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Dragomir and Gosa type inequalities on bmetric spaces
Journal of Inequalities and Applications volume 2019, Article number: 29 (2019)
Abstract
In this paper, we investigate Dragomir and Gosa type inequalities in the setting of bmetric spaces. As an application, we consider some inequalities in bnormed spaces. We prove that the inequalities admit geometrical interpretation.
Introduction and preliminaries
It is a natural trend in fixed point theory to refine a standard metric space structure with a weaker one. One of the interesting extensions of the notion of a metric space is the concept of a bmetric space which was introduced by Czerwik [8].
Definition 1.1
([8])
Let X be a nonempty set and \(s\geq 1\) a given real number. A mapping \(d \colon X \times X\to [0, \infty )\) is said to be a bmetric if for all \(x, y, z \in X\) the following conditions are satisfied:
 \((bM_{1})\) :

\(d(x, y) =0\) if and only if \(x = y\);
 \((bM_{2})\) :

\(d(x, y) = d(y,x)\) (symmetry);
 \((bM_{3})\) :

\(d(x, z)\leq s[d(x, y) + d(y, z)]\) (btriangle inequality).
In this case, the pair \((X, d)\) is called a bmetric space (with constant s).
Clearly, any metric space is a bmetric space (with constant \(s=1\)).
Example 1.2
([10])
Let \(X= [ 0,1 ] \) and let \(d:X\times X\longrightarrow {}[ 0,\infty )\) be defined by \(d ( x,y ) = ( xy ) ^{2}\). Then, clearly, \(( X,d ) \) is a bmetric space with \(s=2\).
The following is another constructive example of bmetric.
Example 1.3
([1])
Let \(X=\{x_{i}: 1\leq i\leq M\}\) for some \(M \in \mathbb{N}\) and \(s\geq 2\). Define \(d: X\times X\to \infty \) as
Consequently, we derive that
for all \(i,j,k \in \{1,M\}\). Thus, \((X,d)\) forms a bmetric for \(s >2\) where the ordinary triangle inequality does not hold.
For more examples for bmetric, we may refer, e.g., to [1,2,3,4,5,6,7, 9, 12] and the corresponding references therein.
Example 1.4
(see, e.g., [6])
The space \(L^{p}[0,1]\) (where \(0< p<1\)) of all real functions \(x(t)\), \(t\in [0,1]\) such that \(\int _{0}^{1} x(t)^{p} \,dt<\infty \), together with the functional
is a bmetric space. Notice that \(s=2^{1/p}\).
Main result
We start this section by recalling an interesting inequality that was proposed by Dragomir and Gosa in [11]. In what follows we investigate their inequality in the setting of a more general structure, namely that of bmetric spaces.
Theorem 2.1
Let \(( X,d ) \) be a bmetric space with constant \(s\geq 1\), and \(x_{i}\in X\), \(p_{i}\geq 0\) (\(i\in \{ 1,2, \dots ,n \} \)) with \(\sum_{i=1}^{n}p_{i}= \frac{1}{s}\). Then we have
The inequality is sharp in the sense that the constant \(c=1\) in front of the infimum cannot be replaced by a smaller constant.
Proof
Using the btriangle inequality, for any \(x\in X\), \(i,j\in \{ 1,2,\dots ,n \} \) we have
If we multiply (2) by \(p_{i}\), \(p_{j}\) and sum over i and j from 1 to n, we get
Note that by symmetry we have
Now, using the condition \(\sum_{i=1}^{n}p_{i}=\frac{1}{s}\), we can easily deduce that
So, from (3) we have
Therefore,
for any \(x\in X\). Using the fact that the infimum is the greatest lower bound, we deduce (1).
Now, suppose that there exists \(c>0\) such that
and choose \(n=2\), \(p_{1}=p\) and \(p_{2}=1p\) where \(p\in (0,1)\). Then,
If we let \(x=x_{1}\) in (4), we get
As \(d ( x_{1},x_{2} )>0\) and \(1p>0\), so \(p\leq c\) for any \(p\in (0,1)\). Using the fact that the supremum is the least upper bound, we deduce that \(c\geq 1\). □
The following corollary is a generalization of Corollary 1 in [11] to the case of a bmetric space.
Corollary 2.2
Let \(( X,d ) \) be a bmetric space with constant \(s\geq 1\), and \(x_{i}\in X\), \(i\in \{ 1,2,\dots ,n \} \), then
The proof follows directly by taking \(p_{i}=\frac{1}{ns}\), \(i\in \{ 1,2,\dots ,n \} \) in the previous theorem.
The above corollary can be interpreted geometrically as follows: The sum of all edges and diagonals of a polygon with n vertices in a bmetric space is less than or equal to \(\frac{n}{s}\)times the sum of the distances from any arbitrary point in the space to its vertices.
The next corollary is a generalization of Corollary 2 in [11] in the framework of bmetric spaces.
Corollary 2.3
Let \(( X,d ) \) be a bmetric space with constant s and \(x_{i}\in X\), \(i\in \{ 1,2,\dots ,n \} \). If there exist \(z\in X\) and \(r>0\) such that the closed ball \(\overline{B} ( z,r ) = \{ y\in X:d ( z,y ) \leq r \} \) contains all the points \(x_{i}\), then for any \(p_{i}\geq 0\) with \(\sum_{i=1}^{n}p_{i}=\frac{1}{s}\) we have
Proof
Using (1) we have
□
Applications
In this section we define a new notion of a bnormed space and study some of its properties.
Definition 3.1
Let X be a vector space over a field K and let \(s\geq 1\) be a constant. A function \(\Vert \cdot \Vert _{b}:X\longrightarrow {}[ 0,\infty )\) is said to be a bnorm if the following conditions hold for every \(x,y\in X\), \(c\in K\):
 (Nb1):

\(\Vert x \Vert _{b}\geq 0\);
 (Nb2):

\(\Vert x \Vert _{b}=0\Longleftrightarrow x=0\);
 (Nb3):

\(\Vert cx \Vert _{b}=c^{\log _{2}s+1} \Vert x \Vert _{b}\) (bhomogeneity);
 (Nb4):

\(\Vert x+y \Vert _{b}\leq s [ \Vert x \Vert _{b}+ \Vert y \Vert _{b} ] \) (bnorm triangle inequality).
In this case \(( X, \Vert \cdot \Vert _{b} ) \) is called a bnormed space with constant s.
Here we give an example of a bnormed space.
Example 3.2
Let \(X=\mathbb{R}\) and define \(\Vert \cdot \Vert _{b}:X \longrightarrow {}[ 0,\infty )\) by \(\Vert x \Vert _{b}=x^{p}\) where \(p\in (1,\infty )\), then, using the relation \(( x+y ) ^{p}\leq 2^{p1} ( x+y ) \), we can easily deduce that \(( X, \Vert \cdot \Vert _{b} ) \) is a bnormed space with constant \(s=2^{p1}\).
Remark 3.3
Let \(( X, \Vert \cdot \Vert _{b} ) \) be a bnormed space with constant \(s\geq 1\), \(x_{i}\in X\), \(i\in \{ 1,\dots ,n \}\). Then it is easy to prove the following generalized btriangle inequality:
Remark 3.4
Any bnorm with \(s\geq 1\) defines a bmetric as follows:
The question now is the following: Is any bmetric induced from a bnorm? The following remark can answer this question.
Remark 3.5
Let X be a vector space over a field K. Any bmetric \(d:X\times X\longrightarrow {}[ 0,\infty )\) with constant \(s\geq 1\) induced from a bnorm must satisfy the following properties for each \(x,y,z\in X\), \(c\in K\):

(i)
\(d ( x+z,y+z ) =d ( x,y ) \) (translation invariance);

(ii)
\(d ( cx,cy ) =c^{\log _{2}s+1}d ( x,y ) \) (bhomogeneity).
Proposition 3.6
A bhomogeneous translation invariant bmetric \(d:X\times X\longrightarrow {}[ 0,\infty )\) with constant \(s\geq 1\) can define a bnorm \(\Vert \cdot \Vert _{b}:X\longrightarrow {}[ 0, \infty )\) as follows:
Proof
Clearly, (Nb1) and (Nb2) are satisfied.
As d is homogeneous, \(\Vert cx \Vert =d ( cx,0 ) =c^{\log _{2}s+1}d ( x,0 ) =c^{\log _{2}s+1} \Vert x \Vert _{b}\).
As d is translation invariant,
which prove (Nb3) and (Nb4), respectively. □
Now, we rewrite inequality (1) in the sense of bnormed spaces and obtain some corollaries.
If \(( X, \Vert \cdot \Vert _{b} ) \) is a bnormed space with constant \(s\geq 1\), \(x_{i}\in X\), and \(p_{i}\geq 0\), \(i\in \{ 1,\dots ,n \} \) with \(\sum_{i=1}^{n}p_{i}=\frac{1}{s} \), then by (1) we have
The following proposition is a generalization of Proposition 2 in [11] to the case of a bnormed space.
Proposition 3.7
Let \(( X, \Vert \cdot \Vert _{b} ) \) be a bnormed space with constant \(s\geq 1\), \(x_{i}\in X\) and \(p_{i} \geq 0\), \(i\in \{ 1,\dots ,n \} \) with \(\sum_{i=1} ^{n}p_{i}=\frac{1}{s}\). Let \(x_{p}=\sum_{i=1}^{n}p_{i}x_{i}\), then
Proof
As the infimum is a lower bound, the second part of inequality (6) is trivial. For the first part, we use a generalized bnorm inequality as follows:
which completes the proof. □
We have the following corollary, which has a nice geometric interpretation.
Corollary 3.8
Let \(( X, \Vert \cdot \Vert _{b} ) \) be a bnormed space with constant \(s\geq 1\) and \(x_{i}\in X\), \(i\in \{ 1,\dots ,n \} \). If \(\overline{x}=\frac{x_{1}+\cdots +x_{n}}{n}\) is the gravity center of the vectors \(\{ x_{1}, \dots ,x_{n} \} \), then we have
Geometrically, the last corollary means that the sum of the edges and diagonals of a polygon with n vertices in a bnormed space is less than or equal to ntimes the sum of the distances from the gravity center to its vertices and greater than or equal to \(\frac{n}{2s^{n}}\)times this quantity.
Conclusion
Similarly, we can generalize more inequalities on metric and normed spaces.
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Karapınar, E., Noorwali, M. Dragomir and Gosa type inequalities on bmetric spaces. J Inequal Appl 2019, 29 (2019). https://doi.org/10.1186/s1366001919799
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DOI: https://doi.org/10.1186/s1366001919799
Keywords
 Dragomir and Gosa type inequalities
 bmetric space
 Inequality