Abstract. Mercer [5] gave a generalization of Levinson's inequality that replaces the assumption of symmetry of the two sequences with a weaker assumptions of equality of variances. Witkowski [10] further loosened this assumption and extended the result to the class of 3-convex functions.We generalize these results to a newly defined, larger class of functions. We also prove the converse in case the function is continuous. In particular, we show that if Levinson's inequality holds under Mercer's assumptions, then the function is 3-convex.Mathematics subject classification (2010): 26D15.
The authors introduce the concept of harmonically ( , )-convex functions in second sense and establish some Ostrowski type inequalities of these classes of functions.
Graph theory has provided a very useful tool, called topological index, which is a number from the graph M with the property that every graph N isomorphic to M value of a topological index must be same for both M and N. Topological index is a descriptor in graph theory which is used to quantify the physio-chemical properties of the chemical graph. In this paper, we computed closed forms of M-polynomials for line graphs of H-naphtalenic nanotubes and chain silicate network. From M-polynomial, we obtained some topological indices based on degrees.
We define a new generalized class of harmonically preinvex functions named harmonically(p,h,m)-preinvex functions, which includes harmonic(p,h)-preinvex functions, harmonicp-preinvex functions, harmonich-preinvex functions, andm-convex functions as special cases. We also investigate the properties and characterizations of harmonically(p,h,m)-preinvex functions. Finally, we establish some integral inequalities to show the applications of harmonically(p,h,m)-preinvex functions.
Harmonic convexity is very important new class of non-convex functions, it gained prominence in the Theory of Inequalities and Applications as well as in the rest of Mathematics's branches. The harmonic convexity of a function is the basis for many inequalities in mathematics. Furthermore, harmonic convexity provides an analytic tool to estimate several known definite integrals like b a e x x n dx, b a e x 2 dx, b a sin x x n dx and b a cos xx n dx ∀n ∈ N, where a, b ∈ (0, ∞). In this article, some un-weighted inequalities of Hermite-Hadamard type for harmonic log-convex functions defined on real intervals are given.
In this article, we present an orthogonal
L
-contraction mapping concepts and prove a fixed point theorem on orthogonal complete Branciari metric spaces. As an application, we apply our major results to solving integral equations.
<abstract><p>In the present paper, we introduce the notion of a $ \mathcal{C}^{\star} $-algebra valued bipolar metric space and prove coupled fixed point theorems. Some of the well-known outcomes in the literature are generalized and expanded by the results shown. An example and application to support our result is presented.</p></abstract>
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