We study Carleson inequalities in a framework of parabolic Hardy spaces. Similar results for parabolic Bergman spaces are discussed in [NSY1] (see also [NSY2]), where $\tau$-Carleson measures play an important roll. In the present case, $T_{\tau}$-Carleson measures are useful. We give an relation between these measures.

For every $q\in(0,1)$ and $0\le \alpha \lt 1$ we define a class of analytic functions, the so-called $q$-starlike functions of order $\alpha$, on the open unit disk. We study this class of functions and explore some inclusion properties with the well-known class of starlike functions of order $\alpha$. The paper is also devoted to the discussion on the Herglotz representation formula for analytic functions $zf'(z)/f(z)$ when $f(z)$ is $q$-starlike of order $\alpha$. As an application we also discuss the Bieberbach conjecture problem for the $q$-starlike functions of order $\alpha$.

In this paper, we discuss with $n$-dimensional complete orientable linear Weingarten hypersurface in locally symmetric manifold and obtain some rigidity results.

From Realization Lemma established by N. Tanaka, differential systems may be regarded as systems of first order differential equations. We characterize the geometric structure of systems of second order partial differential equations of several unknown functions in terms of differential systems and seek a system of equations the Lie algebra of all infinitesimal automorphisms of which is simple.

In 1927 Littlewood constructed a bounded holomorphic function on the unit disc, having no tangential boundary limits almost everywhere. This theorem was the complement of a positive theorem of Fatou (1906), establishing almost everywhere non-tangential convergence of bounded holomorphic functions. There are several generalizations of Littlewood's theorem whose proofs are based on the specific properties of holomorphic functions. Applying real variable methods, we extend these theorems to general convolution operators.

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The annihilator graph of $R$ is defined as the undirected graph $AG(R)$ with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $ann_R(xy)\neq ann_R(x)\cup ann_R(y)$. In this paper, we study the affinity between annihilator graph and zero-divisor graph associated with a commutative ring. For instance, for a non-reduced ring $R$, it is proved that the annihilator graph and the zero-divisor graph of $R$ are identical to the join of a complete graph and a null graph if and only if $ann_R(Z(R))$ is a prime ideal if and only if $R$ has at most two associated primes. Among other results, under some assumptions, we give necessary and sufficient conditions under which $AG(R)$ is a star graph.

The solution space of a constant coefficient ODE gives rise to a natural real analytic curve in Euclidean space. We give necessary and sufficient conditions on the ODE to ensure that this curve is a proper embedding of infinite length or has finite total first curvature. If all the roots of the associated characteristic polynomial are simple, we give a uniform upper bound for the total first curvature and show the optimal uniform upper bound must grow at least linearly with the order $n$ of the ODE. We then examine the case where multiple roots are permitted. We present several examples illustrating that a curve can have finite total first curvature for positive/negative time and infinite total first curvature for negative/positive time as well as illustrating that other possibilities may occur.