Authors:Vijay Gupta; Neha Malik Pages: 3 - 17 Abstract: Abstract In this paper, the exact order of approximation and Voronovskaja type theorems with quantitative estimate for complex genuine Pólya-Durrmeyer polynomials attached to analytic functions on compact disks are obtained. Our results show that extension of the complex genuine Pólya-Durrmeyer polynomials from real intervals to compact disks in the complex plane extends approximation properties (with quantitative estimates). PubDate: 2017-03-01 DOI: 10.1007/s40315-016-0167-x Issue No:Vol. 17, No. 1 (2017)

Authors:Abhijit Banerjee; Sanjay Mallick Pages: 19 - 45 Abstract: Abstract The purpose of the paper is to introduce a new class of strong uniqueness polynomials with the best possible answer to the question posed by the first author (Banerjee, Ann Acad Sci Fenn Math 40:465-474, 2015). We find the corresponding unique range set which improves a result due to Frank–Reinders (Complex Var Theory Appl 37(1):185–193, 1998). At the time of characterising the strong uniqueness polynomial, we also encounter the cases where the polynomial is not necessarily critically injective and find its degree and the corresponding unique range set. Furthermore, we find some connections between this new class of polynomials and most of the polynomials generating URSM introduced so far in the literature of value distribution theory. As an application of our results, we will show that the theorems improve many previous results. Lastly, we rectify the gap in the last section of An (Acta Math Vietnam 27(3), 251–256, 2002, p. 255, l. 11–14) and thus improve the result obtained by An (Acta Math Vietnam 27(3), 251–256, 2002) significantly. PubDate: 2017-03-01 DOI: 10.1007/s40315-016-0174-y Issue No:Vol. 17, No. 1 (2017)

Authors:Tran Van Tan; Nguyen Van Thin Pages: 47 - 63 Abstract: Abstract A well-known result of Lappan states that a meromorphic function f in the unit disc \(\mathbb {D}\) is normal if and only if there is a subset \(E\subset \widehat{\mathbb {C}}\) consisting of five points such that \(\sup \{(1- z ^2) f^{\#}(z): z \in f^{-1}(E)\} < \infty ,\) where \(f^{\#}(z)\) is the spherical derivative of f at z. An analogous result for normal families is due to Hinkkanen and Lappan: a family \(\mathcal F\) of meromorphic functions in a domain \(D\subset \mathbb {C}\) is normal if and only if for each compact subset \(K\subset D,\) there are a subset \(E\subset \widehat{\mathbb {C}}\) consisting of five points and a positive constant M such that \(\sup \{f^{\#}(z): f\in \mathcal F, z \in f^{-1}(E)\}<M.\) In this paper, we extend the above-mentioned results to the case where the set E contains fewer points. In particular, the Pang–Zalcman’s theorem on normality of a family \(\mathcal F\) of holomorphic functions f in a domain D, \(f^nf^{(k)}(z)\ne a\) (for some given constant a), is also extended to the case where the spherical derivative of \(f^nf^{(k)}\) is bounded on the zero set of \(f^nf^{(k)}-a\) . PubDate: 2017-03-01 DOI: 10.1007/s40315-016-0168-9 Issue No:Vol. 17, No. 1 (2017)

Authors:Nada Alhabib; Lasse Rempe-Gillen Pages: 65 - 100 Abstract: Abstract In 1988, Mayer proved the remarkable fact that \(\infty \) is an explosion point for the set \(E(f_a)\) of endpoints of the Julia set of \(f_a:\mathbb {C}\rightarrow \mathbb {C}; \mathrm{e}^z+a\) with \(a<-1\) ; that is, the set \(E(f_a)\) is totally separated (in particular, it does not have any non-trivial connected subsets), but \(E(f_a)\cup \{\infty \}\) is connected. Answering a question of Schleicher, we extend this result to the set \(\tilde{E}(f_a)\) of escaping endpoints in the sense of Schleicher and Zimmer, for any parameter \(a\in \mathbb {C}\) for which the singular value a belongs to an attracting or parabolic basin, has a finite orbit, or escapes to infinity under iteration (as well as many other classes of parameters). Furthermore, we extend one direction of the theorem to much greater generality, by proving that the set \(\tilde{E}(f)\cup \{\infty \}\) is connected for any transcendental entire function f of finite order with bounded singular set. We also discuss corresponding results for all endpoints in the case of exponential maps; to do so, we establish a version of Thurston’s no wandering triangles theorem for exponential maps. PubDate: 2017-03-01 DOI: 10.1007/s40315-016-0169-8 Issue No:Vol. 17, No. 1 (2017)

Authors:Ricardo Abreu Blaya; Juan Bory Reyes; Alí Guzmán Adán; Uwe Kähler Pages: 101 - 119 Abstract: Abstract The aim of this paper is to introduce, in the framework of Clifford analysis, the notions of \(\varphi \) -hyperdifferentiability and \(\varphi \) -hyperderivability for \(\psi \) -hyperholomorphic functions where ( \(\varphi ,\psi \) ) are two arbitrary orthogonal bases (called structural sets) of a Euclidean space. In this study we will also show how to exchange the integral sign and the \(\varphi \) -hyperderivative of the \(\psi \) -Cliffordian Cauchy-type integral. Thereby, we generalize, in a natural way, the corresponding quaternionic antecedent as well as the standard Clifford predecessor. PubDate: 2017-03-01 DOI: 10.1007/s40315-016-0172-0 Issue No:Vol. 17, No. 1 (2017)

Authors:Huaying Huang; Boyong Long Pages: 121 - 127 Abstract: Abstract Any Beltrami coefficient \(\mu \) on R represents a point \([\mu ]_T\) in Teichmüller space T(R) of a hyperbolic Riemann surface R and a point \([\mu ]_B\) in the tangent space of T(R) at the base point. The problem is whether \(\mu \) represents a Strebel point is equivalent to \(\mu \) represents an infinitesimal Strebel point. We prove that for any \(\mu \) , there exists an element \(\nu \) , such that \([\nu ]_T=[\mu ]_T\) and \([\nu ]_B\) is an infinitesimal Strebel point, when R is the unit disk. PubDate: 2017-03-01 DOI: 10.1007/s40315-016-0171-1 Issue No:Vol. 17, No. 1 (2017)

Authors:Maike Thelen Pages: 129 - 138 Abstract: Abstract It is known that unimodular eigenvalues of an operator can give information about its dynamical behaviour. Recently, some situations have been characterized in which the Taylor shift operator is hypercyclic. The aim of this article is to use an eigenvalue criterion to find assumptions that guarantee the frequent hypercyclicity of the Taylor shift operator. As a conclusion, we also obtain holomorphic functions with smooth boundary values. PubDate: 2017-03-01 DOI: 10.1007/s40315-016-0173-z Issue No:Vol. 17, No. 1 (2017)

Authors:Seung-Yeop Lee; Andres Saez Pages: 139 - 149 Abstract: Abstract We find a new lower bound for the maximal number of zeros of harmonic polynomials, \(p(z)+\overline{q(z)}\) , when \(\deg p = n\) and \(\deg q = n-2\) . PubDate: 2017-03-01 DOI: 10.1007/s40315-016-0175-x Issue No:Vol. 17, No. 1 (2017)

Authors:Olivier Sète; Jörg Liesen Pages: 151 - 177 Abstract: Abstract The Faber–Walsh polynomials are a direct generalization of the (classical) Faber polynomials from simply connected sets to sets with several simply connected components. In this paper, we derive new properties of the Faber–Walsh polynomials, where we focus on results of interest in numerical linear algebra, and on the relation between the Faber–Walsh polynomials and the classical Faber and Chebyshev polynomials. Moreover, we present examples of Faber–Walsh polynomials for two real intervals as well as for some non-real sets consisting of several simply connected components. PubDate: 2017-03-01 DOI: 10.1007/s40315-016-0176-9 Issue No:Vol. 17, No. 1 (2017)

Authors:Thi Hoai An Ta; Viet Phuong Nguyen Abstract: Abstract Consider meromorphic functions f, g, and \(\alpha ,\) where \(\alpha \) is a small function with respect to f and g. Let Q be a polynomial of one variable. We give suitable conditions on the degree of Q and on the number of zeros and the multiplicities of the zeros of \(Q'\) so as to be able to conclude uniqueness results if differential polynomials of the form \((Q(f))^{(k)}\) and \((Q(g))^{(k)}\) share \(\alpha \) counting multiplicities. We do not assume that Q has a large order zero, nor do we place restrictions on the zeros and poles of \(\alpha .\) Thus, our work improves on many prior results that either assume Q has a high order zero or place restrictions on the small function \(\alpha \) . PubDate: 2017-04-25 DOI: 10.1007/s40315-017-0198-y

Authors:Jochen Becker; Christian Pommerenke Abstract: Abstract Let the function \(\varphi \) be holomorphic in the unit disk \(\mathbb {D}\) and let \(\varphi (\mathbb {D})\subset \mathbb {D}\) . We consider points \(\zeta \in \partial \mathbb {D}\) where \(\varphi \) has an angular limit \(\varphi (\zeta )\in \partial \mathbb {D}\) and study the behaviour of \((\varphi (z)-\varphi (\zeta ))/(z-\zeta )\) as z tends to \(\zeta \) in various ways. In particular, there is a result connecting \( \varphi '(\zeta _{\nu }) \) and \( \varphi (\zeta _{\mu })-\varphi (\zeta _{\nu }) \) for three points \(\zeta _{\nu }\) . Expressed as a positive semidefinite quadratic form, this result could, perhaps, be generalized to n points \(\zeta _{\nu }\) . PubDate: 2017-04-24 DOI: 10.1007/s40315-017-0199-x

Authors:Juha-Matti Huusko; María J. Martín Abstract: Abstract In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that \(\limsup _{ z \rightarrow 1} S(f)(z) (1- z ^2)^2 < 2\) , then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that \(\limsup _{ z \rightarrow 1} f''(z)/f'(z) \, (1- z ^2)< 1\) . In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic. PubDate: 2017-03-28 DOI: 10.1007/s40315-017-0197-z

Authors:Andreas Schweizer Abstract: Abstract Let \(\mathcal{{F}}\) be a family of meromorphic functions on a domain D. We present a quite general sufficient condition for \(\mathcal{{F}}\) to be a normal family. This criterion contains many known results as special cases. The overall idea is that certain comparatively weak conditions on \(\mathcal{{F}}\) by local arguments lead to somewhat stronger conditions, which in turn lead to even stronger conditions on the limit function g in the famous Zalcman Lemma. Ultimately, the defect relations for g force normality of \(\mathcal{{F}}\) . PubDate: 2017-03-25 DOI: 10.1007/s40315-017-0196-0

Authors:Federico Piazzon Abstract: Abstract The Bernstein–Markov property is an asymptotic quantitative assumption on the growth of uniform norms of polynomials or rational functions on a compact set with respect to \(L^2_\mu \) -norms, where \(\mu \) is a positive finite measure. We consider two variants of the Bernstein–Markov property for rational functions with restricted poles and compare them with the polynomial Bernstein–Markov property to find some sufficient conditions for the latter to imply the former. Moreover, we recover a sufficient mass-density condition for a measure to satisfy the rational Bernstein–Markov property on its support. Finally we present, as an application, a meromorphic \(L^2\) version of the Bernstein–Walsh Lemma. PubDate: 2017-02-11 DOI: 10.1007/s40315-017-0194-2

Authors:David Minda Abstract: Abstract In this paper, precise versions of several intuitive properties of quotients of hyperbolic metrics are established. Suppose that \(\Omega _j\) is a hyperbolic region in \(\mathbb {C}_\infty = \mathbb {C}\cup \{\infty \}\) with hyperbolic metric \(\lambda _j\) , \(j=1,2\) , and \(\Omega _1 \subsetneq \Omega _2\) . First, it is shown that \(\lambda _1/ \lambda _2 \approx 1\) on compact subsets of \(\Omega _1\) that are not too close to \(\partial \Omega _1\) . Second, \(\lambda _1/ \lambda _2 \approx 1\) when z is near \((\partial \Omega _1 \;\cap \; \partial \Omega _2 ) {\setminus } F_b\) , where \(F = \partial \Omega _1 \;\cap \;\Omega _2\) and \(F_b = {{\mathrm{cl}}}(F)\;\cap \;\Omega _2\) . The main tools used in establishing these results are sharp elementary bounds for \(\lambda _1(z)/ \lambda _2(z)\) in terms of the hyperbolic distance relative to \(\Omega _2\) from z to \(\partial \Omega _1 \;\cap \;\Omega _2\) that were first established and employed in complex dynamics. PubDate: 2017-02-11 DOI: 10.1007/s40315-017-0195-1

Authors:Duc Quang Si; Phuong An Do Abstract: Abstract In this article, we establish some new second main theorems for meromorphic mappings of \({\mathbb {C}}^m\) into \({\mathbb {P}}^{n}({\mathbb {C}})\) and moving hypersurfaces with truncated counting functions in the case where the meromorphic mappings may be algebraically degenerate. Our results are improvements of some recent results on second main theorem in the two cases of moving hyperplanes and of moving hypersurfaces. As an application, a unicity theorem for meromorphic mappings sharing moving hypersurfaces is given. PubDate: 2017-01-30 DOI: 10.1007/s40315-017-0192-4

Authors:N. Bosuwan Abstract: Abstract In this paper, convergence theorems of row sequences of vector valued Padé-orthogonal approximants (simultaneous Padé-orthogonal approximants) corresponding to a measure supported on a general compact subset of the complex plane are proved. These theorems are natural extensions of Montessus de Ballore’s theorem for row sequences of (scalar) Padé-orthogonal approximants in Bosuwan et al. (Jaen J Approx 5:179–208, 2013). PubDate: 2017-01-30 DOI: 10.1007/s40315-017-0190-6

Authors:Raymond Cheng; Javad Mashreghi; William T. Ross Abstract: Abstract Bounds are obtained for the zeros of an analytic function on a disk in terms of the Taylor coefficients of the function. These results are derived using the notion of Birkhoff–James orthogonality in the sequence space \(\ell ^p\) with \(p \in (1, \infty )\) , along with an associated Pythagorean theorem. It is shown that these methods are able to reproduce, and in some cases sharpen, some classical bounds for the roots of a polynomial. PubDate: 2017-01-27 DOI: 10.1007/s40315-017-0191-5

Authors:D. Aharonov; D. Bshouty Pages: 677 - 688 Abstract: Abstract The famous Bieberbach Conjecture from 1916 on the coefficients of normalized univalent functions defined in the unit disk (Bieberbach, S.-B. Preuss Akad Wiss 138:940–955, 1916) that was finally proved by de Branges (Acta Math 154:137–152, 1985) some 70 years later, diverted the attention of many complex analysts to other subjects. Those who continued to explore de Branges method and push it as far as possible were not aware of where it may lead. Surprisingly enough, a paper that fell in our hands (Dong Acta Sci Nat Univ Norm Hunan 14:193–197, 1991) contained a way to tackle one of the problems of Bombieri (Research problems in function theory, The Athlone Press, University of London, London, 1967) on the behavior of the coefficients of univalent functions. We shall give an account of the history of the problem and a revised version of it. PubDate: 2016-12-01 DOI: 10.1007/s40315-016-0165-z Issue No:Vol. 16, No. 4 (2016)