A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw's double inequality

In the article, the sufficient and necessary conditions such that a class of functions which involve the psi function ψ$\psi$ and the ratio Γ(x+t)/Γ(x+s)$\Gamma (x+t)/\Gamma (x+s)$ are logarithmically completely monotonic are established, the best bounds for the ratio Γ(x+t)/Γ(x+s)$\Gamma (x+t)/\Gamma (x+s)$ are given, and some comparisons with known results are carried out, where s and t   are two real numbers and x>-min{s,t}$x>-\min \{s,t\}$.     

A class of logarithmically completely monotonic functions and the best bounds in the first Kershaw's double inequality

In the article, the logarithmically complete monotonicity of a class of functions involving Euler's gamma function are proved, a class of the first Kershaw-type double inequalities are established, and the first Kershaw's double inequality and Wendel's inequality are generalized, refined or extended. Moreover, an open problem is posed.     

A Hardy-type inequality in two dimensions

Necessary and sufficient conditions are given for a weighted norm inequality for the sum of two-dimensional Hardy-type integral operators with not necessarily non-negative coefficients.     

Hypersurfaces with constant mean curvature and prescribed area

Summary We consider—in the setting of geometric measure theory—hypersurfacesT (of codimension one) with prescribed boundaryB in Euclideann+1 space which maximize volume (i.e.T together with a fixed hypersurfaceT ₀ encloses oriented volume) subject to a mass constraint. We prove existence and optimal regularity of solutionsT of such variational problems and we show that, on the regular part of its support,T is a classical hypersurface of constant mean curvature. We also prove that the solutionsT become more and more spherical as the valuem of the mass constraint approaches ∞. This work was done at the Centre for Mathematics and its Applications at the Australian National University, Canberra while the author was a visiting member This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

A new lower bound in the second Kershaw's double inequality

In the paper, a new and elegant lower bound in the second Kershaw's double inequality is established, some alternative simple and polished proofs are given, several deduced functions involving the gamma and psi functions are proved to be decreasingly monotonic and logarithmically completely monotonic, and some remarks and comparisons are stated.     

An inequality relating algebraic and trigonometric derivatives

Using the correspondence x↔ cos θ, where −1≤x ≤ 1 and 0 ≤ θ ≤ π, a function f(x) defined on [−1, 1] can be represented as a 2π-periodic function F(θ), and then the derivative f′(x) corresponds to . From these observations, weighted-norm estimates for first and higher derivatives by x will be obtained, using a generalized Hardy inequality. The results in turn imply the generalized Hardy inequality upon which they depend and will hold true in any weighted norm for which the generalized Hardy is true.     

A weighted version of a rearrangement inequality

We show that if a positive absolutely continous measure causes a special relative isoperimetric inequality to hold, then Dirichlet-type integrals of sufficiently smooth real-valued functions decrease under an appropriate equimeasurable rearrangement.

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Sunto Si dimostra che se una misura, positiva e assolutamente continua, rende valida una certa disuguaglianza isoperimetrica relativa, allora integral del tipo di Dirichlet di funzioni sufficientemente regolari descrescono per effetto di un opportuno riordinamento equidistribuito.

     

On two inequalities similar to Opial's inequality in two independent variables

In the present paper we establish two new integral inequalities similar to Opial's inequality in two independent variables. The inequalities established in this paper are similar to the analogues of Calvert's generalizations of Opial's inequality, in two independent variables and contains in the special case the analogue of Opial's inequality given by G. S. Yang in two independent variables.     

A fractional differential inequality with application

We establish a fractional differential inequality using desingularization techniques combined with some generalizations of algebraic Bihari-type inequalities. We use this inequality to prove global existence and determine the asymptotic behavior of solutions for a family of fractional differential equations.     

The best constant in a weighted Hardy-Littlewood-Sobolev inequality

We prove the uniqueness for the solutions of the singular nonlinear PDE system:

(1)

In the special case when and , we classify all the solutions and thus obtain the best constant in the corresponding weighted Hardy-Littlewood-Sobolev inequality.

     

Stability for hypersurfaces of constant mean curvature with free boundary

The partitioning problem for a smooth convex bodyB ³ consists in to study, among surfaces which divideB in two pieces of prescribed volume, those which are critical points of the area functional.We study stable solutions of the above problem: we obtain several topological and geometrical restrictions for this kind of surfaces. In the case thatB is a Euclidean ball we obtain stronger results.Antonio Ros is partially supported by DGICYT grant PB91-0731 and Enaldo Vergasta is partially supported by CNPq grant 202326/91-8.     

The Hermite-Hadamard inequality for log-convex functions

We discuss the existence of a strengthening of Hermite-Hadamard inequality in the case of log-convex functions. Unlike the classical case, which belongs to the field of linear functional analysis, this analogue involves nonlinear means such as the geometric mean and the logarithmic mean.     

On the cases of equality in Bobkov's inequality and Gaussian rearrangement

We determine all of the cases of equality in a recent inequality of Bobkov that implies the isoperimetric inequality on Gauss space. As an application we determine all of the cases of equality in the Gauss space analog of the Faber-Krahn inequality. Received June 21, 1999 / Accepted June 12, 2000 / Published online November 9, 2000     

On the de Bruijn-Newman constant

If λ(0) denotes the infimum of the set of real numbers λ such that the entire function Ξλ represented by

     

An uncertainty inequality for Fourier-Dunkl series

An uncertainty inequality for the Fourier-Dunkl series, introduced by the authors in [Ó. Ciaurri, J.L. Varona, A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform, Proc. Amer. Math. Soc. 135 (2007) 2939-2947], is proved. This result is an extension of the classical uncertainty inequality for the Fourier series.     

On the isoperimetric inequality for the buckling of a clamped plate

In two dimensions, we prove the existence of a domain minimizing the buckling load of a clamped plate among all domains of given measure. In order to prove that the optimal set is the disk, we discuss possible ways of applying an idea of Willms and Weinberger if the regularity of the optimal set is not a priori assumed.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).