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Iason Efraimidis

Publications and source records attributed to Iason Efraimidis.

15 recordsLinked to original sources
arXiv2026

Harmonic mappings, univalence criteria and a theorem of Lehtinen

The harmonic inner radius $σ_H(Ω)$ of a planar domain $Ω$ is the largest constant with which a univalence criterion via the Schwarzian derivative holds for harmonic mappings. We show that $σ_H(Ω)\leqσ_H(\mathbb{D})\leq 3/2$ for the unit disk $\mathbb{D}$ and for every domain $Ω$ that omits an open set. This is an analogue of a theorem of Lehtinen in the setting of holomorphic functions. We provide two related univalence criteria for harmonic mappings.

math.CV↗
arXiv2025

On Hardy spaces, univalent functions and the second coefficient

We consider normalized univalent functions with prescribed second Taylor coefficient $a_2$. For convex functions $f$ we study the Hardy spaces to which $f$ and $f'$ belong, refining in particular on a theorem of Eenigenburg and Keogh, and give a sharp asymptotic estimate and an explicit uniform bound for their coefficients. Relating the lower order of a convex function to the angle at infinity of its range we deduce that its range lies always in some sector of aperture $|a_2|π$. We give sharp smoothness conditions on the boundary for convex functions with prescribed second coefficient. We find the sharp Hardy space estimates for $f$ and $f'$ when $f$ belongs to other geometric subclasses, such as those of starlike, close-to-convex, convex in one direction, convex in the positive direction and typically real funtions. We extend a theorem of Lohwater, Piranian and Rudin, in which a univalent function whose derivative has radial limits almost nowhere is constructed, by showing that this pathological behavior can be obtained for any prescribed value of the second coefficient, in particular, manifesting itself arbitrarily close to the Koebe function.

math.CV↗
arXiv2024

Hyperbolic convexity of holomorphic level sets

We prove that the sublevel set $\big\{z\in\mathbb D\colon k_{\mathbb D}\big(z,z_0\big)-k_{\mathbb D}\big(f(z),w_0\big)<μ\big\}$, ${μ\in\mathbb R}$, is geodesically convex with respect to the Poincaré distance $k_{\mathbb D}$ in the unit disk $\mathbb D$ for every ${z_0,w_0\in\mathbb D}$ and every holomorphic ${f:\mathbb D\to\mathbb D}$ if and only if ${μ\leqslant0}$. An analogous result is established also for the set $\{z\in\mathbb D \colon 1-|f(z)|^2<λ(1-|z|^2)\}$, ${λ>0}$. This extends a result of Solynin (2007) and solves a problem posed by Arango, Mej\'ıa and Pommerenke (2019). We also propose several open questions aiming at possible extensions to more general settings.

math.CV↗
arXiv2024

Korenblum's principle for Bergman spaces with radial weights

We show that the Korenblum maximum (domination) principle is valid for weighted Bergman spaces $A^p_w$ with arbitrary (non-negative and integrable) radial weights $w$ in the case $1\le p<\infty$. We also notice that in every weighted Bergman space the supremum of all radii for which the principle holds is strictly smaller than one. Under the mild additional assumption $\liminf_{r\to 0^+} w(r)>0$, we show that the principle fails whenever $0<p<1$.

math.CV↗
arXiv2022

Estimates for truncated area functionals on the Bloch space

Recently, Kayumov \cite{K} obtained a sharp estimate for the $n$-th truncated area functional for normalized functions in the Bloch space for $n\le 5$ and then, together with Wirths \cite{KW1}, extended the result for $n=6$. We prove that for the functions with non-negative Taylor coefficients, the same sharp estimate is valid for all $n$. For arbitrary functions, we obtain an estimate that is asymptotically of the same order but slightly larger (roughly by a factor of $4/e$). We also consider related weighted estimates for functionals involving the powers $n^t$, $t>0$, and show that the exponent $t=1$ represents the critical case for the expected sharp estimate.

math.CV↗
arXiv2021

Ahlfors-Weill extensions for harmonic mappings

We provide two new formulas for quasiconformal extension to $\overline{\mathbb{C}}$ for harmonic mappings defined in the unit disk and having sufficiently small Schwarzian derivative. Both are generalizations of the Ahlfors-Weill extension for holomorphic functions.

math.CV↗
arXiv2021

Quasiconformal extension for harmonic mappings on finitely connected domains

We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal extension to the whole plane if its Schwarzian derivative is small. We also make the observation that a univalence criterion for harmonic mappings holds on uniform domains.

math.CV↗
arXiv2020

Schwarzian derivatives for pluriharmonic mappings

A pre-Schwarzian and a Schwarzian derivative for locally univalent pluriharmonic mappings in ${\mathbb C}^n$ are introduced. Basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these operators. It is shown that the pre-Schwarzian is stable only with respect to rotations of the identity. A characterization is given for the case when the pre-Schwarzian derivative is holomorphic. Furthermore, it is shown that if the Schwarzian derivative of a pluriharmonic mapping vanishes then the analytic part of this mapping is a Möbius transformation. Some observations are made related to the dilatation of pluriharmonic mappings and to the dilatation of their affine transformations, revealing differences between the theories in the plane and in higher dimensions. An example is given that rules out the possibility for a shear construction theorem to hold in ${\mathbb C}^n$, for $n\geq2$.

math.CV↗
arXiv2020

Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains

If $Ω$ is a simply connected domain in $\overline{\mathbb C}$ then, according to the Ahlfors-Gehring theorem, $Ω$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in $Ω$ in relation to the growth of their Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also show that the mappings satisfying this criterion admit a homeomorphic extension to $\overline{\mathbb C}$ and, under the additional assumption of quasiconformality in $Ω$, they admit a quasiconformal extension to $\overline{\mathbb C}$. The Ahlfors-Gehring theorem has been extended to finitely connected domains $Ω$ by Osgood, Beardon and Gehring, who showed that a Schwarzian criterion for univalence holds in $Ω$ if and only if the components of $\partialΩ$ are either points or quasicircles. We generalize this theorem to harmonic mappings.

math.CV↗
arXiv2017

Some more counterexamples for Bombieri's conjecture on univalent functions

We disprove a conjecture of Bombieri regarding univalent functions in the unit disk in some previously unknown cases. The key step in the argument is showing that the global minimum of the real function $\big(n\sin{x}-\sin(nx)\big)/\big(m\sin{x}-\sin(mx)\big)$ is attained at $x = 0$ for integers $m>n\geq2$ when $m$ is odd and $n$ is even, $m$ is sufficiently big and $0.5 \leq n/m \leq 0.8194$.

math.CV↗
arXiv2017

On the failure of Bombieri's conjecture for univalent functions

A conjecture of Bombieri states that the coefficients of a normalized univalent function $f$ should satisfy $$ \liminf_{f\to K} \frac{n-{\rm Re\,}a_n}{m-{\rm Re\,}a_m} = \min_{t\in{\mathbb R}} \, \frac{n\sin t -\sin(nt)}{m\sin t -\sin(mt)}, $$ when $f$ approaches the Koebe function $K(z)=\frac{z}{(1-z)^2}$. Recently, Leung disproved this conjecture for $n=2$ and for all $m\geq3$ and, also, for $n=3$ and for all odd $m\geq5$. Complementing his work we disprove it for all $m>n\geq2$ which are simultaneously odd or even and, also, for the case when $m$ is odd, $n$ is even and $n\leq \frac{m+1}{2}$. We mostly make use of trigonometry, but also employ Dieudonné's criterion for the univalence of polynomials.

math.CV↗
arXiv2017

Applications of Livingston-type inequalities to the generalized Zalcman functional

We obtain sharp estimates for a generalized Zalcman coefficient functional with a complex parameter for the Hurwitz class and the Noshiro-Warschawski class of univalent functions as well as for the closed convex hulls of the convex and starlike functions by using an inequality from [6]. In particular, we generalize an inequality proved by Ma for starlike functions and answer a question from his paper [16]. Finally, we prove an asymptotic version of the generalized Zalcman conjecture for univalent functions and discuss various related or equivalent statements which may shed further light on the problem.

math.CV↗
arXiv2015

A generalization of Livingston's coefficient inequalities for functions with positive real part

For functions $p(z) = 1 + \sum_{n=1}^\infty p_n z^n$ holomorphic in the unit disk, satisfying $ {\rm Re}\, p(z) > 0$, we generalize two inequalities proved by Livingston in 1969 and 1985, and simplify their proofs. One of our results states that $|p_n -w p_k p_{n-k}|\leq 2\max\{1, |1-2w|\}, w\in\mathbb{C}$. Another result involves certain determinants whose entries are the coefficients $p_n$. Both results are sharp. As applications we provide a simple proof of a theorem of J.E. Brown and various inequalities for the coefficients of holomorphic self-maps of the unit disk.

math.CV↗
arXiv2014

On the generalized Zalcman functional for some classes of univalent functions

We prove three sharp estimates for the generalized Zalcman coefficient functional: one for the Hurwitz class, another for the Noshiro-Warschawski class, and yet another for the functions in the closed convex hull of convex univalent functions. In each case the extremal functions are identified. We also observe that an asymptotic version of the Zalcman conjecture is true.

math.CV↗