The aim of this paper is twofold. The first aim is to describe the entire solutions of the partial differential equation (PDE) $u_{z_1}^2+2Bu_{z_1}u_{z_2}+u_{z_2}^2=e^g$, where B is a constant and g is a polynomial or an entire function in $\mathbb {C}^2$. The second aim is to consider the entire solutions of another PDE, which is a generalization of the well-known PDE of tubular surfaces.

We prove that for $0<p<+\infty $ and $-1<\alpha <+\infty ,$ a conformal map defined on the unit disk belongs to the weighted Bergman space $A_{\alpha }^p$ if and only if a certain integral involving the hyperbolic distance converges.

For an inner function u, we discuss the dual operator for the compressed shift $P_u S|_{{\mathcal {K}}_u}$, where ${\mathcal {K}}_u$ is the model space for u. We describe the unitary equivalence/similarity classes for these duals as well as their invariant subspaces.

Following recent investigations of vanishing coefficients in infinite products, we show that such instances are very rare when the infinite product is among a family of theta-quotients of modulus five. We also prove that a general family of products of theta functions of modulus five can always be effectively 5-dissected.

A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with $2N$ crossings grows exponentially when $N$ grows, but the long-standing problem on the precise asymptotics is still out of reach.

We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as $N$ tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator.

The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics.

In this paper we study a class ${\mathcal{Z}}_{H}$ of harmonic mappings on the open unit disk $\mathbb{D}$ in the complex plane that is an extension of the classical (analytic) Zygmund space. We extend to the elements of this class a characterisation that is valid in the analytic case. We also provide a similar result for a closed separable subspace of ${\mathcal{Z}}_{H}$ which we call the little harmonic Zygmund space.

The main aim of this paper is to investigate the invariant properties of uniform domains under flattening and sphericalization in nonlocally compact complete metric spaces. Moreover, we show that quasi-Möbius maps preserve uniform domains in nonlocally compact spaces as well.

We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight λ depends on the independent variable z. We prove that for an increasing radial weight λ(z) the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight λ(z) we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial symmetry for every such weight.

According to a conjecture by Yang, if $f(z)f^{(k)}(z)$ is a periodic function, where $f(z)$ is a transcendental entire function and $k$ is a positive integer, then $f(z)$ is also a periodic function. We propose related questions, which can be viewed as difference or differential-difference versions of Yang’s conjecture. We consider the periodicity of a transcendental entire function $f(z)$ when differential, difference or differential-difference polynomials in $f(z)$ are periodic. For instance, we show that if $f(z)^{n}f(z+\unicode[STIX]{x1D702})$ is a periodic function with period $c$, then $f(z)$ is also a periodic function with period $(n+1)c$, where $f(z)$ is a transcendental entire function of hyper-order $\unicode[STIX]{x1D70C}_{2}(f)<1$ and $n\geq 2$ is an integer.

We present a complete characterisation of the radial asymptotics of degree-one Mahler functions as $z$ approaches roots of unity of degree $k^{n}$, where $k$ is the base of the Mahler function, as well as some applications concerning transcendence and algebraic independence. For example, we show that the generating function of the Thue–Morse sequence and any Mahler function (to the same base) which has a nonzero Mahler eigenvalue are algebraically independent over $\mathbb{C}(z)$. Finally, we discuss asymptotic bounds towards generic points on the unit circle.

Given integers $g,n\geqslant 0$ satisfying $2-2g-n<0$, let ${\mathcal{M}}_{g,n}$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $g$ with $n$ cusps. We study the global behavior of the Mirzakhani function $B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$ which assigns to $X\in {\mathcal{M}}_{g,n}$ the Thurston measure of the set of measured geodesic laminations on $X$ of hyperbolic length ${\leqslant}1$. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of ${\mathcal{M}}_{g,n}$ and deduce that $B$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $B$ to statistics of counting problems for simple closed hyperbolic geodesics.

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in k (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like k for growing k.

We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.

For $f$ analytic in the unit disk $\mathbb{D}$, we consider the close-to-convex analogue of a class of starlike functions introduced by R. Singh [‘On a class of star-like functions’, Compos. Math.19(1) (1968), 78–82]. This class of functions is defined by $|zf^{\prime }(z)/g(z)-1|<1$ for $z\in \mathbb{D}$, where $g$ is starlike in $\mathbb{D}$. Coefficient and other results are obtained for this class of functions.

We study topological properties of the escaping endpoints and fast escaping endpoints of the Julia set of complex exponential $\exp (z)+a$ when $a\in (-\infty ,-1)$ . We show neither space is homeomorphic to the whole set of endpoints. This follows from a general result stating that for every transcendental entire function $f$ , the escaping Julia set $I(f)\cap J(f)$ is first category.

Assume that $\unicode[STIX]{x1D6FA}$ and $D$ are two domains with compact smooth boundaries in the extended complex plane $\overline{\mathbf{C}}$. We prove that every quasiconformal mapping between $\unicode[STIX]{x1D6FA}$ and $D$ mapping $\infty$ onto itself is bi-Lipschitz continuous with respect to both the Euclidean and Riemannian metrics.

Let $\mathbb{Z}$ and $\mathbb{Z}^{+}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in \mathbb{Z}^{+}$, let $t(a,b,c,d;n)$ be the number of representations of $n$ by $\frac{1}{2}ax(x+1)+\frac{1}{2}by(y+1)+\frac{1}{2}cz(z+1)+\frac{1}{2}dw(w+1)$ with $x,y,z,w\in \mathbb{Z}$. Using theta function identities we prove 13 transformation formulas for $t(a,b,c,d;n)$ and evaluate $t(2,3,3,8;n)$, $t(1,1,6,24;n)$ and $t(1,1,6,8;n)$.

The classical Gauss–Lucas theorem states that the critical points of a polynomial with complex coefficients are in the convex hull of its zeros. This fundamental theorem follows from the fact that if all the zeros of a polynomial are in a half plane, then the same is true for its critical points. The main result of this work replaces the half plane with a sector as follows.

We show that if the coefficients of a monic polynomial $p(z)$ are in the sector {tei𝜓 : 𝜓∈ [0, 𝜙], t⩾0}, for some $\unicode[STIX]{x1D719}\in [0,\unicode[STIX]{x1D70B})$, and the zeros are not in its interior, then the critical points of $p(z)$ are also not in the interior of that sector.

In addition, we give a necessary condition for a polynomial to satisfy the premise of the main result.

Asymptotic expansions of the Gauss hypergeometric function with large parameters, $F(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D716}_{1}\unicode[STIX]{x1D70F},\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D716}_{2}\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D716}_{3}\unicode[STIX]{x1D70F};z)$ as $|\unicode[STIX]{x1D70F}|\rightarrow \infty$, are known for many special cases, but not for one that the author encountered in recent work on fluid mechanics: $\unicode[STIX]{x1D716}_{2}=0$ and $\unicode[STIX]{x1D716}_{3}=\unicode[STIX]{x1D716}_{1}z$. This paper gives the leading term for that case if $\unicode[STIX]{x1D6FD}$ is not a negative integer and $z$ is not on the branch cut $[1,\infty )$, and it shows how subsequent terms can be found.

In 1993, N. Danikas and A. G. Siskakis showed that the Cesàro operator ${\mathcal{C}}$ is not bounded on $H^{\infty }$; that is, ${\mathcal{C}}(H^{\infty })\nsubseteq H^{\infty }$, but ${\mathcal{C}}(H^{\infty })$ is a subset of $BMOA$. In 1997, M. Essén and J. Xiao gave that ${\mathcal{C}}(H^{\infty })\subsetneq {\mathcal{Q}}_{p}$ for every $0<p<1$. In this paper, we characterize positive Borel measures $\unicode[STIX]{x1D707}$ such that ${\mathcal{C}}(H^{\infty })\subseteq M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ and show that ${\mathcal{C}}(H^{\infty })\subsetneq M({\mathcal{D}}_{\unicode[STIX]{x1D707}_{0}})\subsetneq \bigcap _{0<p<\infty }{\mathcal{Q}}_{p}$ by constructing some measures $\unicode[STIX]{x1D707}_{0}$. Here, $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ denotes the Möbius invariant function space generated by ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$, where ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$ is a Dirichlet space with superharmonic weight induced by a positive Borel measure $\unicode[STIX]{x1D707}$ on the open unit disk. Our conclusions improve results mentioned above.