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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
$p:\mathbb{C}\rightarrow \mathbb{C}$
be a polynomial. The Gauss–Lucas theorem states that its critical points,
$p^{\prime }(z)=0$
, are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose that
$p$
has
$n+m$
roots, where
$n$
are inside the unit disk,
then
$p^{\prime }$
has
$n-1$
roots inside the unit disk and
$m$
roots at distance at least
$(dn-m)/(n+m)>1$
from the origin and the involved constants are sharp. We also discuss a pairing result: in the setting above, for
$n$
sufficiently large, each of the
$m$
roots has a critical point at distance
${\sim}n^{-1}$
.
Steady two-dimensional fluid flow over an obstacle is solved using complex variable methods. We consider the cases of rectangular obstacles, such as large boulders, submerged in a potential flow. These may arise in geophysics, marine and civil engineering. Our models are applicable to initiation of motion that may result in subsequent transport. The local flow depends on the obstacle shape, slowing down in confining corners and speeding up in expanding corners. The flow generates hydrodynamic forces, drag and lift, and their associated moments, which differ around each face. Our model replaces the need for ill-defined drag and lift coefficients with geometry-dependent functions. We predict smaller flow velocities to initiate motion. We show how a joint-bound boulder can be transported against gravity, and analyse the influence of a wake region behind an isolated boulder.
We obtain an asymptotic formula for the persistence probability in the positive real line of a random polynomial arising from evolutionary game theory. It corresponds to the probability that a multi-player two-strategy random evolutionary game has no internal equilibria. The key ingredient is to approximate the sequence of random polynomials indexed by their degrees by an appropriate centered stationary Gaussian process.
We give an equality condition for a symmetrization inequality for condensers proved by F.W. Gehring regarding elliptic areas. We then use this to obtain a monotonicity result involving the elliptic area of the image of a holomorphic function f.
Let
${\mathcal{P}}_{{\mathcal{H}}}^{0}(M)$
denote the class of normalised harmonic mappings
$f=h+\overline{g}$
in the unit disk
$\mathbb{D}$
satisfying
$\text{Re}\,(zh^{\prime \prime }(z))>-M+|zg^{\prime \prime }(z)|$
, where
$h^{\prime }(0)-1=0=g^{\prime }(0)$
and
$M>0$
. Let
${\mathcal{B}}_{{\mathcal{H}}}^{0}(M)$
denote the class of sense-preserving harmonic mappings
$f=h+\overline{g}$
in the unit disk
$\mathbb{D}$
satisfying
$|zh^{\prime \prime }(z)|\leq M-|zg^{\prime \prime }(z)|$
, where
$M>0$
. We discuss the coefficient bound problem, the growth theorem for functions in the class
${\mathcal{P}}_{{\mathcal{H}}}^{0}(M)$
and a two-point distortion property for functions in the class
${\mathcal{B}}_{{\mathcal{H}}}^{0}(M)$
.
Assume a point
$z$
lies in the open unit disk
$\mathbb{D}$
of the complex plane
$\mathbb{C}$
and
$f$
is an analytic self-map of
$\mathbb{D}$
fixing 0. Then Schwarz’s lemma gives
$|f(z)|\leq |z|$
, and Dieudonné’s lemma asserts that
$|f^{\prime }(z)|\leq \min \{1,(1+|z|^{2})/(4|z|(1-|z|^{2}))\}$
. We prove a sharp upper bound for
$|f^{\prime \prime }(z)|$
depending only on
$|z|$
.
This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.
Let
${\mathcal{S}}$
be the family of analytic and univalent functions
$f$
in the unit disk
$\mathbb{D}$
with the normalization
$f(0)=f^{\prime }(0)-1=0$
, and let
$\unicode[STIX]{x1D6FE}_{n}(f)=\unicode[STIX]{x1D6FE}_{n}$
denote the logarithmic coefficients of
$f\in {\mathcal{S}}$
. In this paper we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families
${\mathcal{F}}(c)$
and
${\mathcal{G}}(c)$
of functions
$f\in {\mathcal{S}}$
defined by
for some
$c\in (0,3]$
and
$c\in (0,1]$
, respectively. We obtain the sharp upper bound for
$|\unicode[STIX]{x1D6FE}_{n}|$
when
$n=1,2,3$
and
$f$
belongs to the classes
${\mathcal{F}}(c)$
and
${\mathcal{G}}(c)$
, respectively. The paper concludes with the following two conjectures:
∙If
$f\in {\mathcal{F}}(-1/2)$
, then
$|\unicode[STIX]{x1D6FE}_{n}|\leq 1/n(1-(1/2^{n+1}))$
for
$n\geq 1$
, and
The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form
$\unicode[STIX]{x1D6FC}/L(\unicode[STIX]{x1D6FA})$
, and
$\unicode[STIX]{x1D6FC}$
lies between
$-2\unicode[STIX]{x1D70B}$
and
$2\unicode[STIX]{x1D70B}$
. Corollaries include Szegő’s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock’s inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter.
The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.
Dorff et al. [‘Convolutions of harmonic convex mappings’, Complex Var. Elliptic Equ.57(5) (2012), 489–503] formulated a question concerning the convolution of two right half-plane mappings, where the normalisation of the functions was considered incorrectly. In this paper, we reformulate the problem correctly and provide a solution to it in a more general form. We also obtain two new theorems which correct and improve related results.
We apply the Schwarz lemma to find general formulas for the third coefficient of Carathéodory functions dependent on a parameter in the closed unit polydisk. Next we find sharp estimates of the Hankel determinant
$H_{2,2}$
and Zalcman functional
$J_{2,3}$
over the class
${\mathcal{C}}{\mathcal{V}}$
of analytic functions
$f$
normalised such that
$\text{Re}\{(1-z^{2})f^{\prime }(z)\}>0$
for
$z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$
, that is, the subclass of the class of functions convex in the direction of the imaginary axis.
Let
$\unicode[STIX]{x1D707}$
be a positive finite Borel measure on the unit circle and
${\mathcal{D}}(\unicode[STIX]{x1D707})$
the associated harmonically weighted Dirichlet space. In this paper we show that for each closed subset
$E$
of the unit circle with zero
$c_{\unicode[STIX]{x1D707}}$
-capacity, there exists a function
$f\in {\mathcal{D}}(\unicode[STIX]{x1D707})$
such that
$f$
is cyclic (i.e.,
$\{pf:p\text{ is a polynomial}\}$
is dense in
${\mathcal{D}}(\unicode[STIX]{x1D707})$
),
$f$
vanishes on
$E$
, and
$f$
is uniformly continuous. Next, we provide a sufficient condition for a continuous function on the closed unit disk to be cyclic in
${\mathcal{D}}(\unicode[STIX]{x1D707})$
.
Suppose that
$D\subset \mathbb{C}$
is a simply connected subdomain containing the origin and
$f(z_{1})$
is a normalized convex (resp., starlike) function on
$D$
. Let
where
$p_{j}\geqslant 1$
,
$N=1+n_{1}+\cdots +n_{k}$
,
$w_{1}\in \mathbb{C}^{n_{1}},\ldots ,w_{k}\in \mathbb{C}^{n_{k}}$
and
$\unicode[STIX]{x1D706}_{D}$
is the density of the hyperbolic metric on
$D$
. In this paper, we prove that
is a normalized convex (resp., starlike) mapping on
$\unicode[STIX]{x1D6FA}_{N}(D)$
. If
$D$
is the unit disk, then our result reduces to Gong and Liu via a new method. Moreover, we give a new operator for convex mapping construction on an unbounded domain in
$\mathbb{C}^{2}$
. Using a geometric approach, we prove that
$\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)$
is a spiral-like mapping of type
$\unicode[STIX]{x1D6FC}$
when
$f$
is a spiral-like function of type
$\unicode[STIX]{x1D6FC}$
on the unit disk.
Large-scale sublinearly Lipschitz maps have been introduced by Yves Cornulier in order to precisely state his theorems about asymptotic cones of Lie groups. In particular, Sublinearly bi-Lipschitz Equivalences (SBE) are a weak variant of quasi-isometries, with the only requirement of still inducing bi-Lipschitz maps at the level of asymptotic cones. We focus here on hyperbolic metric spaces and study properties of boundary extensions of SBEs, reminiscent of quasi-Möbius (or quasisymmetric) mappings. We give a dimensional invariant of the boundary that allows to distinguish hyperbolic symmetric spaces up to SBE, answering a question of Druţu.
Let
$f$
be analytic in
$\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$
and given by
$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$
. We give sharp bounds for the initial coefficients of the Taylor expansion of such functions in the class of strongly Ozaki close-to-convex functions, and of the initial coefficients of the inverse function, together with some growth estimates.
A new criterion for local invertibility of slice regular quaternionic functions is obtained. This paper is motivated by the need to find a geometrical interpretation for analytic conditions on the real Jacobian associated with a slice regular function f. The criterion involves spherical and Cullen derivatives of f and gives rise to several geometric implications, including an application to related conformality properties.
The aim of this note is to give a simple topological proof of the well-known
result concerning continuity of roots of polynomials. We also consider a
more general case with polynomials of a higher degree approaching a given
polynomial. We then examine the continuous dependence of solutions of linear
differential equations with constant coefficients.
While the existence of conformal mappings between doubly connected domains is characterized by their conformal moduli, no such characterization is available for harmonic diffeomorphisms. Intuitively, one expects their existence if the domain is not too thick compared to the codomain. We make this intuition precise by showing that for a Dini-smooth doubly connected domain Ω* there exists a ε > 0 such that for every doubly connected domain Ω with ModΩ* < ModΩ < ModΩ* + ε there exists a harmonic diffeomorphism from Ω onto Ω*.