This note addresses monotonic growths and logarithmic convexities of the weighted ($(1-t^2)^\alpha dt^2$, $-\infty <\alpha <\infty$, $0< t< 1$) integral means $\mathsf{A}_{\alpha,\beta}(f,\cdot)$ and $\mathsf{L}_{\alpha,\beta}(f,\cdot)$ of the mixed area $(\pi r^2)^{-\beta}A(f,r)$ and the mixed length $(2\pi r)^{-\beta}L(f,r)$($0\le\beta\le 1$ and $0< r< 1$) of $f(r\mathbb D)$ and $\partial f(r\mathbb D)$ under a holomorphic map $f$ from the unit disk $\mathbb D$ into the finite complex plane $\mathbb C$.

In this paper, we survey some recent progress on analytic theory of partial differential equations.

In the article, we discuss basic concepts of the residue theory of logarithmic and multi-logarithmic differential forms, and describe some aspects of the theory, developed by the author in the past few years. In particular, we introduce the notion of logarithmic differential forms with the use of the classical de Rham lemma and give an explicit description of regular meromorphic differential forms in terms of residues of logarithmic or multi-logarithmic differential forms with respect to hypersurfaces, complete intersections or pure-dimensional Cohen-Macaulay spaces. Among other things, several useful applications are considered, which are related with the theory of holonomic $\mathscr{D}$-modules, the theory of Hodge structures, the theory of residual currents and others.

Let $\mathbb{K}$ be a complete algebraically closed $p$-adic field of characteristic zero. We apply results in algebraic geometry and a new Nevanlinna theorem for $p$-adic meromorphic functions in order to prove results of uniqueness in value sharing problems, both on $\mathbb{K}$ and on $\mathbb{C}$. Let $P$ be a polynomial of uniqueness for meromorphic functions in $\mathbb{K}$ or $\mathbb{C}$ or in an open disk. Let $f$, $g$ be two transcendental meromorphic functions in the whole field $\mathbb{K}$ or in $\mathbb{C}$ or meromorphic functions in an open disk of $\mathbb{K}$ that are not quotients of bounded  analytic functions. We show that if $f'P'(f)$ and $g'P'(g)$ share a small function $\alpha$ counting multiplicity, then $f=g$, provided that the multiplicity order of zeros of $P'$ satisfy certain inequalities. A breakthrough in this paper consists of replacing inequalities $n\geq k+2$ or $n\geq k+3$ used in previous papers by Hypothesis (G). In the $p$-adic context, another consists of giving a lower bound for a sum of $q$ counting functions of zeros with $(q-1)$ times the characteristic function of the considered meromorphic function.

In this note, we discuss a class of so-called generalized sampling functions. These functions are defined to be the inverse Fourier transform of a family of piecewise constant functions that are either square integrable or Lebegue integrable on the real number line. They are in fact the generalization of the classic sinc function. Two approaches of constructing the generalized sampling functions are reviewed. Their properties such as cardinality, orthogonality, and decaying properties are discussed. The interactions of those functions and Hilbert transformer are also discussed.

In this article, we introduce some results with respect to the integrality and exact solutions of some 2nd order algebraic DEs. We obtain the sufficient and necessary conditions of integrable and the general meromorphic solutions of these equations by the complex method, which improves the corresponding results obtained by many authors. Our results show that the complex method provides a powerful mathematical tool for solving a large number of nonlinear partial differential equations in mathematical physics.

In this paper, we study an ODE of the form$$b_0 u^{(4)} + b_1 u'' + b_2 u + b_3 u^3 + b_4 u^5 = 0, \  ' = \frac{d}{d z},$$ which includes, as a special case,  the stationary case of the cubic-quintic Swift-Hohenberg equation. Based on Nevanlinna theory and Painlevé analysis, we first show that all its meromorphic solutions are elliptic or degenerate elliptic. Then we obtain them all explicitly by the method introduced in [7].

In this paper we present the most important definitions and results of the theory of parabolic-like mappings, and we will give an example. The proofs of the results can be found in [2,4] and [3].

The Strebel point is a Teichmüller equivalence class in the Teichmüller space that has a certain rigidity in the extremality of the maximal dilatation. In this paper, we give a sufficient condition in terms of the Schwarzian derivative for a Teichmüller equivalence class of the universal Teichmüller space under which the class is a Strebel point. As an application, we construct a Teichmüller equivalence class that is a Strebel point and that is not an asymptotically conformal class.

The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros has been recently generalized onto subharmonic functions with the Riesz measure on a half-line in $\mathbb{R}^n$, $n\geq 3$. Here we extend the Drasin complement to the Valiron-Titchmarsh theorem and show that if $u$ is a subharmonic function of this class and of order $0<\rho<1$, then the existence of the limit $\lim_{r \to \infty} \log u(r)/N(r),$ where $N(r)$ is the integrated counting function of the masses of $u$, implies the regular asymptotic behavior for both $u$ and its associated measure.