Edward Neuman

Inequalities involving modified Bessel functions of the first kind II

The intrinsic properties, including logarithmic convexity (concavity), of the modified Bessel functions of the first kind and some other related functions are obtained. Several inequalities involving functions under discussion are established.

Inequalities and bounds for elliptic integrals

Computable lower and upper bounds for the symmetric elliptic integrals and for Legendres incomplete integral of the first kind are obtained. New bounds are sharper than those known earlier. Several inequalities involving integrals under discussion are derived.

One- and two-sided inequalities for Jacobian elliptic functions and related results

Bounds and inequalities involving Jacobian elliptic functions are obtained. The main results are established with the aid of new bounds for the incomplete elliptic integral R F . It is also shown that some known inequalities for the trigonometric and hyperbolic functions are the limiting cases of the results proven in this paper.

Bounds for symmetric elliptic integrals

Lower and upper bounds for the four standard incomplete symmetric elliptic integrals are obtained. The bounding functions are expressed in terms of the elementary transcendental functions. Sharp bounds for the ratio of the complete elliptic integrals of the second kind and the first kind are also derived. These results can be used to obtain bounds for the product of these integrals. It is shown that an iterative numerical algorithm for computing the ratios and products of complete integrals has the second order of convergence.

On the Ky Fan inequality and related inequalities II

Refinements of the inequalities of Ky Fan [3], Wang and Wang [16], Sándor and Trif [12], and Sándor [14] are obtained. Generalizations and new proofs of some of these inequalities are also included. Mathematics subject classification (2000): 26D15, 26D99.

Moments of Dirichlet splines and their applications to hypergeometric functions

Abstract Dirichlet averages of multivariate functions are employed for a derivation of basic recurrence formulas for the moments of multivariate Dirichlet splines. An algorithm for computing the moments of multivariate simplex splines is presented. Applications to hypergeometric functions of several variables are discussed.

Convex interpolating splines of arbitrary degree III

For given data (xi, fi)i=0n (x0<x1<...<xn) we consider the possibility of finding a spline functions of arbitrary degreek (k≧3) with preassigned smoothnessl, where 1≦l≦[(k-1)/2]. The splines should be such thats(xi)=fi (i=0, 1,...,n) ands is convex or nondecreasing and convex on [x0,xn]. An explicit formula for this function as well as the conditions that guarantee the required properties are established. An algorithm for the determination of the splines and the error bounds is also included.

Inequalities for Jacobian elliptic functions and Gauss lemniscate functions

Abstract A new proof of inequalities involving Jacobian elliptic functions and their inverse functions are obtained. Similar results for the Gauss lemniscate functions are also established. Upper bounds for the inverse Jacobian elliptic functions and for the Gauss arc lemniscate functions are derived.

Inequalities for hyperbolic functions

Abstract Several inequalities involving hyperbolic functions are derived. Some of them are obtained with the aid of Stolarsky and Gini means.

Product formulas and bounds for Jacobian elliptic functions with applications

Product formulas for Jacobian elliptic functions s n, s c, s d and their reciprocals are established. Applications to Legendre’s incomplete elliptic integral of the first kind and to the arc lemniscate sine function are given. Lower and upper bounds for elliptic functions and elliptic integrals mentioned above are also derived.