in this paper, the notion of n, pn - summability to generalize the concept of statisticalconvergence is used. we call this new method weighted statistically convergence. we also establish itsrelationship with statistical convergence, c,1-summability and strong   n n, p -summability.

We will introduce new means of Cauchy’s type M r,l f, μ defined, for example, as M s r,l f, μ l l− s /r r − s M r f, μ −Mr s f, μ /M l f, μ −Ml s f, μ 1/ r−l , in the case when l / r / s, l, r / 0. We will show that this new Cauchy’s mean is monotonic, that is, the following result. Theorem. Let t, r, u, v ∈ R, such that t ≤ v, r ≤ u. Then for M r,l f, μ , one has M t,r ≤ M v,u. We will also gi...

provided that f : a, b → is a convex function 1, page 137 , 2, page 1 . This result for convex functions plays an important role in nonlinear analysis. These classical inequalities have been improved and generalized in a number of ways and applied for special means including Stolarsky type, logarithmic, and p-logarithmic means. A generalization of H.H inequalities was obtained in 3–5 , 2, page ...

In this paper we introduce Levinson means of Cauchy’s type. We show that these means are monotonic.