This paper is largely of expository nature. We generalize the determinantal and tracial inequalities, originating from Bloomfield-Watson and Knott, from the standpoint of majorization of eigenvalues, and observe the results as estimates of singular values of modified off-diagonal blocks of a block matrix representation in terms of the eigenvalues of the original matrix.

This is a survey work or an overview of recent developements of generalizations of the F. and M. Riesz theorem and Riesz sets on groups, emphasizing recent works on non-Abelian cases.

We survey results on linear operators leaving invariant different kinds of generalized numerical ranges and numerical radii.

A survey of some recent results on the shape of the numerical ranges associated with Lie groups, mainly convexity and star-shapedness, is given. Some questions are asked.

We explain that boundary value problems which satisfy radial solutions are reduced to a canonical form after a suitable change of variables. We introduce structure theorems to the canonical form to equations with power nonlinearities with the homogeneous Dirichlet boundary condition. By virtue of this fact, we can understand known results systematically, make clear unknown structure of various equations. As applications, we can investigate the structure of radial solutions including all solutions with singularity at r = 0 and r = 1 of Matukuma’s equation.

This is a summary report of my recent articles. Nevanlinna theory asserts that each meromorphic mapping f of Cm into Pn(C) has few defects. However, it seems that meromorphic mappings with defects are very few. In this report, we shall show that for any given transcendental meromorphic mapping of Cm into Pn(C); there is a small deformation of f which has no Nevanlinna deficient hyperplanes in Pn(C); and also in the case m = 1; there is a small deformation of f which has no Nevanlinna deficient hypersurfaces of degree · d for each given positive integer d; or deficient rational moving targets. Furthermore, we shall show that mappings without Nevanlinna defects are dense in a space of transcendental meromorphic mappings.

We demonstrate how Epstein's method using theory of Pick func-tions improves the existing results and also proves new ones on the joint con-cavity of trace functions of the form Tr (F(A1, . . . ,Ak)), where F(A1, . . .,Ak)is a matrix-valued function of positive semidefinite matrices A1, . . . ,Ak.

We study bounded and compact weighted composition operators, induced by a fixed analytic function and an analytic self-map of the open unit disk, between H1 and the Bloch space.

Let A be a C¤ -algebra and let ± be a nondegenerate coaction of a locally compact group G on A. Suppose that ± is pointwise unitary and that bA is the Hausdorff spectrum of A. Then it is shown that A has the weak Banach-Saks property and G is discrete if and only if the crossed product A £ ± G by ± has the weak Banach-Saks property.

Suppose $F$ is a nonzero function in the Hardy space $H^1$. We study the set $\{f; f \mbox{ is outer and } |F| \le \mbox{Re } f \, \mbox{ a. e. on } \partial D\}$, where $\partial D$ is the unit circle. When $F$ is a strongly outer function in $H^1$ and $\gamma$ is a positive constant, we describe the set $\{f; f \mbox{ is outer and } |F| \le \gamma \mbox{ Re } f \mbox{ and } |F^{-1}| \le \gamma \mbox{ Re } \, \, (f^{-1}) \, \, \mbox{ a. e. on } \partial D\}$. Suppose $W$ is a Helson-Szeg¨o weight. As an application, we parametrize real-valued functions $v$ in ${\mathcal L}^\infty(\partial D)$ such that the difference between $\log W$ and the harmonic conjugate function $\tilde{v}$ of $v$ belongs to ${\mathcal L}^\infty(\partial D)$ and $||v||_\infty$ is strictly less than $\pi/2$ using a contractive function $\alpha$ in $H^\infty$ such that $(1 + \alpha)=(1 - \alpha)$ is equal to the Herglotz integral of $W$.

It is known that the fractional integral $I_\alpha (0 \lt \alpha \le n)$ is bounded from $L^p({\mathbb R}^n)$ to $L^q ({\mathbb R}^n)$ when $p \gt 1$ and $n/p - \alpha = n/q \gt 0$, from $L^p({\mathbb R}^n)$ to BMO$({\mathbb R}^n)$ when $p \gt 1$ and $n/p - \alpha = 0$, from $L^p({\mathbb R}^n)$ to $\mbox{Lip}_\beta({\mathbb R}^n)$ when $p \gt 1$ and $-1 \lt n/p - \alpha = -\beta \lt 0$, from BMO$({\mathbb R}^n)$ to $\mbox{Lip}_\alpha({\mathbb R}^n)$ when $0 \lt \alpha \lt 1$, and from $\mbox{Lip}_\beta({\mathbb R}^n)$ to $\mbox{Lip}_\gamma ({\mathbb R}^n )$ when $0 \lt \alpha + \beta = \gamma \lt 1$. We introduce generalized fractional integrals and extend the above boundedness to the Orlicz spaces and $\mbox{BMO}_\phi$.

An approximateminimum, for the minimization of a function f over a feasible set S, is a point » such that f (x) ¸ f (») ¡ ² for all feasible x near the minimum point p of f on S. This concept is relevant when the problem data, or the computation, are approximate. Under regularity assumptions, an approximate minimum is a local minimum of a perturbation of the given problem. This depends on the property of a strict local minimum, that a small perturbation moves the minimum point only by a small amount.

In this paper we introduce a Stratonovich type stochastic integralwith respect to the fractional Brownian motion with Hurst parameter less than1/2. Using the techniques of the Malliavin calculus, we provide sufficientconditions for a process to be integrable. We deduce an It^o formula andwe apply these results to study stochastic differential equations driven by afractional Brownian motion with Hurst parameter less than 1/2.

We study Poincar´e inequality on unbounded domains and its applications to the semilinear elliptic equation $-\Delta u = u^p$.

In this note, we consider the dynamics of the family of transcendental entire functions f (z; a; b;c) = abz + ebz + c. In particular, while studying the case when they have a Baker domains, we investigate the area of Julia sets and the Hausdorff convergence of Julia sets.

minimax theorem involving two functions is derived, where the convexity assumptions on two functions are given by mixed conditions, and in which we answer the open question raised by Forg´o and Jo´o in [2]. Simons’s upward-downward minimax theorem and Lin-Quan’s two functions symmetric minimax theorem are generalized.

The question of which ratios of products of principal minors are bounded over all matrices in a given class has been of interest historically. This question is settled herein for the class of tridiagonal sign-symmetric Pmatrices, which essentially lies in each of the classes: positive definite, invertible totally nonnegative and M-matrices. It happens that all bounded ratios are bounded by one.

In this paper we give some criteria for the propagation of algebraic dependence of dominant meromorphic mappings from an analytic finite covering space X over the complex m-space into a projective algebraic manifold and give their applications. We study this problem under a condition on the existence of meromorphic mappings separating the fibers of X.