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Estimates for the Logarithmic Derivative of a Meromorphic Function, Plus Similar Estimates
Journal of The London Mathematical Society
... . It now follows, using the Wiman-Valiron theory [12] and estimates for logarithmic derivatives [8] applied to g, that f has finite order. It also follows that f has an unbounded sequence of poles, since otherwise f ′ /f is rational at infinity. ...
... Set T 1 = f ′ /f . Since g k and T 1 have finite order, standard estimates [8] give ...
... The following lemma, in which transcendentally fast means faster than any power of z, gives a sufficient condition for an analytic solution of N[y] = 0 to annihilate a solution of L[y] = 0. Proof. If g tends to zero transcendentally fast on a sector, then F/f = g −k tends to infinity transcendentally fast there; since f has finite order by Lemma 5.3, this contradicts standard estimates [8] for logarithmic derivatives f (j) /f . Next, if N[y] = 0 has a fundamental set of canonical formal solutions with the same exponential part κ, then κ is a polynomial in z, by Lemma 4.1 and Abel's identity. ...
The paper determines all meromorphic functions with finitely many zeros in the plane having the property that a linear differential polynomial in the function, of order at least 3 and with rational functions as coefficients, also has finitely many zeros.
... However, in order to fulfill our later application to establishing the Theorem 2.1, one needs more detailed information concerning the exceptional set that arises from removing the zeros and poles from applying the Cartan lemma (below) than quoting Gundersen [9, §7] directly. So we judge it is appropriate to offer a full proof of the Lemma 2.4 based on that in [9] but tailored to our need here. In fact, our construction of the exceptional disks which have larger radii than those constructed in [9] by 3η. ...
... So we judge it is appropriate to offer a full proof of the Lemma 2.4 based on that in [9] but tailored to our need here. In fact, our construction of the exceptional disks which have larger radii than those constructed in [9] by 3η. This is to guarantee the fact that we need below, namely that whenever z lies outside E (3η) , then the line segment [z, z + η] lies outside E (η) . ...
... Let us first recall Cartan's theorem [5] which we adopt from [9]. ...
The paper gives a precise asymptotic relation between higher order logarithmic difference and logarithmic derivatives for meromorphic functions with order strictly less then one. This allows us to formulate a useful Wiman-Valiron type estimate for logarithmic difference of meromorphic functions of small order. We then apply this estimate to prove a classical analogue of Valiron about entire solutions to linear differential equations with polynomials coefficients for linear difference equations.
... Gundersen used R-sets and E-sets (but under different terminology) in finding sharp estimates for logarithmic derivatives of meromorphic functions [6]. Applying these findings, Chiang and Feng obtained pointwise estimates for logarithmic differences of meromorphic functions [3], while Wen and Ye estimated logarithmic q-differences of meromorphic functions [18]. ...
... The paper [6] is frequently cited in the theory of complex differential equations. The extremal cases K (x) ≡ x and K (x) ≡ 1 in Corollary 4.1 below correspond to Theorems 3 and 4 in [6], respectively. ...
... The paper [6] is frequently cited in the theory of complex differential equations. The extremal cases K (x) ≡ x and K (x) ≡ 1 in Corollary 4.1 below correspond to Theorems 3 and 4 in [6], respectively. In fact, [6,Theorem 4] is slightly improved because r ε in the upper bound is replaced with log α r . ...
The well-known E -set introduced by Hayman in 1960 is a countable collection of Euclidean discs in the complex plane, whose subtending angles at the origin have a finite sum. An important special case of an E -set is known as the R -set, for which the sum of the diameters of the discs is finite. These sets appear in numerous papers in the theories of complex differential and functional equations. A given E -set (hence an R -set) has the property that the set of angles θ for which the ray arg ( z ) = θ meets infinitely many discs in the E -set has linear measure zero. This paper offers a continuous transition from E -sets to R -sets and then to much thinner sets. In addition to rays, plane curves that originate from the zero distribution theory of exponential polynomials will be considered. It turns out that almost every such curve meets at most finitely many discs in the collection in question. Analogous discussions are provided in the case of the unit disc D , where the curves tend to the boundary ∂ D tangentially or non-tangentially. Finally, these findings will be used for improving well-known estimates for logarithmic derivatives, logarithmic differences and logarithmic q -differences of meromorphic functions, as well as for improving standard results on exceptional sets.
... We say F has logarithmic density if log dens(F ) = log dens(F ). The proofs of our results highly rely on the estimation of logarithmic derivatives, which is due to Gundersen [12]. 12]). ...
... The proofs of our results highly rely on the estimation of logarithmic derivatives, which is due to Gundersen [12]. 12]). Let f be a transcendental meromorphic function of finite order ρ(f ). ...
... Case 2. Assume meas(E * ) > 0, then there exist some sectors in which the indicator of A(z) satisfying h(θ) < 0. Hence, there must exist an interval I A ∈ [0, 2π) such that h(θ) < 0 for all θ ∈ I A . By Lemma 2.1, there exists a set E 1 ⊂ (1, ∞) with finite logarithmic measure such that for all z satisfying |z| = r ∈ E 1 ∪ [0, 1], (12) holds. For given 0 < c < 1, set ...
In this paper we discuss the classical problem of finding conditions on the entire coefficients A(z) and B(z) guaranteeing that all non-trivial solutions of f + A(z)f + B(z)f = 0 are of infinite order. We assume A(z) is an entire function of completely regular growth and B(z) satisfies three different conditions, then we obtain three results respectively. The three conditions are (1) B(z) has a dynamical property with a multiply connected Fatou component, (2) B(z) satisfies T (r, B) ∼ log M (r, B) outside a set of finite logarithmic measure, (3) B(z) is extremal for Den-joy's conjecture.
... The following lemma is given by Gundersen [5], plays very crucial role in proving our results. It provides an estimate of logarithmic derivative of finite order transcendental meromorphic function. ...
... It provides an estimate of logarithmic derivative of finite order transcendental meromorphic function. Lemma 1. [5] Let f be a transcendental meromorphic function with finite order and (k, j) be a finite pair of integers that satisfies k > j ≥ 0. Let > 0 be a given constant. Then, the following three statements hold: ...
... Suppose f be any non-trivial solution of finite order of equation (2). Then, using Lemma 1, we get there exists a set F ⊂ [0, ∞] of finite linear measure such that for all |z| ∈ F, we get (5). Let ω and δ be fixed constant such that max{ρ(a 1 ), . . . ...
... The following result due to Gundersen [15] plays an important role in the theory of complex differential equations. ...
... Lemma 3.1. ([15]) Let f be a transcendental meromorphic function, and let χ > 1 be a given constant. Then there exist a set E 1 ⊂ (1, ∞) with finite logarithmic measure and a constant B > 0 that depends only on χ and i, j (0 ≤ i < j ≤ k), such that for all z satisfying |z| = r / ∈ [0, 1] ∪ E 1 , we have ...
In this paper, we deal with the growth and oscillation of solutions of higher order linear differential equations. Under the conditions that there exists a coefficient which dominates the other coefficients by its lower -order and lower -type, we obtain some growth and oscillation properties of solutions of such equations which improve and extend some recently results of the author and Biswas \cite{b8}.
... In addition, we define the logarithmic measure of a set E ⊂ (1, +∞) by mes l (E) = E dt t and the linear measure of a set F ⊂ (0, +∞) by mes(F) = F dt. The following result due to Gundersen [8] plays an important role in the theory of complex differential equations. Theorem 1.1. ...
... Theorem 1.1. [8] Let f be a transcendental meromorphic function of finite order ρ := ρ( f ). Let ε > 0 be a constant, and k, j be integers such that k > j ≥ 0. ...
In this paper, we give some estimates on the growth of the logarithmic derivative of meromorphic functions by considering the concept of ϕ-order. We discuss their relationship with the growth of solutions of certain complex differential equations.
... The following logarithmic derivative estimation was found in [12] from Gundersen. Proof. ...
... ([12]) Let f be a transcendental meromorphic function, and let ξ > 1 be a given constant. Then there exist a set E ⊂ (1. ...
The main aim of this paper is to study the growth of solutions of higher order linear differential equations using the concepts of -order and -type. We obtain some results which improve and generalize some previous results of Kinnunen \cite{13}, Long et al. \cite{L} as well as Bela\"{\i}di \cite{b3}, \cite{b5}.
... We say F has logarithmic density if log dens(F ) = log dens(F ). The proofs of our results highly rely on the estimation of logarithmic derivatives, which is due to Gundersen [12]. 12]). ...
... The proofs of our results highly rely on the estimation of logarithmic derivatives, which is due to Gundersen [12]. 12]). Let f be a transcendental meromorphic function of finite order ρ(f ). ...
The growth of solutions of second order complex differential equations f + A(z)f + B(z)f = 0 with transcendental entire coefficients is considered. Assuming that A(z) has a finite deficient value and that B(z) has either Fabry gaps or a multiply connected Fatou component, it follows that all solutions are of infinite order of growth.
... The growth of the solutions to complex differential equations is an important area of study in complex analysis. In particular, the solutions of complex differential equations with specific coefficients have infinite growth order, such as [9,12,13,19,40]. Based on the above properties regarding the Borel direction of entire functions of infinite order, we can consider the common Borel directions of the solutions of these complex differential equations and their (anti)derivatives. ...
In this paper, we study the common Borel directions of an entire function f and its (anti)derivatives, as well as the common Julia limiting directions of f and its (anti)derivatives, where f is an infinite-order solution of linear complex differential equations. Our main results provide lower bounds for the measure of the intersection of both the set of common Borel directions and the set of the common Julia limiting directions of these solutions.
... To prove the remaining assertions, write Z = Z r e T +iB on K ± , where |B| = π/8 and T is real. The formulas (10) and (11) imply that there exists E q ∈ C such that ...
Suppose that a transcendental meromorphic function in the plane has finitely many critical values, while its multiple points have bounded multiplicities, and its inverse function has finitely many transcendental singularities. Using the Wiman-Valiron method it is shown that the Schwarzian derivative does not have a direct transcendental singularity over infinity, and does not have infinity as a Borel exceptional value.
... The following lemma due to G. G. Gundersen [3], which is used to prove Theorem 2.1, is an estimation of the logarithmic measure about the total moduli of the zeros and poles of a meromorphic function. Its statement and proof have been embedded as a part of the proof of [3, Theorem 3]. ...
We extend the difference analogue of Cartan's second main theorem for the case of slowly moving periodic hyperplanes, and introduce two different natural ways to find a difference analogue of the truncated second main theorem. As applications, we obtain a new Picard type theorem and difference analogues of the deficiency relation for holomorphic curves.
... Finally, there exists a set E 1 ⊆ R of measure 0 such that for all θ ∈ R \ E 1 the estimate (4.20) holds as z = re iθ tends to infinity. [13]. ✷ Proof. ...
Lower bounds are given for the number of non-real zeros of a second order linear differential polynomial with constant coefficients in a real entire function with finitely many non-real zeros.
... Lemma 6.8 (see [7]). Let z 1 , z 2 , · · · be an infinite sequence of complex numbers that has no finite limit point, and that is ordered by increasing moduli. ...
By extending the idea of a difference operator with a fixed step to varying-steps difference operators, we have established a difference Nevanlinna theory for meromorphic functions with the steps tending to zero (vanishing period) and a difference Nevanlinna theory for finite order meromorphic functions with the steps tending to infinity (infinite period) in this paper. We can recover the classical little Picard theorem from the vanishing period theory, but we require additional finite order growth restriction for meromorphic functions from the infinite period theory. Then we give some applications of our theories to exhibit connections between discrete equations and and their continuous analogues.
... The following new version of Cartan's lemma will be needed. The proof is influenced by that of [11,Lemma 2]. Lemma 4.5. ...
A meromorphic solution of a complex linear differential equation (with meromorphic coefficients) for which the value zero is the only possible finite deficient/deviated value is called a standard solution. Conditions for the existence and the number of standard solutions are discussed for various types of deficient and deviated values.
... First, we denote the Lebesgue linear measure of a set E ⊂ [0, +∞) by m (E) = F dt, and the logarithmic measure of a set F ⊂ (1, +∞) by m l (F ) = F dt t . The following result due to Gundersen [13] plays an important role in the theory of complex differential equations. Then there exist a set E 1 ⊂ (1, ∞) with finite logarithmic measure and a constant B > 0 that depends only on χ and i, j (0 ≤ i < j ≤ k), such that for all ...
In this paper, we deal with the growth and oscillation of solutions of higher order linear differential equations. Under the conditions that there exists a coefficient which dominates the other coefficients by its lower (α, β, γ)-order and lower (α, β, γ)-type, we obtain some growth and oscillation properties of solutions of such equations which improve and extend some recently results of the author and Biswas [4].
... for a B ∈ R, except for a set E in r = |z| of finite linear measure, since f (m) /f is of polynomial growth except for some set E in r = |z| of finite linear measure, see Gundersen [4,Corollary 3], such that ...
Consider the linear differential equation where the coefficients are exponential polynomials. It is known that every solution is entire. This paper will show that all transcendental solutions of finite growth order are of completely regular growth. This problem was raised in Heittokangas et al.[8, p.33], which involves an extensive question about Gol'dberg-Ostrovski\v{i}'s Problem [5, p.300]. Moreover, we define functions in a generalized class concluding exponential polynomial functions, which are also of completely regular growth.
... By substituting (9), (10) and (40) into (30), for all z satisfying |z| = r m / ...
The purpose of this paper is the study of the growth of solutions of higher order linear differential equations and where are meromorphic functions of finite order and are polynomials with degree . Under some others conditions, we extend the previous results due to Hamani and Belaïdi [1].
... Lemma 16. ( [14]) Let f (z) be a nontrivial entire function, and let κ > 1 and ε > 0 be given constants. Then there exist a constant c > 0 and a set E 1 ⊂ [0, +∞) having finite linear measure such that for all z satisfying |z| = r / ∈ E 1 , we have ...
In this paper, we wish to investigate the complex higher order linear differential equations in which the coefficients are entire functions of (α, β)-order and obtain some results which improve and generalize some previous results of Tu et al. [33] as well as Belaïdi [2, 3, 4].
... We changed the conditions on A(z) to have multiply connected Fatou component. Lemma 5 is given by Gundersen [4]. He generalized the estimates of logarithmic derivatives of transcendental meromorphic function of finite order. ...
In this article, we study about the solutions of second order linear differential equations by considering several conditions on the coefficients of homogenous linear differential equation and its associated non-homogenous linear differential equation.
... In this section we present some lemmas which will be needed in the sequel. 11]). Let f (z) be a nontrivial entire function, and let κ > 1 and ε > 0 be given constants. ...
In this article, we investigate the complex higher order linear differential equations in which the coefficients are entire functions of (α, β, γ)-order and obtain some results which improve and generalize some previous results of Tu et al. [29] as well as Belaidi [1, 2, 3].
... The following lemma is used to prove Theorem 1.7 for the case of q = 1. The following logarithmic derivative estimation was found in [6] from Gundersen. ...
The fast growing solutions of the following linear differential equation ( ∗ ) is investigated by using a more general scale [ p , q ] , φ -order, f ( k ) + A k - 1 ( z ) f ( k - 1 ) + · · · + A 0 ( z ) f = 0 , ( ∗ ) where A i ( z ) are entire functions in the complex plane, i = 0 , 1 , … , k - 1 . The growth relationships between entire coefficients and solutions of the equation ( ∗ ) is found by using the concepts of [ p , q ] , φ -order and [ p , q ] , φ -type, which extend and improve some previous results.
... , see [11]. We next recall four results that all are from the seminal paper [6]. ...
The classical problem of finding conditions on the entire coefficients A j (j = 0, 1, · · · , k − 1) ensuring that all nontrivial solutions to higher order differential equations f (k) + A k−1 f (k−1) + · · · + A 1 (z)f + A 0 (z)f = 0 are of infinite lower order is being discussed in this paper. In particular, we assume that the coefficients (or most of them) are Mittag-Leffler functions.
... The next lemma was proved by Gundersen [9] and will be used to obtain the hyper order 2 (f ) of the meromorphic solution f. ...
In this paper, we discuss the growth of meromorphic solutions of some classes of homogeneous and nonhomogeneous differential equations. We will prove that if meromorphic functions F , A j , D j and polynomials P j with degree n ≥ 1 ( j = 0 , 1 , ⋯ , k - 1 ) satisfy some conditions, then the equation f ( k ) + ( A k - 1 e P k - 1 + D k - 1 ) f ( k - 1 ) + ⋯ + ( A 1 e P 1 + D 1 ) f ′ + ( A 0 e P 0 + D 0 ) f = F ( z ) ( k ≥ 2 ) when F ≡ 0 , all solutions f ≢ 0 have infinite order and hyper order σ 2 ( f ) ≥ n . This is a continue work of [Gan and Sun in Adv. Math. 36 (1), 51–60 (2007)] and [Chen and Xu in Electron. J. Qual. Theory Differ. Equ. 1 : 1–13 (2009)]
In this paper, we study the value distribution of zeros of certain nonlinear difference polynomials of entire functions of finite order.
Sitting at the top level of the Askey-scheme, Wilson polynomials are regarded as the most general hypergeometric orthogonal polynomials. Instead of a differential equation, they satisfy a second order Sturm-Liouville type difference equation in terms of the Wilson divided-difference operator. This suggests that in order to better understand the distinctive properties of Wilson polynomials and related topics, one should use a function theory that is more natural with respect to the Wilson operator. Inspired by the recent work of Halburd and Korhonen, we establish a full-fledged Nevanlinna theory of the Wilson operator for meromorphic functions of finite order. In particular, we prove a Wilson analogue of the lemma on logarithmic derivatives, which helps us to derive Wilson operator versions of Nevanlinna's Second Fundamental Theorem, some defect relations and Picard's Theorem. These allow us to gain new insights on the distributions of zeros and poles of functions related to the Wilson operator, which is different from the classical viewpoint. We have also obtained a relevant five-value theorem and Clunie type theorem as applications of our theory, as well as a pointwise estimate of the logarithmic Wilson difference, which yields new estimates to the growth of meromorphic solutions to some Wilson difference equations and Wilson interpolation equations.
In this paper, we study a problem to determine the forms of a non-constant entire function f when it shares a set S of 4 distinct elements with its k-th derivatives. This work is motivated by the work of Chang et al. (Arch Math 89: 561–569, 2007) which pertains to a set of three elements. We have comprehensively extended the same result, as this type of extension has not been addressed earlier.
We study the solutions of the fractional differential equation LC D α z f ′ (z) + A(z) LC D β z f (z) + B(z)f (z) = 0 where LC D α z and LC D β z are the Liouville-Caputo fractional derivatives of orders n − 1 < α, β ≤ n ∈ N * , and z is complex number, A(z), B(z) be entire functions. We find conditions on the coefficients so that every solution that is not identically zero has infinite order.
In this paper, we consider entire solutions of higher order homogeneous differential equations with the entire coefficients having the same order, and prove that the entire solutions are of infinite lower order. The properties on the radial distribution, the limit direction of the Julia set and the existence of a Baker wandering domain of the entire solutions are also discussed.
"In this paper, we investigate the relations between the growth of entire coefficients and that of solutions of complex homogeneous and non-homogeneous linear difference equations with entire coefficients of -order by using a slow growth scale, the -order, where is a non-decreasing unbounded function. We extend some precedent results due to Zheng and Tu (2011) [15] and others."
Given an unbounded non-decreasing positive function φ, we studied what the relations are between the growth order of any solution of a complex linear differential–difference equation whose coefficients are entire or meromorphic functions of finite φ-order. Our findings extend some earlier well-known results.
A study of linear differential equations and their coefficients is presented in order to analyze their application to neural networks. Since neural networks have more layers, it is possible to solve the complicated functions using the differential equation notation for the input dimension, output dimension, and vector of parameters. A system with differentiable layers is predictable as its state can be predicted and its solutions can be determined using minimum loss functions.
This paper consists of three parts: First, letting , , and be nonzero polynomials such that and have the same degree and distinct leading coefficients 1 and , respectively, we solve entire solutions of the Tumura–Clunie type differential equation , where is an integer, is a differential polynomial in f of degree with coefficients having polynomial growth. Second, we study the oscillation of the second-order differential equation and prove that for some integer if this equation admits a nontrivial solution such that . This partially answers a question of Ishizaki. Finally, letting and be constants and l and s be relatively prime integers such that , we prove that l=2 if the equation admits two linearly independent solutions and such that . In particular, we precisely characterize all solutions such that when l=2 and l=4 .
By looking at the situation when the coefficients Pj(z) (j = 1, 2, ⋯, n − 1) (or most of them) are exponential polynomials, we investigate the fact that all nontrivial solutions to higher order differential equations f(n) + Pn−1(z)f(n−1) + ⋯ + P0(z)f = 0 are of infinite order. An exponential polynomial coefficient plays a key role in these results.
An exponential polynomial is a finite linear sum of terms P(z)eQ(z), where P(z)P(z) and Q(z)Q(z) are polynomials. The early results on the value distribution of exponential polynomials can be traced back to Georg Pólya's paper published in 1920, while the latest results have come out in 2022. Despite of over a century of research work, many intriguing problems on value distribution of exponential polynomials still remain unsolved. The role of exponential polynomials and their quotients in the theories of linear/non‐linear differential equations, oscillation theory and differential‐difference equations will also be discussed. Thirteen open problems are given to motivate the readers for further research in these topics.
In this paper, we investigate the [p,q]-order of growth of solutions to certain linear differential equations with entire coefficients and analytic coefficients in the unit disc by using the Nevanlinna theory of meromorphic functions. This work is an improvement and generalization of some previous results by the second author.
We consider the di?erential equation f'' +A(z)f' +B(z)f = 0, where A(z) and B(z) are entire complex functions. We improve various restrictions on coe?cients A(z) and B(z) and prove that all non-trivial solutions are of in?nite order.
In this study, we show that all non-trivial solutions of f′′+A(z)f′+B(z)f=0 have infinite order, provided that the entire coefficient A(z) has certain restrictions and B(z) has multiply-connected Fatou component. We also extend these results to higher order linear differential equations.
This article is devoted to the study of solutions of non-homogenous linear differential equations having entire coefficients. We get all non-trivial solutions of infinite order of equation , by restricting the conditions on coefficients.
Let A be a transcendental entire function of order < 1. If w1and w2are two linearly independent solutions of the differential equation y″+ Ay= 0, then at least one of w1w2has the property that the exponent of convergence of its zeros is = 1.
A representation for the logarithmic derivative of an entire function f of finite order, parametrically in terms of some zeros and critical points of f, is derived from the Hadamard representation and applied to Lucas' type theorems and to growth estimates of .
Synopsis
We show that if B(z) is either (i) a transcendental entire function with order (B) ≠1, or (ii) a polynomial of odd degree, then every solution f ≠0 to the equation f ″ + e −z f ′ + B(z)f = 0 has infinite order. We obtain a partial result in the case when B(z) is an even degree polynomial. Our method of proof and lemmas for case (i) of the above result have independent interest.
{Das Hauptergebnis der Arbeit ist folgender Satz: Im Einheitskreis sei eine unendliche Menge von Systemen von je p regulären Funktionen ohne Nullstellen gegeben, für die identisch erfüllt ist. Man kann eine Teilmenge auswählen, die eine der beiden folgenden Eigenschaften besitzt:par I. Die Nummern 1,dots,p zerfallen in zwei Sorten derart, daß (1) der Quotient zweier beliebiger Funktionen erster Art konvergiert, (2) der Quotient einer beliebigen Funktion zweiter Art durch eine Funktion erster Art gegen Null konvergiert, (3) als Folge von (2) der Quotient der Summe aller Funktionen erster Art durch eine beliebige derselben gegen Null konvergiert. Es gibt stets mindestens zwei Nummern erster Art. Die Nummern zweiter Art können fehlen.par II. Es gibt zwei Gruppen von Nummern, deren jede mindestens zwei Nummern umfaß t, dazu evtl. einige Nummern, die keiner der beiden Gruppen angehören. In jeder Gruppe kann man wieder zwei Nummernsorten unterscheiden. Für diese gelten wieder die Aussagen (1), (2), (3) wie unter I, nur daß (3) keine Folge von (2) mehr ist.par Besonders eingebend werden die Fälle und erörtert. Unter den unmittelbaren Anwendungen sei hervorgehoben das Studium von Familien von Funktionenpaaren, derart, daß alle Funktionen frei von Nullstellen sind, und daß die Summe eines jeden Paares den Wert Eins ausläß t; oder derart, daß die Summe gegen Null konvergiert. In diesem letzteren Falle z. B. kann man eine Teilfolge auswählen, in der entweder die beiden Funktionen f und g des Paares einzeln gegen Null streben, oder gegen konvergiert.par Weiter ergeben sich gewisse Verallgemeinerungen der Sätze von it Schottky und it Landau.par Es folgen Anwendungen auf den it Pólya-Nevanlinnaschen Unitätssatz der in der Umgebung von z=infty meromorphen Funktionen, wobei namentlich die Fälle von 2, 3, 4 Stellensorten diskutiert werden. Die Beweise stützen sich auf einen hier erstmalig bewiesenen Satz von it A. Bloch. Verf. gibt ihm die folgende präzise Fassung: In der z-Ebene seien n beliebige Stellen gegeben; h sei eine gegebene positive Zahl. Die Stellen z, für die gilt, gehören dem Innern von höchstens n Kreisen an, deren Radiensumme 2eh nicht übertrifft. Dieser Satz wird verschiedentlich verallgemeinert und auch auf einige funktionentheoretische Fragen angewendet. Hervorgehoben sei z. B. die folgende Verschärfung eines Satzes von it S. Mandelbrojt: Es sei f(z) in regulär und daselbst ; gegeben seien zwei beliebige positive Zahlen und gamma. Dann gehört dazu eine von f unabhängige positive Zahl A so, daß fraclog vert f(x)vertlog vert f(y)vert} leqq A für jedes Wertepaar aus gilt, ausgenommen die x-Stellen aus gewissen Kreisen der Radiensumme gamma.par Besprechung: P. Montel; Bulletin sc. Math. 53 (1929), 44-48.
Meromorphic functions on the complex plane which have the same inverse images counting multiplicities for four values are Mobius transforms of each other. The aim of this paper is to give an extension of this statement to moving targets.
On a solution of w" 1s-' t ' *(az+b)w : 0', Kodai Math
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A general theorem concerning the growth ofsolutions offirst-order algebraic differeatial equations
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equations', C omp o s it io M a t h. 25 (19'1 2) 61-7 A.
Sur 1es systimes de lonctions holomorphes i varidtds lin6aires laculaires et leurs applications
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Onthequestionofwhether/"+e-"f'+n917=
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G. G. Guvornsex,'Onthequestionofwhether/"+e-"f'+n917=
Ocanadmitasolution/* 0offinite order
Ocanadmitasolution/* 0offinite
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