On boundary critical points for semigroups of analytic functions

Let X be a one-to-one analytic function of the unit disk D into itself, with O(O) = 0. The origin is an attracting fixed point for X, if X is not a rotation. In addition, there can be fixed points on &19D where X has a finite angular derivative. These boundary fixed points must be repelling (abbreviated b.r.f.p.). The Koenigs function of X is a one-to-one analytic function a defined on D0 such that X = -'(Ao}), where A = O'(0). If OK is the first iterate of / that does have b.r.f.p., we compute the Hardy number of a, h(a) = sup{p > 0: a E HP(D)}, in terms of the smallest angular derivative of OK at its b.r.f.p.. In the case when no iterate of X has b.r.f.p., then a E np . When COk acts on H2(D), by a result of C. Cowen and B. MacCluer, the spectrum of C0, is determined by A and the essential spectral radius of CQ, re(C4s). Also, by a result of P. Bourdon and J. Shapiro, and our earlier work, re (CO) can be computed in terms of h(o). Hence, our result implies that the spectrum of CQ is determined by the derivative of X at the fixed point 0 E D and the angular derivatives at b.r.f.p. of X or some iterate of k.

The rational Bernstein bases and the multirational blossoms

The classical Gauss—Lucas theorem describes the location of the critical points of a polynomial. There is also a hyperbolic version, due to Walsh, in which the role of polynomials is played by finite Blaschke products on the unit disk. We consider similar phenomena for generic inner functions, as well as for certain “locally inner” self-maps of the disk. More precisely, we look at a unit-norm function f ∈ H∞ that has an angular derivative on a set of positive measure (on the boundary) and we assume that its inner factor, I, is nontrivial. Under certain conditions to be discussed, it follows that f′ must also have a nontrivial inner factor, say J, and we study the relationship between the boundary singularities of I and J. Examples are furnished to show that our sufficient conditions cannot be substantially relaxed.

Let α e [0,1]p be an analytic function in the unit disc E q and h be analytic and univalent functions in E, Φ be an analytic function in a certain domain D⊃ q (E). The authors study the differential subordinations and where z⊂E Examples of the application of these results are given.

Let f be an analytic function in the open unit disk that assumes the values 0 and 1 at least once and an unequal number of rimes in a disk of radius 1 10-5 entered at the origin. We prove that fmust assume the value 0 or 1 elsewhere in the unit disk. This result is motivated by a control engineering design problem which is also presented.

Introduction.We extend the basic result on the subordination of analytic functions to analytic functions with values in a Banach space.The fundamental form of this theorem is as follows [2, p. 421]: Let/and g be two analytic (scalarvalued) functions defined on the unit disc D with g univalent and the range off contained in the range of g.Then fand g are analytically related, i.e. there is an analytic function to mapping D into itself with/(z)=g(cu(z)) for all z in D. This result extends to vector-valued analytic functions (for definitions see [1]) but there are subtleties: (1) the proof for scalar-valued functions consists of noting that co(z)=g~1(f(z)) is analytic; for vector-valued functions not only is the proper notion of analyticity for g'1 unavailable but, as we show below, g"1 need not even be continuous, and (2) the result for scalar-valued functions is true for arbitrary domains (usually stated for D only to avoid superfluous generality) while for vector-valued functions we cannot allow punctures in the domain of g.The basic theorem on subordination has as a simple consequence the interesting result that iff and g are nonconstant analytic (scalar-valued) functions on D with intersecting ranges then they are locally analytically related, i.e. there is a neighborhood Vin D and an analytic function oe mapping F into D with/(z)=g(co(z)) for z in V.This result also extends to vector-valued analytic functions but in this case the appropriate hypothesis is that/and g have ranges intersecting in an uncountable set.For f and g scalar-valued, if the ranges off and g intersect at all, then they intersect in an uncountable set by the open mapping theorem; so, as in the basic theorem, it is the failure of the open mapping theorem for vector-valued analytic functions which lends interest to the generalization and makes the proof more delicate.We will show that for vector-valued/and g their ranges may intersect in a set containing an infinite number of accumulation points and yet not be locally analytically related.I. Global theorem.Theorem.Let Xbe a complex Banach space and fand g be two analytic functions mapping the unit disc D into X.If g is one-to-one with the range of g containing the range off, then there is an analytic function u> mapping the disc into itself with f(z)=g(w(z))for all z in D.

A variational trial wave function of the form ψ=f(r1)f(r2)g(r12) has been applied to the ground state of He I. In Sec. I, f(r) is taken to be an arbitrary function and g(r12) is taken as linear. The form of f(r) and the slope of g(r12) are chosen to yield the minimum energy. The energy, —5.79613RHehc, and the form of f are compared with the results of earlier work and with the results obtained with analytic functions. A table of f(r) and the charge distribution is given. The charge distribution is compared with that obtained from a Hartree-Fock wave function and the difference is discussed. In Sec. II, f(r) is taken as an exponential and g(r12) is taken to be an arbitrary function. The coefficient in the exponent in f(r) and the form of g(r12) are chosen to yield the minimum energy. The energy, —5.78252RHehc, and the form of f are compared with the results obtained with analytic wave functions. The behavior of g is shown in a table and by its asymptotic series. The probable effectiveness of ψ if both f(r) and g(r12) are taken as arbi trary is discussed.

We study the two-point correlation K n m ( z , w ) K^m_n(z,w) between zeros and critical points of Gaussian random holomorphic sections s n s_n over Kähler manifolds. The critical points are points ∇ h n s n = 0 \nabla _{h^n} s_n=0 , where ∇ h n \nabla _{h^n} is the smooth Chern connection with respect to the Hermitian metric h n h^n on line bundle L n L^n . The main result is that the rescaling limit of K n m ( z 0 + u n , z 0 + v n ) K^m_n(z_0+\frac u{\sqrt n}, z_0+\frac v{\sqrt n}) for any z 0 ∈ M z_0\in M is universal as n n tends to infinity. In fact, the universal rescaling limit is the two-point correlation between zeros and critical points of Gaussian analytic functions for the Bargmann–Fock space of level 1 1 . Furthermore, in the length scale of order n − 1 2 n^{-\frac 12} , there is a “repulsion" between zeros and critical points for the short range, and a “neutrality" for the long range.

Functions f in the class $ K_H $ are convex, univalent, harmonic, and sense preserving in the unit disk. Such functions can be expressed as $ f = h + \\overline {g} $ where h and g are analytic functions. If $ f \\in K_H $ has $ h(0) = 0, g(0) = 0, h'(0) = 1$ , and $ g'(0) = 0 $ , then $ f \\in K_H^0 $ . For $ f \\in K_H^0 $ and } analytic in the unit disk, an integral representation for $ f\ ilde {*}\\varphi = h*\\varphi + \\overline {g*\\varphi } $ is found. With } a strip mapping, $ f\ ilde {*}\\varphi $ is shown to be in $ K_H^0 $ . In a 1958 paper, Pólya and Schoenberg conjectured that if f and g are conformal mappings of the unit disk onto convex domains, then the Hadamard product f 2 g of f and g has the same property. It is known that the analogue of that result for harmonic mappings is false. In this paper, some examples are given in which the property of convexity is preserved for Hadamard products of certain convex harmonic mappings. In addition, an integral formula is used to determine the geometry of the Hadamard product from the geometry of the factors. This is true in particular for the convolution of strip mappings with certain functions $ f_n \\in K_H^0 $ which take the unit disk to regular n -gons.

Suppose $\\varphi$ is an analytic self-map of open unit disk $\\mathbf{D}$ and $\\psi$ is an analytic function on $\\mathbf{D}$. Then a weighted composition operator induced by $\\varphi$ with weight $\\psi$ is given by $(W_{\\psi, \\varphi}f)(z)= \\psi(z)f(\\varphi(z))$, for $z \\in \\mathbf{D}$ and $f$ analytic on $\\mathbf{D}$. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators $W_{\\psi, \\varphi}$. In terms of fixed points in the closed unit disk $\\overline{\\mathbf{D}}$, conditions under which $W_{\\psi, \\varphi}$ is compact are given. Necessary conditions for the compactness of $C_{\\varphi}$ are given in terms of the angular derivative $\\varphi^{\\prime}(\\zeta)$ where $\\zeta$ is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten $p$-class $S_{p}(H^s(\\beta_1), H^q(\\beta_2))$, where the inducing map has supremum norm strictly smaller than~$1$.

Let f be an analytic function in the unit disc |z|<1 on the complex plane mathbb {C}. This paper is devoted to obtaining the correspondence between f(z) and zf'(z) at the point w, 0<|w|=R< 1, such that |f(w)|=min {|f(z)|: f(z)inpartial f(|z|leq R) }. We present several applications of the main result. A part of them improve the previous results of this type.

We investigate the generalization of the starlikeness of complex order and the generalization of convexity of complex order for the analytic functions in the unit disc D = {z: \z\ &lt; 1}.

Growth analysis of analytic functions is very important part of research inthe field of complex analysis and many researchers are involved in this areaduring past decades. Collecting ideas from Heittokangas et al. (Meromorphicfunctions of finite $\varphi $-order and linear q-difference equations, J.Difference Equ. Appl., 27 (9) (2021), 1280-1309) and Bela\"{\i}di etal. (Study of complex oscillation of solutions of a second order lineardifferential equation with entire coefficients of $(\alpha ,\beta ,\gamma )$%-order, WSEAS Trans. Math., 21 (2022), 361-370), here in this paper,we have defined the $(\alpha ,\beta ,\gamma )$-Nevanlinna order and\ $%(\alpha ,\beta ,\gamma )$-Nevanlinna type of an analytic function $f$ in theunit disc $U$. We have also established some growth properties of thecomposition of two analytic functions in the unit disc on the basis of their$(\alpha ,\beta ,\gamma )$-Nevanlinna order, $(\alpha ,\beta ,\gamma )$%-Nevanlinna lower order, $(\alpha ,\beta ,\gamma )$-Nevanlinna type and $%(\alpha ,\beta ,\gamma )$-Nevanlinna weak type as compared to the growth oftheir corresponding left and right factors, where $\alpha ,\beta ,\gamma $are continuous non-negative functions defined on $(-\infty ,+\infty )$.

This paper is devoted to an investigation of a topological ring of analytic functions. Specifically, this ring, denoted by R, is the set of functions analytic on the unit disc with the usual addition and scalar multiplication, the Hadamard product for its ring multiplication, and the compact-open topology. The ring R is identified algebrai cally with a subring RA of the ring of continuous functions on the non-negative integers X. The operations in fT are the usual pointwise operations, ana the structure of R is determined by considering its isomorph iT. In Chapter I we are concerned with the problems of identifying the maximal ideal space of R and describing the maximal ideals intrinsically. We first show, using theorems on general rings of continuous functions, that the maximal ideals are in one-to-one correspondence with the points of the Stone-Cech compactification px of X. We next give an intrinsic description of the maximal ideals, using the properties of the power series expansions for analytic functions. Using this description we strengthen the prev ious theorem appreciably and show that the maximal ideal space with the hul1-kernel topology is homeomorphic to pX. Finally, the Hadamard product is used to give a simple iv characterization of the dual space of the topological linear space of analytic functions on the unit disc. This dual space is isomorphic to the set of functions in R whose radius of convergence exceeds one, which is exactly the intersection of the maximal ideals corresponding to points of pX -X (the dense maximal ideals of R) . In Chapter II we continue the investigation of the maximal ideals by studying the structure of their associated residue class rings. The complex number field 0 is isomorphically embedded in R/M, where M is a maximal ideal of R. If M corresponds to a point of X, then R/M and the isomorph 0* of 0 are identical; whereas, if M corresponds to a point of ^X -X, then R/M is a transcendental extension of 0* having transcendence degree c, the cardinality of the continuum. Moreover, we show, in the second case, that R/M is algebraically closed. Using theorems on transcendental extensions and algebraically closed fields, we show that, in either case, R/M and 0 are isomorphic fields. The two classes of maximal ideals are distinguished by the fact that their residue class rings admit radically different types of complex-valued isomorphisms. In Chapter III we are concerned primarily with the structure of the closed ideals of R. The basic tool used is the rotational completeness theorem for analytic v functions, which we proved using the methods and results of harmonic analysis. We show that the closure of every principal ideal is principal, give a necessary and suffi cient condition that a principal ideal be closed, and show that every closed ideal is a principal ideal generated by an idempotent element of R. Using these theorems we indicate connections with the general theory of dual rings, of which R is an example, and raise several questions for further i investigation in the direction of releasing some of the restrictions with which most of the results so far have been obtained.

For an analytic function ऒ on the unit disk , we show that the L2 integral mean of ऒ on c &lt; |z| &lt; r with respect to the weighted area measure (1 − |z|2)αd A(z) is a logarithmically convex function of r on (c, 1), where −3 ≤ ∞ ≤ 0 and c ∈ [0, 1). Moreover, the range [−3, 0] for ∞ is best possible. When c = 0, our arguments here also simplify the proof for several results we obtained in earlier papers.