In this paper, we establish integral inequalities of Hermite-Hadamard type for multiplicativelys-preinvex functions. We also obtain some new inequalities involving multiplicative integralsby using some properties of multiplicatively s-preinvex and preinvex functions.

The goal of the article is to introduce an order on a simple closed curve. To do this, we x two points on the curve and devide it to two arcs. We prove that such a decomposition is unique. Other auxiliary theorems about arcs are proven for preparation of the proof of the above. For simplicity, we use the following convention: a, b, c, s, r are real numbers, n is a natural number, p, q are point...

We study the relation of the ideal class group of a Dedekind domain A to that of As, where S is a multiplicatively closed subset of A. We construct examples of (a) a Dedekind domain with no principal prime ideal and (b) a Dedekind domain which is not the integral closure of a principal ideal domain. We also obtain some qualitative information on the number of non-principal prime ideals in an ar...

For simplicity, we use the following convention: a, b, c, s, r are real numbers, n is a natural number, p, q are points of E2 T , and P is a subset of the carrier of E2 T . The following propositions are true: (1) If a = a+b 2 , then a = b. (2) If r ¬ s, then r ¬ r+s 2 and r+s 2 ¬ s. (3) Let T1 be a non empty topological space, P be a subset of the carrier of T1, A be a subset of the carrier of...

Let D be a non empty set and let E be a complex-membered set. One can verify that every element of D→̇E is complex-valued. Let D be a non empty set, let E be a complex-membered set, and let F1, F2 be elements of D→̇E. Then F1 + F2 is an element of D→̇C. Then F1 − F2 is an element of D→̇C. Then F1 · F2 is an element of D→̇C. Then F1/F2 is an element of D→̇C. Let D be a non empty set, let E be a comple...

we first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a banach space and next show that if the banach space is having the opial condition, then the fixed points set of such a mapping with the convex range is nonempty. in particular, we establish that if the banach space is uniformly convex, and the range of such a mapping is bounded, closed and con...

This thesis consists of three contributions to the theory of complex approximation on Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E, then it is “usually” not possible to approximate f uniformly by functions holomorphic on all of R. In Chapter 2, we show, however, that for every open Riemann surface...

In this article, we have proved the Darboux's theorem. This theorem is important to prove the Riemann integrability. We can replace an upper bound and a lower bound of a function which is the definition of Riemann integration with convergence of sequence by Darboux's theorem. We adopt the following rules: x, y are real numbers, i, j, k are natural numbers, and p, q are finite sequences of eleme...

1 Definitions Definition 1. A graded ring is a ring S together with a set of subgroups Sd, d ≥ 0 such that S = ⊕ d≥0 Sd as an abelian group, and st ∈ Sd+e for all s ∈ Sd, t ∈ Se. One can prove that 1 ∈ S0 and if S is a domain then any unit of S also belongs to S0. A homogenous ideal of S is an ideal a with the property that for any f ∈ a we also have fd ∈ a for all d ≥ 0. A morphism of graded r...