Gamma function inequalities

We prove various new inequalities for Euler’s gamma function. One of our theorems states that the double-inequality $$\alpha \cdot \Bigl(\frac{1}{\Gamma\,(\sqrt{x})}+\frac{1}{\Gamma\,(\sqrt{y})}\Bigr) {\kern-1pt}<{\kern-1pt} \frac{1}{\Gamma\,( \sqrt{x+y-xy})}+ \frac{1}{\Gamma\,( \sqrt{xy})} {\kern-1pt} <{\kern-1pt} \beta \cdot \Bigl( \frac{1}{\Gamma\,(\sqrt{x})}+\frac{1}{\Gamma\,(\sqrt{y})}\Bigr)$$ is valid for all real numbers x,y?∈?(0,1) with the best possible constant factors $\alpha=1/\sqrt{2}=0.707...$ and β?=?1.     

Error inequalities for a generalized trapezoid rule

A generalized trapezoid rule is derived. Various error bounds for this rule are established.     

Error inequalities for an optimal quadrature formula

An optimal 3-point quadrature formula of closed type is derived. It is shown that the optimal quadrature formula has a better error bound than the well-known Simpson’s rule. A corrected formula is also considered. Various error inequalities for these formulas are established. Applications in numerical integration are given.     

Error bounds for solutions of linear equations and inequalities

Given a system of linear equations and inequalities inn variables, a famous result due to A. J. Hoffman (1952) says that the distance of any point in ⁿ to the solution set of this system is bounded above by the product of a positive constant and the absolute residual. We shall discuss explicit representations of this constant in dependence upon the pair of norms used for the estimation. A method for computing a special form of Hoffman constants is proposed. Finally, we use these results in the analysis of Lipschitz continuity for solutions of parametric quadratic programs.     

Error estimates for the approximation of semicoercive variational inequalities

Summary. An abstract error estimate for the approximation of semicoercive variational inequalities is obtained provided a certain condition holds for the exact solution. This condition turns out to be necessary as is demonstrated analytically and numerically. The results are applied to the finite element approximation of Poisson's equation with Signorini boundary conditions and to the obstacle problem for the beam with no fixed boundary conditions. For second order variational inequalities the condition is always satisfied, whereas for the beam problem the condition holds if the center of forces belongs to the interior of the convex hull of the contact set. Applying the error estimate yields optimal order of convergence in terms of the mesh size . The numerical convergence rates observed are in good agreement with the predicted ones. Received August 16, 1993 / Revised version received March 21, 1994     

Error inequalities for quintic and biquintic discrete Hermite interpolation

In this paper we shall develop a class of discrete Hermite interpolates in one and two independent variables. Further, we offer explicit error bounds in ?_∞ norm for the quintic and biquintic discrete Hermite interpolates. Some numerical examples are included to illustrate the results obtained.     

Some inequalities for the gamma function

Several inequalities involving gamma function are obtained. They are established using elementary properties of logarithmically convex functions.     

Differential inequalities and the Okamura function

Summary The theory of differential inequalities and Liapunov functions are used to characterize the integral stability, integral boundedness and integral extendability of solution of differential equations. The Liapunov functions are modifications of the Okamura function. Entrata in Redazione il giorno 8 marzo 1972.     

Error estimates for the finite element solution of variational inequalities

Summary We study the mixed finite element approximation of variational inequalities, taking as model problems the so called obstacle problem and unilateral problem. Optimal error bounds are obtained in both cases.Supported in part by National Science Foundation grant MCS 75-09457, and by Office of Naval Research grant N00014-76-C-0369     

Error estimates for the finite element solution of variational inequalities

Summary We analyze the convergence of finite element approximations of some variational inequalities namely the obstacle problem and the unilateral problem. OptimalO(h) andO(h^3/2–) error bounds for the obstacle problem (for linear and quadratic elements) and anO(h) error bound for the unilateral problem (with linear elements) are proved.Supported in part by the Institut de Recherche d'Informatique et d'Automatique and by National Science Foundation grant MCS 75-09457     

Square function inequalities for non-commutative martingales

We prove a non-commutative version of the weak-type (1,1) boundedness of square functions of martingales. More precisely, we prove that there is an absolute constantK with the following property: ifM is a semifinite von Neumann algebra with a faithful normal traceτ and (M _n ) _n=1 ^∞ is an increasing filtration of von Neumann subalgebras of (M then for any martingalex= _n=1 ^∞ inL ¹(M,τ), adapted to (M _n ) _n=1 ^∞ , there is a decomposition into two sequences (x _n ) _n=1 ^∞ and (z _n ) _n=1 ^∞ withx _n=y _n+z _nfor everyn≥1 and such that . This generalizes a result of Burkholder from classical martingale theory to non-commutative martingales. We also include some applications to martingale Hardy spaces. Supported in part by NSF grant DMS-0096696.     

Error analysis of finite element methods for mildly nonlinear variational inequalities

The finite element method is applied through the use of a variational inequality to an obstacle problem involving nonhomogeneous boundary data. For piecewise linear conforming trial functions energy norm error bounds are derived.     

Error bounds and strong upper semicontinuity for monotone affine variational inequalities

Global error bounds for possibly degenerate or nondegenerate monotone affine variational inequality problems are given. The error bounds are on an arbitrary point and are in terms of the distance between the given point and a solution to a convex quadratic program. For the monotone linear complementarity problem the convex program is that of minimizing a quadratic function on the nonnegative orthant. These bounds may form the basis of an iterative quadratic programming procedure for solving affine variational inequality problems. A strong upper semicontinuity result is also obtained which may be useful for finitely terminating any convergent algorithm by periodically solving a linear program.This material is based on research supported by Air Force Office of Scientific Research Grant AFOSR-89-0410 and National Science Foundation Grants CCR-9101801 and CCR-9157632.     

Mean inequalities in certain Banach function spaces

Boundedness of the Hardy operator and its adjoint is characterized between Banach function spaces Xq and Lp. By applying a limiting procedure, corresponding boundedness of the geometric mean operator is also derived.     

New inequalities for the Hurwitz zeta function

We establish various new inequalities for the Hurwitz zeta function. Our results generalize some known results for the polygamma functions to the Hurwitz zeta function.     

Martingale inequalities in rearrangement invariant function spaces

The Burkholder-Davis-Gundy equivalence of the square function and maximal function of a martingale is extended to the setting of rearrangement invariant function spaces. Supported in part by NSF DMS-8703815 and U.S.-Isreal Binational Science Foundation. Supported in part by U.S.-Israel Binational Science Foundation.     

Error bounds for nondifferentiable convex inequalities under a strong Slater constraint qualification

A global error bound is given on the distance between an arbitrary point in then-dimensional real spaceR ⁿ and its projection on a nonempty convex set determined bym convex, possibly nondifferentiable, inequalities. The bound is in terms of a natural residual that measures the violations of the inequalities multiplied by a new simple condition constant that embodies a single strong Slater constraint qualification (CQ) which implies the ordinary Slater CQ. A very simple bound on the distance to the projection relative to the distance to a point satisfying the ordinary Slater CQ is given first and then used to derive the principal global error bound. This material is based on research supported by National Science Foundation Grant CCR-9322479 and Air Force Office of Scientific Research grant F49620-97-1-0326.     

Some inequalities for the generalized linear distortion function

In this paper,we establish several inequalities for the the generalized linear distortion function λ(a,K) by using the monotonicity and convexity of certain combinations λ(a,K).