Double Inequality Research Articles

A Sharp Double Inequality for Sums of Powers

It is established that the sequences and are strictly increasing and converge to and , respectively. It is shown that there holds the sharp double inequality .

On the Circumradius of a Special Class of n-Simplices

An n-simplex is called circumscriptible (or edge-incentric) if there is a sphere which is tangent to all its n(n + 1)/2 edges. We obtain a closed formula for the radius of the circumscribed sphere of a circumscriptible n-simplex, and we prove a double inequality involving the circumradius and the edge-inradius of such simplices. With these results a part of a problem posed by the authors is solved.

How inequitable is the global distribution of responsibility, capability, and vulnerability to climate change: A comprehensive indicator-based assessment

While it is generally asserted that those countries who have contributed least to anthropogenic climate change are most vulnerable to its adverse impacts some recently developed indices of vulnerability to climate change come to a different conclusion. Confirmation or rejection of this assertion is complicated by the lack of an agreed metric for measuring countries’ vulnerability to climate change and by conflicting interpretations of vulnerability. This paper presents a comprehensive semi-quantitative analysis of the disparity between countries’ responsibility for climate change, their capability to act and assist, and their vulnerability to climate change for four climate-sensitive sectors based on a broad range of disaggregated vulnerability indicators. This analysis finds a double inequity between responsibility and capability on the one hand and the vulnerability of food security, human health, and coastal populations on the other. This double inequity is robust across alternative indicator choices and interpretations of vulnerability. The main cause for the higher vulnerability of poor nations who have generally contributed little to climate change is their lower adaptive capacity. In addition, the biophysical sensitivity and socio-economic exposure of poor nations to climate impacts on food security and human health generally exceeds that of wealthier nations. No definite statement can be made on the inequity associated with climate impacts on water supply due to large uncertainties about future changes in regional water availability and to conflicting indicators of current water scarcity. The robust double inequity between responsibility and vulnerability for most climate-sensitive sectors strengthens the moral case for financial and technical assistance from those countries most responsible for climate change to those countries most vulnerable to its adverse impacts. However, the complex and geographically heterogeneous patterns of vulnerability factors for different climate-sensitive sectors suggest that the allocation of international adaptation funds to developing countries should be guided by sector-specific or hazard-specific criteria despite repeated requests from participants in international climate negotiations to develop a generic index of countries’ vulnerability to climate change.

An optimal double inequality among the one-parameter, arithmetic and harmonic means

For \(p\in\mathbb{R}\), the one-parameter mean \(J_{p}(a,b)\), arithmetic mean \(A(a,b)\), and harmonic mean \(H(a,b)\) of two positive real numbers \(a\) and \(b\) are defined by\begin{equation*}J_{p}(a,b)=\begin{cases}\tfrac{p(a^{p+1}-b^{p+1})}{(p+1)(a^p-b^p)}, & a\neq b,p\neq 0,-1,\\\tfrac{a-b}{\log{a}-\log{b}}, & a\neq b,p=0,\\\tfrac{ab(\log{a}-\log{b})}{a-b}, & a\neq b,p=-1,\\a, & a=b,\end{cases}\end{equation*}\(A(a,b)=\tfrac{a+b}{2}\), and \(H(a,b)=\tfrac{2ab}{a+b}\),respectively. In this paper, we answer the question: For \(\alpha\in(0,1)\), what are the greatest value \(r_{1}\) and the least value \(r_{2}\) such that the double inequality \(J_{r_{1}}(a,b)<\alpha A(a,b)+(1-\alpha)H(a,b)<J_{r_{2}}(a,b)\) holds for all \(a,b>0\) with \(a\neq b\)?

A Double Inequality for the Equivalent Impulse Bandwidth

In this paper, we derive a double inequality that permits one to obtain a lower and an upper limit for the value of the impulse bandwidth of a measuring receiver. The limits are the reciprocal of the integral of the relative envelope of the impulse response (lower limit) and the integral of the relative frequency response (upper limit) of the intermediate frequency (IF) filter. Since the limits are relative quantities, their evaluation does not require the use of a calibrated generator, the only significant sources of error being receiver's vertical scale nonlinearity and noise proximity. Here, the deviation between the impulse bandwidth and its limits is quantified for practical IF filter configurations. The dominant contributions to measurement uncertainty are identified and suggestions for reducing their magnitude are also offered.

Are the affluent prepared to pay for the planet? Explaining willingness to pay for public and quasi-private environmental goods in Switzerland

A large number of ‘environmental justice’ studies show that wealthier people are less affected by environmental burdens and also consume more resources than poorer people. Given this double inequity, we ask, to what extent are affluent people prepared to pay to protect the environment? The analyses are couched within the compensation/affluence hypothesis, which states that wealthier persons are able to spend more for environmental protection than their poorer counterparts. Further, we take into account various competing economic, psychological and sociological determinants of individuals’ willingness to pay (WTP) for both public environmental goods (e.g., general environmental protection) and quasi-private environmental goods (e.g., CO2-neutral cars). Such a comprehensive approach contrasts with most other studies in this field that focus on a limited number of determinants and goods. Multivariate analyses are based on a general population survey in Switzerland (N = 3,369). Although income has a positive and significant effect on WTP supporting the compensation hypothesis, determinants such as generalized interpersonal trust that is assumed to be positively associated with civic engagement and environmental concern prove to be equally important. Moreover, we demonstrate for the first time that time preferences can considerably influence survey-based WTP for environmental goods; since investments in the environment typically pay off in the distant future, persons with a high subjective discount rate are less likely to commit.

Estimating gamma function by digamma function

The aim of this paper is to refine some recent results stated by Alzer and Grinshpan [H. Alzer, A.Z. Grinshpan, Inequalities for the gamma and q -gamma functions J. Approx. Theory 144 (2007) 67–83] and Batir [N. Batir, An interesting double inequality for Euler’s gamma function J. Inequal. Pure Appl. Math. 5(4) (2004) article 97] concerning the problem of approximation of the gamma function in terms of the digamma function.

Some inequalities for f-divergence measures generated by 2n-convex functions

A double Jensen type inequality for 2n-convex functions is obtained and applied to establish upper and lower bounds for the f-divergence measure in Information Theory. Some particular inequalities of interest are stated as well.

Two optimal double inequalities between power mean and logarithmic mean

For p ∈ R the power mean M p ( a , b ) of order p , the logarithmic mean L ( a , b ) and the arithmetic mean A ( a , b ) of two positive real values a and b are defined by M p ( a , b ) = { ( a p + b p 2 ) 1 p , p ≠ 0 , a b , p = 0 , L ( a , b ) = { b − a log b − log a , a ≠ b , a , a = b and A ( a , b ) = a + b 2 , respectively. In this article, we answer the questions: What are the greatest values p and r , and the least values q and s , such that the inequalities M p ( a , b ) ≤ 1 3 A ( a , b ) + 2 3 L ( a , b ) ≤ M q ( a , b ) and M r ( a , b ) ≤ 2 3 A ( a , b ) + 1 3 L ( a , b ) ≤ M s ( a , b ) hold for all a , b > 0 ?

On an integral inequality and application to Poisson’s equation

Let n ∈ N , m ± : = max / min { 1 , 8 n + 4 } and let I ( x ) = ⨍ B n ( 1 − | y | 2 ) ( | x − y | | y | | x − y ∗ | ) − n / 2 d y , where B n be the unit ball in R n . It is proved the double sharp inequality m − ⩽ I n ( x ) ⩽ m + . As an application, we obtain the following: if u is a solution to homogeneous Dirichlet’s problem of Poisson’s equation Δ u = g , g ∈ L ∞ , in the unit disk B 2 , then there holds the inequality | ∇ u | ⩽ 2 3 | g | ∞ .

An Optimal Double Inequality for Means

For , the generalized logarithmic mean , arithmetic mean and geometric mean of two positive numbers and are defined by , ; , , , ; , , ; , , ; and , respectively. In this paper, we give an answer to the open problem: for , what are the greatest value and the least value , such that the double inequality holds for all ?

An Optimal Double Inequality between Power-Type Heron and Seiffert Means

For , the power-type Heron mean and the Seiffert mean of two positive real numbers and are defined by , ; , and , ; , , respectively. In this paper, we find the greatest value and the least value such that the double inequality holds for all with .

Optimal Power Mean Bounds for the Weighted Geometric Mean of Classical Means

For , the power mean of order of two positive numbers and is defined by , for , and , for . In this paper, we answer the question: what are the greatest value and the least value such that the double inequality holds for all and with ? Here , , and denote the classical arithmetic, geometric, and harmonic means, respectively.

The Optimal Convex Combination Bounds of Arithmetic and Harmonic Means for the Seiffert's Mean

We find the greatest value and least value such that the double inequality holds for all with . Here , , and denote the arithmetic, harmonic, and Seiffert's means of two positive numbers and , respectively.

A general form of Jordan-type double inequality for the generalized and normalized Bessel functions

In this note, a general form of Jordan-type double inequality involving the generalized and normalized Bessel functions is presented, and then some recent results concerning generalized and sharp work of Jordan’s inequality are extended. At the same time, the applications of the results above give two new infinite series for sin x / x and sinh x / x .

An optimal limiting 2𝐷 Sobolev inequality

The main goal of this paper is to prove an optimal limiting Sobolev inequality in two dimensions for Hölder continuous functions. Additionally, from this inequality we derive the double logarithmic inequality\[‖u‖L∞⩽‖∇u‖L22παln⁡(1+62πα‖u‖C˙α‖∇u‖L2ln⁡(1+2πα‖u‖C˙α‖∇u‖L2))\|u\|_{L^{\infty }} \leqslant \frac {\|\nabla u\|_{L^2}}{\sqrt {2\pi \alpha }} \sqrt {\ln \Bigl (1+6\sqrt {2\pi \alpha } \tfrac {\|u\|_{{\left .\rm \! \dot C\right .^{\!\alpha }}}}{\|\nabla u\|_{L^{2}}} \sqrt {\ln (1+\sqrt {2\pi \alpha } \tfrac {\|u\|_{{\left .\rm \! \dot C\right .^{\!\alpha }}}} {\|\nabla u\|_{L^{2}}} )\!}\, \Bigr )}\]for functionsu∈W01,2(B1)u\in W^{1,2}_0(B_1)on the unit diskB1B_1inR2\mathbb R^2,α∈(0,1].\alpha \in (0,1].

Inequalities for the Euler–Mascheroni constant

Let γ = 0.577215 … be the Euler–Mascheroni constant, and let R n = ∑ k = 1 n 1 k − log ( n + 1 2 ) . We prove that for all integers n ≥ 1 , 1 24 ( n + a ) 2 ≤ R n − γ < 1 24 ( n + b ) 2 with the best possible constants a = 1 24 [ − γ + 1 − log ( 3 / 2 ) ] − 1 = 0.55106 … and b = 1 2 . This refines the result of D. W. DeTemple, who proved that the double inequality holds with a = 1 and b = 0 .

Approximating real Pochhammer products: a comparison with powers

Abstract Accurate estimates of real Pochhammer products, lower (falling) and upper (rising), are presented. Double inequalities comparing the Pochhammer products with powers are given. Several examples showing how to use the established approximations are stated.

On doubling inequalities for elliptic systems

We prove doubling inequalities for solutions of elliptic systems with an iterated Laplacian as diagonal principal part and for solutions of the Lame' system of isotropic linearized elasticity. These inequalities depend on global properties of the solutions.

A CLASS OF COMPLETELY MONOTONIC FUNCTIONS INVOLVING DIVIDED DIFFERENCES OF THE PSI AND TRI-GAMMA FUNCTIONS AND SOME APPLICATIONS

A class of functions involving divided differences of the psi and tri-gamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving the ratio of two gamma functions and originating from the establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in <TEX>$\mathbb{R}^{n-1}$</TEX> and <TEX>$\mathbb{R}^n$</TEX> respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.