This article examines integral inequalities dealing with functions of the form “a function raised to the power of another function” under varying monotonicity and convexity assumptions. First, we assess the validity of a referenced theorem on the subject. Specifically, we present a counterexample and identify a gap in its proof. We then propose an alternative version of the theorem with more flexible convexity assumptions. In addition, we establish new lower and upper bounds for the same integral using refined Hermite–Hadamard integral inequalities. A complementary variant is also discussed. Thus, our results fill gaps in the literature and extend existing results on integral inequalities under classical assumptions.
Language:
EN
| Published:
30-09-2005
|
Abstract
| pp. 23-29
Two dual sequence functions describing some kind of local convexity and dimension of subspaces of linear metric spaces are introduced. It is shown that the functions give a useful tool in the investigations of fixed point properties of the Schauder type.
In this paper, we deal with the question: under what conditions n distinct points Pi(xi,yi) (i=1,...,n) provided x1<...<xn form a convex polygon? One of the main findings of the paper can be stated as follows: Let P1(x1,y1),..., Pn(xn,yn) be n distinct points (n≥3) with x1<...<xn. Then \overline{P1P2},...,\overline{PnP1} form a convex n-gon lying in the half-space \underline{H} = {(x,y)| x∈ℝ and y≤y1 + [(x-x1)/(xn-x1)](yn-y1)} ⊆ ℝ2 if and only if the following inequality holds (yi-yi-1)/(xi-xi-1) ≤ (yi+1-yi)/(xi+1-xi) for all i∈{2,...,n−1}. Based on this result, we establish a connection between the property of sequential convexity and convex polygon. We show that in a plane if any n points are scattered in such a way that their horizontal and vertical distances preserve some specific monotonic properties, then those points form a 2-dimensional convex polytope.
Language:
EN
| Published:
08-01-2025
|
Abstract
| pp. 248-268
The sequence spaces ruℓ∞(O, ∇q), ruℓp(O, ∇q), ruc(O, ∇q), ruc0(O, ∇q), rumφ(O, ∇q, p), runφ(O, ∇q, p), rumφ(O, ∇q), runφ(O, ∇q) are defined by the Orlicz function in this article. We examine all of its characteristics, including symmetry, solidity, and completeness. A few geometric properties on convexity on the space rumφ(O, ∇q, p) are also examined in this article.
Language:
EN
| Published:
30-09-1996
|
Abstract
| pp. 7-12
Let I be an interval and M,N: I×I→I some means with the strict internality property. Suppose that ϕ: I→ℝ is a non-constant and continuous solution of the functional equation ϕ(M(x,y)) + ϕ(N(x,y)) = ϕ(x) + ϕ(y). Then ϕ is one-to-one; moreover for every lower semicontinuous function f: I→ℝ satisfying the inequality f(M(x,y)) + f(N(x,y)) ≤ f(x) + f(y), the function f◦ϕ-1 is convex on ϕ(I). This is a generalization of an earlier result of Zs. Páles. An application to the a-Wright convex function is given.
Teodoro Lara
,
Nelson Merentes
,
Edgar Rosales
,
Ambrosio Tineo
Language:
EN
| Published:
31-01-2018
|
Abstract
| pp. 237-245
In this research we deal with algebraic properties and characterizations of convex functions in the context of a time scale; this notion of convexity has been studied for some other authors but the setting of properties are establish here. Moreover, characterizations, a separation theorem and an inequality of Jensen type for this class of functions are shown as well.
2018-01-31
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English