On certain subclasses of meromorphic functions associated with certain differential operators

In this work, we study some subordination and convolution properties of certain subclasses of meromorphic functions which are defined by a previously mentioned differential operator.     

On the eigenvalues of certain canonical higher-order ordinary differential operators

We consider the operator of taking the 2pth derivative of a function with zero boundary conditions for the function and its first p−1 derivatives at two distinct points. Our main result provides an asymptotic formula for the eigenvalues and resolves a question on the appearance of certain regular numbers in the eigenvalue sequences for p=1 and p=3.     

Pointwise convergence of eigenfunction expansions associated with ordinary differential operators

With an ordinary differential expression L = ∑ⁿ_k=0P_kD^k on an open interval I?$\text{r}$ is associated a selfadjoint operator H in a Hilbert space, possibly beyond $\text{K}$=L²(l). The set $\text{D}$H∩$\text{K}$ only depends on the generalized spectral family associated with H. It is shown that the (differentiated) eigenfunction expansion given by H converges uniformly on compact subintervals of l for functions in $\text{D}$(H)∩$\text{L}$ In case H is a semibounded selfadjoint operator in $\text{K}$=L²T, a similar result is proved for functions in $\text{D}$|H|, which is the set of all $\text{K}$∈$\text{K}$ for which there exists a sequence f_n∈(H) such that f_n→f in $\text{H}$ and (H(f_n ? f_m), f_n ? f_m → 0 as n, m → ∞.     

Hypoelliptic ordinary differential operators

Necessary and sufficient conditions for the hypoellipticity of an ordinary differential operator withC ^∞ coefficients in a neighborhood of a zero of finite order of the leading term are given. A sufficient condition for such an operator to be in a certain Hörmander class is also given.     

Self-adjoint commuting ordinary differential operators

In this paper we study self-adjoint commuting ordinary differential operators of rank two. We find sufficient conditions when an operator of fourth order commuting with an operator of order 4g+2 is self-adjoint. We introduce an equation on potentials V(x),W(x) of the self-adjoint operator \(L=(\partial_{x}^{2}+V)^{2}+W\) and some additional data. With the help of this equation we find the first example of commuting differential operators of rank two corresponding to a spectral curve of higher genus. These operators have polynomial coefficients and define commutative subalgebras of the first Weyl algebra.     

Opial-type inequalities for differential operators

Some continuous and discrete versions of Opial-type inequalities which are readily applicable to differential and difference operators are established. These generalize earlier results of Anastassiou and Pe?ari?, and of Koliha and Pe?ari?.     

Some determinant inequalities for operators

The aim of this note is to study the determinant inequalities for operators in a finite von Neumann algebra. In particular, some determinant inequalities due to Hua, Lin and Zhang are extended to the case of operators in a finite von Neumann algebra.     

Complex symplectic geometry with applications to ordinary differential operators

Complex symplectic spaces, and their Lagrangian subspaces, are defined in accord with motivations from Lagrangian classical dynamics and from linear ordinary differential operators; and then their basic algebraic properties are established. After these purely algebraic developments, an Appendix presents a related new result on the theory of self-adjoint operators in Hilbert spaces, and this provides an important application of the principal theorems.

     

Some properties for certain classes of univalent functions defined by differential inequalities

Let A be the space of functions analytic in the unit disk D = {z:|z| 1}.Let U denote the set of all functions f ∈ A satisfying the conditions f(0) = f'(0)-1 = 0 and|f'(z)(z/f(z))~2-1|1(|z|1).Also,let Ω denote the set of all functions f ∈ A satisfying the conditions f(0) = f'(0)-1 = 0and|zf'(z)-f(z)|1/2(|z|1).In this article,we discuss the properties of U and Ω.