scholarly journals Properties of a Generalized Class of Weights Satisfying Reverse Hölder’s Inequality

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
S. H. Saker ◽  
S. S. Rabie ◽  
R. P. Agarwal
Keyword(s):  
Upper Bounds ◽  
Hölder Inequality ◽  
The Self ◽  
Power Mean ◽  
Lower And Upper Bounds ◽  
Real Numbers ◽  
Positive Real ◽  
Fundamental Properties ◽  
Reverse Hölder ◽  
Special Case

In this paper, we will prove some fundamental properties of the discrete power mean operator M p u n = 1 / n ∑ k = 1 n   u p k 1 / p , for   n ∈ I ⊆ ℤ + , of order p , where u is a nonnegative discrete weight defined on I ⊆ ℤ + the set of the nonnegative integers. We also establish some lower and upper bounds of the composition of different operators with different powers. Next, we will study the structure of the generalized discrete class B p q B of weights that satisfy the reverse Hölder inequality   M q u ≤ B M p u , for positive real numbers p , q , and B such that 0 < p < q and B > 1 . For applications, we will prove some self-improving properties of weights from B p q B and derive the self improving properties of the discrete Gehring weights as a special case. The paper ends by a conjecture with an illustrative sharp example.

scholarly journals On 3-Rainbow Domination Number of Generalized Petersen Graphs P(6k,k)

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1860
Author(s):  
Rija Erveš ◽  
Janez Žerovnik
Keyword(s):  
Domination Number ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Generalized Petersen Graphs ◽  
Rainbow Domination ◽  
Petersen Graphs ◽  
Special Case ◽  
Domination Numbers

We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs P(6k,k). In some cases, for some infinite families, exact values are established; in all other cases, the lower and upper bounds with small gaps are given. We also define singleton rainbow domination, where the sets assigned have a cardinality of, at most, one, and provide analogous results for this special case of rainbow domination.

Generalizing the McClelland Bounds for Total π-Electron Energy

2008 ◽  
Vol 63 (5-6) ◽  
pp. 280-282 ◽  
Author(s):  
Ivan Gutman ◽  
Gopalapillai Indulal ◽  
Roberto Todeschinic
Keyword(s):  
Electron Energy ◽  
Distance Matrix ◽  
Upper Bounds ◽  
Arithmetic Mean ◽  
Lower And Upper Bounds ◽  
Real Numbers ◽  
Laplacian Energy ◽  
Graph Energy

In 1971 McClelland obtained lower and upper bounds for the total π-electron energy. We now formulate the generalized version of these bounds, applicable to the energy-like expression EX = Σni =1 |xi − x̅|, where x1,x2, . . . ,xn are any real numbers, and x̅ is their arithmetic mean. In particular, if x1,x2, . . . ,xn are the eigenvalues of the adjacency, Laplacian, or distance matrix of some graph G, then EX is the graph energy, Laplacian energy, or distance energy, respectively, of G.

scholarly journals Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Dug Hun Hong ◽  
Jae Duck Kim
Keyword(s):  
Lebesgue Measure ◽  
Upper Bound ◽  
Convex Functions ◽  
Hölder Inequality ◽  
Concave Functions ◽  
Lebesgue Integral ◽  
Reverse Hölder ◽  
Special Case

The classical Hölder inequality shows an interesting upper bound for Lebesgue integral of the product of two functions. This paper proposes Hölder type inequalities and reverse Hölder type inequalities for Sugeno integrals under usual multiplication operations for nonincreasing concave or convex functions. One of the interesting results is that the inequality, (S)∫01f(x)pdμ1/p(S)∫01g(x)qdμ1/q≤p-q/p-p-q+1∨q-p/q-q-p+1(S)∫01f(x)g(x)dμ, where 1<p<∞,1/p+1/q=1 and μ is the Lebesgue measure on R, holds if f and g are nonincreasing and concave functions. As a special case, we consider Cauchy-Schwarz type inequalities for Sugeno integrals involving nonincreasing concave or convex functions. Some examples are provided to illustrate the validity of the proposed inequalities.

RECTANGLE AND BOX VISIBILITY GRAPHS IN 3D

1999 ◽  
Vol 09 (01) ◽  
pp. 1-27 ◽  
Author(s):  
SÁNDOR R. FEKETE ◽  
HENK MEIJER
Keyword(s):  
Complete Graph ◽  
Dimensional Space ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
3 Dimensional ◽  
Visibility Graphs ◽  
Axis Parallel ◽  
Parallel Direction ◽  
Visibility Representations ◽  
Special Case

We discuss rectangle and box visibility representations of graphs in 3-dimensional space. In these representations, vertices are represented by axis-aligned disjoint rectangles or boxes. Two vertices are adjacent if and only if their corresponding boxes see each other along a small axis-parallel cylinder. We concentrate on lower and upper bounds for the size of the largest complete graph that can be represented. In particular, we examine these bounds under certain restrictions: What can be said if we may only use boxes of a limited number of shapes? Some of the results presented are as follows: • There is a representation of K8 by unit boxes. • There is no representation of K10 by unit boxes. • There is a representation of K56, using 6 different box shapes. • There is no representation of K184 by general boxes. A special case arises for rectangle visibility graphs, where no two boxes can see each other in the x- or y-directions, which means that the boxes have to see each other in z-parallel direction. This special case has been considered before; we give further results, dealing with the aspects arising from limits on the number of shapes.

scholarly journals Single source unsplittable flows with arc-wise lower and upper bounds

2021 ◽  
Author(s):  
Sarah Morell ◽  
Martin Skutella
Keyword(s):  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Common Source ◽  
Lower And Upper Bounds ◽  
New Approach ◽  
Math Program ◽  
Unsplittable Flow ◽  
The Common ◽  
Special Case ◽  
Single Path

AbstractIn a digraph with a source and several destination nodes with associated demands, an unsplittable flow routes each demand along a single path from the common source to its destination. Given some flow x that is not necessarily unsplittable but satisfies all demands, it is a natural question to ask for an unsplittable flow y that does not deviate from x by too much, i.e., $$y_a\approx x_a$$ y a ≈ x a for all arcs a. Twenty years ago, in a landmark paper, Dinitz et al. (Combinatorica 19:17–41, 1999) proved that there exists an unsplittable flow y such that $$y_a\le x_a+d_{\max }$$ y a ≤ x a + d max for all arcs a, where $$d_{\max }$$ d max denotes the maximum demand value. Our first contribution is a considerably simpler one-page proof for this classical result, based upon an entirely new approach. Secondly, using a subtle variant of this approach, we obtain a new result: There is an unsplittable flow y such that $$y_a\ge x_a-d_{\max }$$ y a ≥ x a - d max for all arcs a. Finally, building upon an iterative rounding technique previously introduced by Kolliopoulos and Stein (SIAM J Comput 31:919–946, 2002) and Skutella (Math Program 91:493–514, 2002), we prove existence of an unsplittable flow that simultaneously satisfies the upper and lower bounds for the special case when demands are integer multiples of each other. For arbitrary demand values, we prove the weaker simultaneous bounds $$x_a/2-d_{\max }\le y_a\le 2x_a+d_{\max }$$ x a / 2 - d max ≤ y a ≤ 2 x a + d max for all arcs a.

Additive Functionals on Lorentz Spaces

1984 ◽  
Vol 27 (1) ◽  
pp. 31-37
Author(s):  
Pratibha G. Ghatage
Keyword(s):  
Sufficient Conditions ◽  
Concave Function ◽  
Additive Functional ◽  
Real Numbers ◽  
Positive Real ◽  
Additive Functionals ◽  
Atomic Measure ◽  
Necessary And Sufficient ◽  
Special Case ◽  
Lorentz Norm

AbstractIf (X, β, μ) is a σ-finite, non-atomic measure space, and ϕ is an increasing non-negative concave function defined on the positive real numbers, we give a set of necessary and sufficient conditions for an additive functional T on the Lorentz space Nϕ to have an integral representation with a Caratheodory kernel. In the special case when T is statistical we classify the functional properties (enjoyed by the kernels) in terms of the Lorentz norm on the space.

scholarly journals KETAKSAMAAN HERMITE-HADAMARD TERHADAP INTEGRAL RIEMANN-STIELTJES

2012 ◽  
Vol 4 (1) ◽  
pp. 59
Author(s):  
Denny Ivanal Hakim ◽  
Hendra Gunawan
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Closed Interval ◽  
Mean Value ◽  
Stieltjes Integral ◽  
Power Mean ◽  
Real Numbers ◽  
Positive Real ◽  
Hadamard Inequality

The Hermite-Hadamard inequality is an inequality for convex functions that gives an estimate for the integral mean value of a convex function on a closed interval by its value at the middle of interval and the average of its values at the endpoints. The Hermite-Hadamard inequality can be generalized by using the Riemann-Stieltjes integral mean value.  An application of the Hermite-Hadamard inequality with respect to Riemann-Stieltjes integral  for estimating the power mean of   positive real numbers by the aritmethic mean is given at the end of discussion.

scholarly journals The general Albertson irregularity index of graphs

2021 ◽  
Vol 7 (1) ◽  
pp. 25-38
Author(s):  
Zhen Lin ◽  
◽  
Ting Zhou ◽  
Xiaojing Wang ◽  
Lianying Miao ◽  
...  
Keyword(s):  
Real Number ◽  
Connected Graph ◽  
Positive Real Number ◽  
Upper Bounds ◽  
Calculation Formula ◽  
Lower And Upper Bounds ◽  
Positive Real ◽  
Irregularity Index ◽  
Degree Of Vertex ◽  
Extremal Value

<abstract><p>We introduce the general Albertson irregularity index of a connected graph $ G $ and define it as $ A_{p}(G) = (\sum_{uv\in E(G)}|d(u)-d(v)|^p)^{\frac{1}{p}} $, where $ p $ is a positive real number and $ d(v) $ is the degree of the vertex $ v $ in $ G $. The new index is not only generalization of the well-known Albertson irregularity index and $ \sigma $-index, but also it is the Minkowski norm of the degree of vertex. We present lower and upper bounds on the general Albertson irregularity index. In addition, we study the extremal value on the general Albertson irregularity index for trees of given order. Finally, we give the calculation formula of the general Albertson index of generalized Bethe trees and Kragujevac trees.</p></abstract>

scholarly journals The reverse Holder inequality for an elementary function

2021 ◽  
Vol 56 (1) ◽  
pp. 28-38
Author(s):  
A.O. Korenovskii
Keyword(s):  
Characteristic Function ◽  
Elementary Function ◽  
Positive Function ◽  
Step Function ◽  
Hölder Inequality ◽  
Reverse Hölder Inequality ◽  
Power Mean ◽  
Reverse Hölder ◽  
Extremal Properties

For a positive function $f$ on the interval $[0,1]$, the power mean of order $p\in\mathbb R$ is defined by \smallskip\centerline{$\displaystyle\|\, f\,\|_p=\left(\int_0^1 f^p(x)\,dx\right)^{1/p}\quad(p\ne0),\qquad\|\, f\,\|_0=\exp\left(\int_0^1\ln f(x)\,dx\right).$} Assume that $0<A<B$, $0<\theta<1$ and consider the step function$g_{A<B,\theta}=B\cdot\chi_{[0,\theta)}+A\cdot\chi_{[\theta,1]}$, where $\chi_E$ is the characteristic function of the set $E$. Let $-\infty<p<q<+\infty$. The main result of this work consists in finding the term \smallskip\centerline{$\displaystyleC_{p<q,A<B}=\max\limits_{0\le\theta\le1}\frac{\|\,g_{A<B,\theta}\,\|_q}{\|\,g_{A<B,\theta}\,\|_p}.$} \smallskip For fixed $p<q$, we study the behaviour of $C_{p<q,A<B}$ and $\theta_{p<q,A<B}$ with respect to $\beta=B/A\in(1,+\infty)$.The cases $p=0$ or $q=0$ are considered separately. The results of this work can be used in the study of the extremal properties of classes of functions, which satisfy the inverse H\"older inequality, e.g. the Muckenhoupt and Gehring ones. For functions from the Gurov-Reshetnyak classes, a similar problem has been investigated in~[4].

scholarly journals Some basic properties of Sombor indices

2021 ◽  
Vol 4 (1) ◽  
pp. 1-3
Author(s):  
Ivan Gutman ◽  
Keyword(s):  
Molecular Structure ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Lower And Upper Bounds ◽  
Basic Properties ◽  
Special Case

The recently introduced class of vertex-degree-based molecular structure descriptors, called Sombor indices (\(SO\)), are examined and a few of their basic properties established. Simple lower and upper bounds for \(SO\) are determined. It is shown that any vertex-degree-based descriptor can be viewed as a special case of a Sombor-type index.

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