On Some Metric Inequalities and Applications

We derive a new inequality in metric spaces and provide its geometric interpretation. Some applications of our result are given, including metric inequalities in Lebesgue spaces, matrices inequalities, multiplicative metric inequalities, and partial metric inequalities. Our main result is a generalization of that obtained by Dragomir and Go s a.


Introduction
In [1], Dragomir and Gosa established the following interesting inequality in metric spaces.

Theorem 1. (Dragomir-Gosa inequality).
Let V be a nonempty set equipped with a metric δ. Let N ≥ 2 be a natural number, fξ i g N i=1 ⊂ ½0,∞Þ, ∑ N i=1 ξ i = 1, and Moreover, the inequality is optimal, in the sense that the multiplicative coefficient C = 1 on the right-hand side of (1) (in front of inf ) cannot be replaced by a smaller real number.
In the special case when ξ i = 1/N, for all i ∈ f1, 2, ⋯, Ng, (1) reduces to Inequality (2) can be considered a polygonal-type inequality. Namely, it has the following geometric interpretation: let Ω be a polygon in a metric space with N vertices and ϑ be an arbitrary point in the space. Then, the sum of all edges and diagonals of Ω is less than N-times the sum of the distances from ϑ to the vertices of Ω.
In the same paper [1], the authors presented some interesting applications of inequality (1) to normed linear spaces and pre-Hilbert spaces.
In this paper, motivated by the above-mentioned work, a generalization of inequality (1) is obtained and its geometric interpretation is provided. Moreover, some applications of our result are given, including metric inequalities in Lebesgue spaces, matrices inequalities, multiplicative metric inequalities, and partial metric inequalities.
(iii) δðϑ, ρÞ ≤ δðϑ, κÞ + δðκ, ρÞ In this case, we say that ðV , δÞ is a metric space. Let ℕ be the set of positive natural numbers. Our main result is the following: Moreover, the inequality is optimal, in the sense that the multiplicative coefficient 1/2 on the right-hand side of (3) cannot be replaced by a smaller real number.
Proof. By the triangle inequality, for all ϑ ∈ V , one has which yields On the other hand, by the binomial theorem, one has where Hence, combining (5) with (6), one obtains Multiplying the above inequality by ξ i ξ j and taking the sum over i and j, it holds that Notice that due to the symmetry of δ and the fact that δ ðu, uÞ = 0, u ∈ V , one has Furthermore, using that ∑ N i=1 ξ i = 1, one obtains i.e., Hence, it follows from (9), (10), and (12) that Notice that the above inequality holds for all ϑ ∈ V . So, taking the infimum over ϑ ∈ V , (3) follows.

Journal of Function Spaces
Suppose now that there exists a certain constant M > 0 such that Taking for all ϑ ∈ V . In particular, for ϑ = ϑ 1 , one deduces that which yields (since Taking the limit as λ ⟶ 1 − in the above inequality, it holds that M ≥ 1/2. The proof is then complete. Remark 3. Taking m = 1 in Theorem 2, (3) reduces to (1).
In the special case m = 2, one deduces form Corollary 4 the following result. Corollary 6. Let V be a nonempty set equipped with a metric δ. Let N ∈ ℕ, N ≥ 2, and fϑ Inequality (20) has the following geometric interpretation.

Corollary 7.
Let Ω be a polygon in a metric space with N vertices, and ϑ be an arbitrary point in the space. Then, the sum of the squares of all edges and diagonals of Ω is less than N-times the sum of the squares of the distances from ϑ to the vertices of Ω plus the square of the sum of the distances from ϑ to the vertices of Ω. Given ρ > 0 and ϑ ∈ V , where ðV , δÞ is a metric space, we denote by B ϑ ðρÞ the closed ball in V with center ϑ and radius ρ, namely, Corollary 8. Let V be a nonempty set equipped with a metric δ.
Proof. Since fϑ i g N i=1 ⊂ B ϑ ðρÞ, one has Hence, using (3), (23), and the fact that ∑ N i=1 ξ i = 1, one deduces that 3 Journal of Function Spaces The proof is complete.

A Metric Inequality in Lebesgue Spaces.
Consider a measure space ðχ, M, μÞ and a real number r ∈ ½1,∞Þ. We denote by L r ðχ, M, μÞ the space of measurable functions f such that ð χ f j j r dμ < ∞: ð25Þ for all f ∈ L r ðχ, M, μÞ. Moreover, this inequality is optimal, in the sense that the multiplicative coefficient 1/2 on the righthand side of (26) cannot be replaced by a smaller real number.

A Matrix Inequality.
We denote by M n ðℝÞ the set of square matrices of size n ∈ ℕ, n ≥ 2, with real number coefficients. Let M ∈ M n ðℝÞ. We denote by ρðMÞ the spectral radius of M, namely, where λ i , i ∈ f1, 2, ⋯, ng, are the eigenvalues of M. We denote by σ max ðMÞ the largest singular value of M, namely, where M t is the transpose of M. For more details on matrix analysis, see, for example, [4,5]. where Moreover, the inequality is optimal, in the sense that the multiplicative coefficient 1/2 on the right-hand side of (31) cannot be replaced by a smaller real number.

A Multiplicative Metric Inequality.
We first recall the notion of multiplicative metric spaces (see [6]). A multiplicative metric on a nonempty set V is a function σ : V × V ⟶ ½1, ∞Þ satisfying the following properties: for all ϑ, ρ, κ ∈ V , In this case, we say that ðV , σÞ is a multiplicative metric space.
Proposition 13. Let V be a nonempty set equipped with a multiplicative metric σ.
Moreover, the inequality is optimal, in the sense that the multiplicative coefficient 1/2 on the right-hand side of (35) cannot be replaced by a smaller real number.
Proof. Consider the function δ : V × V ⟶ ½0,∞Þ defined by It can be easily seen that δ is a metric on V . Then, using (3) with δ as defined above, (35) follows. The optimality of (35) follows from Theorem 2.
A partial metric on a nonempty set V is a function η : V × V ⟶ ½0,∞Þ satisfying the following properties: for all ϑ, ρ, κ ∈ V , In this case, we say that ðV , ηÞ is a partial metric space.
Proposition 15. Let V be a nonempty set equipped with a partial metric η. Let m, N ∈ ℕ, N ≥ 2, where for all ϑ ∈ V . Moreover, the inequality is optimal, in the sense that the multiplicative coefficient 1/2 on the right-hand side of (39) cannot be replaced by a smaller real number.

Conclusion
A new inequality in metric spaces is proved. This inequality is a generalization of that derived by Dragomir and Gosa [1]. Moreover, we provided a geometric interpretation of our main result (see Corollary 7) and discussed some special cases including Lebesgue spaces, matrices inequalities, multiplicative metric inequalities, and partial metric inequalities.

Data Availability
The data used to support the study can be available upon request.