the triangle inequality was relaxed as
where
is a constant.
On the other hand, inequalities involving distance functions are very useful in various areas of mathematics, for instance, in analysis, fixed point theory, operator theory, topology and geometry. Due to this fact, great attention has been paid to the study of inequalities on metric spaces, and examples can be found in [
11,
12,
13,
14,
15,
16,
17,
18].
Let M be a nonempty set and . We say that D is a distance (or metric) on M, if for all ,
- •
,
- •
,
- •
.
In this case, we say that is a metric space.
In [
11], Dragomir and Gosa established a polygonal inequality in the setting of metric spaces and provided some applications in normed linear spaces and inner product spaces. Namely, it was proven that if
is a metric space,
is an integer,
and
, with
, then
Later, in [
15], the above inequality was extended to natural powers of the distance. Namely, it was shown that under the above assumptions, we have
for all integers
. In [
19], Dragomir studied sums of the form
where
. He proved the following:
Some other inequalities of the same type can be found in [
12,
13]. We also refer to [
20], where a continuous version of (
1) was obtained.
In this paper, we first introduce the notion of a generalized distance with respect to a pair of mappings and provide some examples of such distance functions (
Section 2). Let us provide some motivations for introducing such a notion. Let us observe that some of the above-mentioned inequalities involve the power of a (standard) distance function. Now, if
d is a distance function on
M, and if we define mapping
as
we obtain, by the triangle inequality, that
for all
, that is,
where
. Hence, a natural question is whether inequalities of Dragomir type can be extended to mappings
satisfying (
2) for arbitrary
. A positive answer is obtained in
Section 3, where we establish several inequalities of type (
1) involving generalized distance functions (
Section 3). Finally, in
Section 4, some generalized distance inequalities for self-crossing polygons are proved.
3. Inequalities Involving Generalized Distance Functions
The below inequality involving generalized distance functions holds.
Theorem 1. Let D be a distance function on M with respect to , in the sense of Definition 1, where . Let be an integer, and , with . Then, Proof. Let
. By property (iii) in Definition 1, we have
where
. By multiplying the above inequality by
and summing over
i and
j, we obtain
On the other hand, by properties (i)–(ii) in Definition 1, we have
Moreover, we have
and
Hence, it follows from (
10)–(
15) that
Finally, by taking the infimum over
u in (
16), we obtain (
9). □
Now, let us study some special cases of Theorem 1. We first consider the case when
f and
g are symmetric, that is,
In this case, from Theorem 1, we deduce the below result.
Corollary 1. Let D be a distance function on M with respect to , in the sense of Definition 1, where are symmetric. Let be an integer, and , with . Then, By taking in Corollary 1, we deduce the below result.
Corollary 2. Let D be a distance function on M with respect to , in the sense of Definition 1, where is symmetric. Let be an integer, and , with . Then, By taking
in Theorem 1, we obtain the below result.
Corollary 3. Let D be a distance function on M with respect to , in the sense of Definition 1, where . Let be an integer and . Then, If f and g are symmetric, we deduce, by Corollary 3, the below result.
Corollary 4. Let D be a distance function on M with respect to , in the sense of Definition 1, where are symmetric. Let be an integer and . Then, If in Corollary 4, then we deduce the below result.
Corollary 5. Let D be a distance function on M with respect to , in the sense of Definition 1, where is symmetric. Let be an integer and . Then, Next, using the above results, we provide below some upper bounds for the following sum:
where
and
are two distances on
M.
We first consider the case when , .
Corollary 6. For all , let be a distance on M and . Let be an integer, and , with . Then, Proof. By Example 3, since
,
, we know that mapping
defined as
is a distance with respect to
, in the sense of Definition 1, where (iii) holds with constant
. Since
,
, are symmetric, (
17) follows from Corollary 1 by taking
,
and
. □
Next, we consider the case when .
Corollary 7. For all , let be a distance on M and . Let be an integer, and , with . Then, Proof. By Example 3, since
, we know that mapping
is a distance with respect to
, in the sense of Definition 1, where (iii) holds with constant
. Since
,
, are symmetric, (
18) follows from Corollary 1 by taking
,
and
. □
We now consider the case when
. In this case, we deduce the below result obtained in [
19].
Corollary 8. Let d be a metric on M and . Let be an integer, and , with . Then, Proof. By Example 3, since
, we know that
is a distance with respect to
, in the sense of Definition 1, where (iii) holds with constant
. Then, (
19) follows from Corollary 1 by taking
and
. □
Next, we consider the case when , .
Corollary 9. For all , let be a distance on M and . Let be an integer, and , with . Then, Proof. By Example 3, since
,
, we know that mapping
is a distance with respect to
, in the sense of Definition 1, where (iii) holds with constant
. Since
,
, are symmetric, (
20) follows from Corollary 1 by taking
,
and
. □
Finally, we consider the case when
. In this case, we deduce the below result obtained in [
19].
Corollary 10. Let d be a distance on M and . Let be an integer, and , with . Then, Proof. By Example 3, since
, we know that
is a distance with respect to
, in the sense of Definition 1, where (iii) holds with constant
. Then, (
21) follows from Corollary 1 by taking
and
. □
4. Generalized Distance Inequalities for Self-Crossing Polygons
Let
D be a distance on
M with respect to
, in the sense of Definition 1, where
. Let
,
, be the vertices of a possibly self-crossing polygon with unit perimeter with respect to
D. The perimeter with respect to
D is defined as
Let
under the assumption of
The below result holds.
Theorem 2. Let . Let D be a distance on M with respect to , in the sense of Definition 1, where . We havewhere is defined in (22). Proof. Let
be such that
Let
S be the sum of pair-wise distances, that is,
Then,
On the other hand, by property (iii) in Definition 1, we have
By summing over
i, we obtain
On the other hand, we have
Hence, the following holds:
Next, by summing over
j and using (
24), we obtain
which yields (
23). □
Let us consider the special case of Theorem 2 when
where
and
d is a distance on
M. Notice that by Example 3, we know that
D is a distance with respect to
, in the sense of Definition 1, where (iii) holds with
Hence, by Theorem 2, we deduce the below result.
Corollary 11. Let D be the generalized distance defined in (25). Then, for all , the following holds: In the case when , we have the below additional result.
Theorem 3. Let D be the generalized distance defined in (25) with . Then, for all and , with , we have Proof. Let
be fixed. Then, by (
26), for all
, with
we have
Let us suppose that (
27) is not true. Then, there exist
with
such that
On the other hand, we have
Hence, by (
28) and due to the assumption on
n, we obtain
On the other hand,
Thus, we reach a contradiction. □
We next consider the case when
and
where
and
is the Euclidean norm on
. In this case, we obtain the below result.
Theorem 4. Let D be the generalized distance defined in (29). Then, for all : - (i)
- (ii)
If n is even and , then - (iii)
If n is odd and , then
where is defined in (22). Proof. (i) It immediately follows from Corollary 11 that by taking
(ii) Let
n be even and
. Let us consider the self-crossing polygon, where the vertices are defined as follows:
Then,
Furthermore, by (
24), we have
which yields
. Then, by (
26), we deduce that
.
(iii) Let
n be even and
. Let us consider the self-crossing polygon, where the vertices are defined as follows:
Then,
Furthermore, by (
24), we have
This shows that
. Since
, we obtain
. □
5. Conclusions
In this paper, we first introduce the notion of a generalized distance function with respect to a pair of mappings. Namely, given a nonempty set
M, we say that
is a distance with respect to
, where
, if:
- (i)
for all .
- (ii)
for all .
- (iii)
There exists
such that
for all
.
In
Section 2, we provide several examples of generalized distance functions with respect to a pair of mappings. Moreover, motivated by the recent obtained results obtained by Dragomir [
19], several inequalities involving sums of the form
where
and
, with
, are established in
Section 3. In
Section 4, we provide new distance inequalities for self-crossing polygons.
It would be interesting to study the topological properties of distance functions with respect to a pair of mappings, for instance, convergence, Cauchy criterion and completeness. An interesting problem in this direction is to extend the Banach contraction principle [
21] to a set
M equipped with a distance function with respect to a pair of mappings.