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Dragomir and Gosa type inequalities on bmetric spaces
 Erdal Karapınar^{1}Email authorView ORCID ID profile and
 Maha Noorwali^{2}
https://doi.org/10.1186/s1366001919799
© The Author(s) 2019
 Received: 20 November 2018
 Accepted: 17 January 2019
 Published: 31 January 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.
Keywords
 Dragomir and Gosa type inequalities
 bmetric space
 Inequality
1 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])
 \((bM_{1})\) :

\(d(x, y) =0\) 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).
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])
For more examples for bmetric, we may refer, e.g., to [1–7, 9, 12] and the corresponding references therein.
Example 1.4
(see, e.g., [6])
2 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
Proof
The following corollary is a generalization of Corollary 1 in [11] to the case of a bmetric space.
Corollary 2.2
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
Proof
3 Applications
In this section we define a new notion of a bnormed space and study some of its properties.
Definition 3.1
 (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
Remark 3.4
The question now is the following: Is any bmetric induced from a bnorm? The following remark can answer this question.
Remark 3.5
 (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
Proof
Clearly, (Nb1) and (Nb2) are satisfied.
As d is homogeneous, \(\Vert cx \Vert =d ( cx,0 ) =c^{\log _{2}s+1}d ( x,0 ) =c^{\log _{2}s+1} \Vert x \Vert _{b}\).
Now, we rewrite inequality (1) in the sense of bnormed spaces and obtain some corollaries.
The following proposition is a generalization of Proposition 2 in [11] to the case of a bnormed space.
Proposition 3.7
Proof
We have the following corollary, which has a nice geometric interpretation.
Corollary 3.8
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.
4 Conclusion
Similarly, we can generalize more inequalities on metric and normed spaces.
Declarations
Funding
We declare that funding is not applicable for our paper.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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