Some Integral Inequalities Involving Metrics

In this work, we establish some integral inequalities involving metrics. Moreover, some applications to partial metric spaces are given. Our results are extension of previous obtained metric inequalities in the discrete case.

In [14], Dragomir and Gosa established a polygonal type inequality and provided some applications to normed linear spaces and inner product spaces. We recall below the main result obtained in [14]. Let X be a nonempty set and ρ : X × X → [0, +∞) be a metric on X (see [15]), that is, for all a, b, c ∈ X , • ρ(a, b) = 0 if and only if a = b; • ρ(a, b) = ρ(b, a); • ρ(a, b) ≤ ρ(a, c) + ρ(c, b). Let n ≥ 2 be a natural number, {x i } n i=1 ⊂ X , and {µ i } n i=1 ⊂ [0, +∞) with µ 1 + · · · µ n = 1. Then It was shown also that (1) is sharp in the following sense: there exists n ≥ 2 and for some c > 0, then c ≥ 1. Inequality (1) can be interpreted as follows: If P is a polygon having q vertices and M is a point in the space, then the sum of all edges and diagonals of P is less than q-times the sum of the distances from M to the vertices of P.
Recently, inequality (1) has been extended by some authors. In [16], Karapinar and Noorwali derived a b-metric version of (1). In [17], Aydi and Samet proved (under the above assumptions) that where m ≥ 1 is a natural number. In [18] (see also [19]) Dragomir improved inequality (1) by proving that for all α > 0, where In this paper, our goal is to derive continuous versions of inequality (2). In the next section, we recall some basic definitions. In Section 3, we present and prove our obtained results. Finally, in Section 4, some applications to partial metric spaces are provided.

The Case: ω Is Sub-Additive
Proof. Let u ∈ X be fixed. Then, for every t, s ∈ [0, A], Since ω is nondecreasing, the above inequality leads to Due to the sub-additivity of ω, it holds that Multiplying the above inequality by µ(t)µ(s) (notice that µ ≥ 0) and integrating over On the other hand, by (H2), we have Combining the above inequalities, we obtain On the other hand, using Fubini's theorem and the symmetry of ρ, we obtain Therefore, by (7) and (8), we deduce that x(s))) ds dt. (9) Finally, (4) follows from (5), (6) and (9).
Next, we study some special cases of ω. In the case ω(y) = y α , 0 < α ≤ 1, we deduce from Theorem 1 the In the special case when ω(y) = α + y 2 , α > 0, we deduce from Theorem 1 the In the case when ω(y) = arctan y, we deduce from Theorem 1 the Proof. Fix u ∈ X . Since ω is nondecreasing, we have Due to the convexity of ω, the above inequality leads to
In the special case when ω(y) = y α , α > 1, by Theorem 2, we deduce the following result.

The Case: µ ≡ A −1
Consider now the case when Then by Theorems 1-4, we deduce the following inequalities.  (s), u)) ds.
where a α is given by (3).
Similarly, by Theorems 2-4, we obtain the following inequalities.

Conclusions
Metric inequalities provide powerful tools for the study of several problems from different branches of mathematics and sciences. New integral inequalities involving metrics, sub-additive, convex, log-convex, and σ-Lipschitzian functions are established in this work, and some applications to partial metric spaces are provided. The obtained results are continuous versions of some discrete metric inequalities obtained in [14,18,19].