Let ${\mathcal{S}}$ denote the set of all univalent analytic functions $f$ of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ on the unit disk $|z|<1$. In 1946, Friedman [‘Two theorems on Schlicht functions’, Duke Math. J.13 (1946),171–177] found that the set ${\mathcal{S}}_{\mathbb{Z}}$ of those functions in ${\mathcal{S}}$ which have integer coefficients consists of only nine functions. In a recent paper, Hiranuma and Sugawa [‘Univalent functions with half-integer coefficients’, Comput. Methods Funct. Theory13(1) (2013), 133–151] proved that the similar set obtained for functions with half-integer coefficients consists of only 21 functions; that is, 12 more functions in addition to these nine functions of Friedman from the set ${\mathcal{S}}_{\mathbb{Z}}$. In this paper, we determine the class of all normalized sense-preserving univalent harmonic mappings $f$ on the unit disk with half-integer coefficients for the analytic and co-analytic parts of $f$. It is surprising to see that there are only 27 functions out of which only six functions in this class are not conformal. This settles the recent conjecture of the authors. We also prove a general result, which leads to a new conjecture.

It is well known that functions in the analytic Besov space $B_{1}$ on the unit disk $\mathbb{D}$ admit an integral representation

$$\begin{eqnarray}f(z)=\int _{\mathbb{D}}\frac{z-w}{1-z\overline{w}}\,d{\it\mu}(w),\end{eqnarray}$$

${\it\mu}$

$|{\it\mu}|(\mathbb{D})<\infty$

$B_{p}$

$0<p\leq 1$

${\rm\Lambda}_{t}$

$t>1$

For $0<p<\infty$ and $-2\leq {\it\alpha}\leq 0$ we show that the $L^{p}$ integral mean on $r\mathbb{D}$ of an analytic function in the unit disk $\mathbb{D}$ with respect to the weighted area measure $(1-|z|^{2})^{{\it\alpha}}\,dA(z)$ is a logarithmically convex function of $r$ on $(0,1)$.

Thurston introduced shear deformations (cataclysms) on geodesic laminations–deformations including left and right displacements along geodesics. For hyperbolic surfaces with cusps, we consider shear deformations on disjoint unions of ideal geodesics. The length of a balanced weighted sum of ideal geodesics is defined and the Weil–Petersson (WP) duality of shears and the defined length is established. The Poisson bracket of a pair of balanced weight systems on a set of disjoint ideal geodesics is given in terms of an elementary $2$-form. The symplectic geometry of balanced weight systems on ideal geodesics is developed. Equality of the Fock shear coordinate algebra and the WP Poisson algebra is established. The formula for the WP Riemannian pairing of shears is also presented.

We consider solutions to the algebraic differential equation $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f^nf'+Q_d(z,f)=u(z)e^{v(z)}$, where $Q_d(z,f)$ is a differential polynomial in $f$ of degree $d$ with rational function coefficients, $u$ is a nonzero rational function and $v$ is a nonconstant polynomial. In this paper, we prove that if $n\ge d+1$ and if it admits a meromorphic solution $f$ with finitely many poles, then

$$\begin{equation*} f(z)=s(z)e^{v(z)/(n+1)} \quad \mbox {and}\quad Q_d(z,f)\equiv 0. \end{equation*}$$

$f$

$f'p_k(f)+q_m(f)$

$\alpha $

$p_k(f), q_m(f)$

$f$

$k$

$m$

$k\ge m+1$

Suppose that $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ and $E'$ denote real Banach spaces with dimension at least 2 and that $D\subset E$ and $D'\subset E'$ are domains. Let $\varphi :[0,\infty )\to [0,\infty )$ be a homeomorphism with $\varphi (t)\geq t$. We say that a homeomorphism $f: D\to D'$ is $\varphi $-FQC if for every subdomain $D_1 \subset D$, we have $\varphi ^{-1} (k_D(x,y))\leq k_{D'} (f(x),f(y))\leq \varphi (k_D(x,y))$ holds for all $x,y\in D_1$. In this paper, we establish, in terms of the $j_D$ metric, a necessary and sufficient condition for a homeomorphism $f: E \to E'$ to be FQC. Moreover, we give, in terms of the $j_D$ metric, a sufficient condition for a homeomorphism $f: D\to D'$ to be FQC. On the other hand, we show that this condition is not necessary.

We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite-index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.

We show that a computable and conformal map of the unit disk onto a bounded domain $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}D$ has a computable boundary extension if $D$ has a computable boundary connectivity function.

We use Zalcman’s lemma to study a uniqueness question for meromorphic functions where certain associated nonlinear differential polynomials share a nonzero value. The results in this paper extend Theorem 1 in Yang and Hua [‘Uniqueness and value-sharing of meromorphic functions’, Ann. Acad. Sci. Fenn. Math. 22 (1997), 395–406]and Theorem 1 in Fang [‘Uniqueness and value sharing of entire functions’, Comput. Math. Appl. 44 (2002), 823–831]. Our reasoning in this paper also corrects a defect in the reasoning in the proof of Theorem 4 in Bhoosnurmath and Dyavanal [‘Uniqueness and value sharing of meromorphic functions’, Comput. Math. Appl. 53 (2007), 1191–1205].

Let $G$ and $\tilde{G}$ be Kleinian groups whose limit sets $S$ and $\tilde{S}$, respectively, are homeomorphic to the standard Sierpiński carpet, and such that every complementary component of each of $S$ and $\tilde{S}$ is a round disc. We assume that the groups $G$ and $\tilde{G}$ act cocompactly on triples on their respective limit sets. The main theorem of the paper states that any quasiregular map (in a suitably defined sense) from an open connected subset of $S$ to $\tilde{S}$ is the restriction of a Möbius transformation that takes $S$ onto $\tilde{S}$, in particular it has no branching. This theorem applies to the fundamental groups of compact hyperbolic 3-manifolds with non-empty totally geodesic boundaries. One consequence of the main theorem is the following result. Assume that $G$ is a torsion-free hyperbolic group whose boundary at infinity $\partial _{\infty }G$ is a Sierpiński carpet that embeds quasisymmetrically into the standard 2-sphere. Then there exists a group $H$ that contains $G$ as a finite index subgroup and such that any quasisymmetric map $f$ between open connected subsets of $\partial _{\infty }G$ is the restriction of the induced boundary map of an element $h\in H$.

Let ${\it\alpha}\in \mathbb{C}$ in the upper half-plane and let $I$ be an interval. We construct an analogue of Selberg’s majorant of the characteristic function of $I$ that vanishes at the point ${\it\alpha}$. The construction is based on the solution to an extremal problem with positivity and interpolation constraints. Moreover, the passage from the auxiliary extremal problem to the construction of Selberg’s function with vanishing is easily adapted to provide analogous “majorants with vanishing” for any Beurling–Selberg majorant.

We give a short and elementary proof of an inverse Bernstein-type inequality found by S. Khrushchev for the derivative of a polynomial having all its zeros on the unit circle. The inequality is used to show that equally-spaced points solve a min–max–min problem for the logarithmic potential of such polynomials. Using techniques recently developed for polarization (Chebyshev-type) problems, we show that this optimality also holds for a large class of potentials, including the Riesz potentials $1/r^{s}$ with $s>0.$

We study the geometry of the space of measures of a compact ultrametric space $X$, endowed with the $L^{p}$ Wasserstein distance from optimal transportation. We show that the power $p$ of this distance makes this Wasserstein space affinely isometric to a convex subset of $\ell ^{1}$. As a consequence, it is connected by $1/p$-Hölder arcs, but any ${\it\alpha}$-Hölder arc with ${\it\alpha}>1/p$ must be constant. This result is obtained via a reformulation of the distance between two measures which is very specific to the case when $X$ is ultrametric; however, thanks to the Mendel–Naor ultrametric skeleton it has consequences even when $X$ is a general compact metric space. More precisely, we use it to estimate the size of Wasserstein spaces, measured by an analogue of Hausdorff dimension that is adapted to (some) infinite-dimensional spaces. The result we get generalizes greatly our previous estimate, which needed a strong rectifiability assumption. The proof of this estimate involves a structural theorem of independent interest: every ultrametric space contains large co-Lipschitz images of regular ultrametric spaces, i.e. spaces of the form $\{1,\dots ,k\}^{\mathbb{N}}$ with a natural ultrametric. We are also led to an example of independent interest: a space of positive lower Minkowski dimension, all of whose proper closed subsets have vanishing lower Minkowski dimension.

We study the automorphic Green function $\mathop{\rm gr}\nolimits _\Gamma $ on quotients of the hyperbolic plane by cofinite Fuchsian groups $\Gamma $, and the canonical Green function $\mathop{\rm gr}\nolimits ^{\rm can}_X$ on the standard compactification $X$ of such a quotient. We use a limiting procedure, starting from the resolvent kernel, and lattice point estimates for the action of $\Gamma $ on the hyperbolic plane to prove an “approximate spectral representation” for $\mathop{\rm gr}\nolimits _\Gamma $. Combining this with bounds on Maaß forms and Eisenstein series for $\Gamma $, we prove explicit bounds on $\mathop{\rm gr}\nolimits _\Gamma $. From these results on $\mathop{\rm gr}\nolimits _\Gamma $ and new explicit bounds on the canonical $(1,1)$-form of $X$, we deduce explicit bounds on $\mathop{\rm gr}\nolimits ^{\rm can}_X$.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H(\mathbb{D})$ denote the space of holomorphic functions on the unit disc $\mathbb{D}$. Given $p>0$ and a weight $\omega $, the Hardy growth space $H(p, \omega )$ consists of those $f\in H(\mathbb{D})$ for which the integral means $M_p(f,r)$ are estimated by $C\omega (r)$, $0<r<1$. Assuming that $p>1$ and $\omega $ satisfies a doubling condition, we characterise $H(p, \omega )$ in terms of associated Fourier blocks. As an application, extending a result by Bennett et al. [‘Coefficients of Bloch and Lipschitz functions’, Illinois J. Math. 25 (1981), 520–531], we compute the solid hull of $H(p, \omega )$ for $p\ge 2$.

The conformal invariance of Brownian motion is used to give a short proof of the Open Mapping theorem for analytic functions.

In this paper, we investigate the properties of locally univalent and multivalent planar harmonic mappings. First, we discuss coefficient estimates and Landau’s theorem for some classes of locally univalent harmonic mappings, and then we study some Lipschitz-type spaces for locally univalent and multivalent harmonic mappings.

We study the postcritically finite maps within the moduli space of complex polynomial dynamical systems. We characterize rational curves in the moduli space containing an infinite number of postcritically finite maps, in terms of critical orbit relations, in two settings: (1) rational curves that are polynomially parameterized; and (2) cubic polynomials defined by a given fixed point multiplier. We offer a conjecture on the general form of algebraic subvarieties in the moduli space of rational maps on ${ \mathbb{P} }^{1} $ containing a Zariski-dense subset of postcritically finite maps.

We consider a recent work of Pascu and Pascu [‘Neighbourhoods of univalent functions’, Bull. Aust. Math. Soc. 83(2) (2011), 210–219] and rectify an error that appears in their work. In addition, we study certain analogous results for sense-preserving harmonic mappings in the unit disc $\vert z\vert \lt 1$. As a corollary to this result, we derive a coefficient condition for a sense-preserving harmonic mapping to be univalent in $\vert z\vert \lt 1$.

Ritt introduced the concepts of prime and composite polynomials and proved three fundamental theorems on factorizations (in the sense of compositions) of polynomials in 1922. In this paper, we shall give a density estimate on the set of composite polynomials.