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.     

On some integrals over the product of three Legendre functions

The definite integrals \(\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2}[P_{\nu}(x)]^{3}\, \mathrm{d}x\) , \(\int_{-1}^{1}(1-x^{2})^{(\nu-1)/2} [P_{\nu}(x)]^{2}P_{\nu}(-x)\, \mathrm{d}x\) , \(\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2}[P_{\nu+1}(x)]^{3}\,\mathrm{d}x\) , and \(\int_{-1}^{1}x(1-x^{2})^{(\nu-1)/2} [P_{\nu+1}(x)]^{2}P_{\nu +1}(-x)\,\mathrm{d}x \) are evaluated in closed form, where P _ν is the Legendre function of degree ν, and \(\operatorname{Re}\nu>-1\) . Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.     

An inequality on elementary symmetric functions

We prove an inequality on elementary symmetric functions of nonnegative real numbers that answer a problem posed by S. Pierce in this journal.     

An inequality for 3-convex functions

In this paper an inequality for 3-convex functions analogous to well-known Levinson's inequality is proved. Some applications are also given.     

An inequality for differences of distribution functions

Ohne Zusammenfassung     

An inequality for ratios of gamma functions

Let Γ(x) denote Euler's gamma function. The following inequality is proved: for y>0 and x>1 we have

     

Multiple integrals involving product of modified Bessel functions of the second kind

The following two multiple integrals are evaluated

$$\prod\limits_{r = 1}^{m - 1} {\int\limits_0^\infty {\lambda _r^{k_r } K_{\gamma r} (\lambda _r )d\lambda _r K_\mu } \left[ {x\left( {\lambda _1 ...\lambda _{m - 1} } \right)^{ \pm 1} } \right].} $$     

An asymptotic expansion for product integration applied to Cauchy principal value integrals

Product integration methods for Cauchy principal value integrals based on piecewise Lagrangian interpolation are studied. It is shown that for this class of quadrature methods the truncation error has an asymptotic expansion in integer powers of the step-size, and that a method with an asymptotic expansion in even powers of the step-size does not exist. The relative merits of a quadrature method which employs values of both the integrand and its first derivative and for which the truncation error has an asymptotic expansion in even powers of the step-size are discussed.     

An inequality for convex functions involving G-majorization

In this paper we derive a simple inequality involving expectations of convex functions and the notion of G-majorization. The result extends a similar inequality of Marshall and Proschan (1965), J. Math. Anal. Applic. Useful applications of the more general inequality are presented.     

An inequality relating the regulator and the discriminant of a number field

Let K be a number field of degree d with regulator R_K and absolute discriminant D_K. Let r(K) be the rank of the unit group in K, and let p(K) be the maximum of r(k) as k ranges over proper subfields of K. We prove

$\text{R}{}_{\mathrm{K}}\text{> c}{}_{\mathrm{d}}\text{(}\text{log}\text{γ}{}_{\mathrm{d}}\text{D}{}_{\mathrm{K}}\text{)}{}^{\mathrm{r(K)?\rho (K)}}$

for constants c_d, γ_d > 0 depending only on d.     

Polynomial averages converge to the product of integrals

We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges inL ² to the product of the integrals. Such averages are characterized by nilsystems and so we reduce the problem to one of uniform distribution of polynomial sequences on nilmanifolds. Dedicated to Hillel Furstenberg upon his retirement The second author was partially supported by NSF grant DMS-0244994.     

A sharp Hardy-type inequality of Sugeno integrals

This paper improves on previous work presenting a Hardy-type inequality for Sugeno integrals. Indeed, we show that for (S)_∫Df(x)dx?1,