Remarks on the Generalizations of Opial′s Inequality

Some remarks on generalizations of classical prime submodules

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. Suppose that $phi:S(M)rightarrow S(M)cup lbraceemptysetrbrace$ be a function where $S(M)$ is the set of all submodules of $M$. A proper submodule $N$ of $M$ is called an $(n-1, n)$-$phi$-classical prime submodule, if whenever $r_{1},ldots,r_{n-1}in R$ and $min M$ with $r_{1}ldots r_{n-1}min Nsetminusphi(N)$, then $r_{1...

SOME REMARKS ON GENERALIZATIONS OF MULTIPLICATIVELY CLOSED SUBSETS

Let R be a commutative ring with identity and Mbe a unitary R-module. In this paper we generalize the conceptmultiplicatively closed subset of R and we study some propertiesof these genaralized subsets of M. Among the many results in thispaper, we generalize some well-known theorems about multiplicativelyclosed subsets of R to these generalized subsets of M. Alsowe show that some other well-kno...

The Generalizations of Hilbert's Inequality

Generalizations of Hölder’s Inequality

Some generalized Hölder’s inequalities for positive as well as negative exponents are obtained. 2000 Mathematics Subject Classification. 26D07, 26D15.

Generalizations of Popoviciu’s inequality

We establish a general criterion for inequalities of the kind convex combination of f (x1) , f (x2) , ..., f (xn) and f (some weighted mean of x1, x2, ..., xn) ≥ convex combination of f (some other weighted means of x1, x2, ..., xn) , where f is a convex function on an interval I ⊆ R containing the reals x1, x2, ..., xn, to hold. Here, the left hand side contains only one weighted mean, while t...