Band 34, Heft 2

A new concept of generalised orthants in n -dimension is given and it is shown that the n -dimensional orthants are special cases of the generalized orthants in ℝ n defined by us. This generalization of orthants is very helpful in evaluating bounds of certain integrals in n -dimensional space. This fact is illustrated by an example.

The notion of weighted sharing of sets is employed to deal with the problem of uniqueness of meromorphic functions sharing three sets. We obtain two results which radically improve and extend a number of results in [2] and [3].

Some of important univalence criteria for a non-constant meromorphic function f ( z ) on the unit disk 𝔻 involve its pre-Schwarzian or Schwarzian derivative. We consider an appropriate norm for the pre-Schwarzian derivative, and discuss the problem of finding the largest possible r  ∈ (0, 1) for which the pre-Schwarzian norm of the dilation r −1 f ( rz ) is not greater than a prescribed number for normalized univalent functions f ( z ) in the unit disk. Similar results concerning the Schwarzian derivative are also obtained.

We prove two uniqueness theorems for entire functions sharing a nonzero value with higher order derivatives.

In the paper, the authors provide four alternative proofs of an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.

In this paper we extend the local separability property for 1-codimension Reifenberg sets to parabolic space.

In the paper, two sharp inequalities for bounding the psi function ψ and the harmonic numbers H n are established respectively, some known results are improved and some remarks are given.

We consider regular open curves in ℝ n with fixed boundary points, curvature equal to zero at the boundary, subject to a fixed length constraint and moving according to the L 2 -gradient flow of the elastic energy. For this flow we prove a long-time existence result and subconvergence to critical points.

Refinements of the Ostrowski inequality for functions of bounded variation in terms of the cumulative variation function are given. Applications for selfadjoint operators on complex Hilbert spaces are also provided.