A class of second order differential equations

We study the differential equationf″=N(f)f′ ² +M(f)f′+L(f), whereL, M, N are rational functions, and prove that if the differential equation has a transcendental meromorphic solutionf with order,p(f)>2, then the differential equation must be one of nine forms; and, moreover, we construct examples showing the existence of these nine forms with a transcendental meromorphic solution.     

Convolution properties of some classes of analytic functions

Let A denote the class of functions which are analytic in |z|<1 and normalized so that f(0)=0 and f′(0)=1, and let R(α, β)⊂A be the class of functions f such thatRe[f′(z)+αzf″(z)]>β,Re α>0, β<1. We determine conditions under which (i) f ∈ R(α₁, β₁), g ∈ R(α₂, β₂) implies that the convolution f×g of f and g is convex; (ii) f ∈ R(0, β₁), g ∈ R(0, β₂) implies that f×g is starlike; (iii) f≠A such that f′(z)[f(z)/z]^μ-1 ≺ 1 + λz, μ>0, 0<λ<1, is starlike, and (iv) f≠A such that f′(z)+αzf″(z) ≺ 1 + λz, α>0, δ>0, is convex or starlike. Bibliography: 16 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 138–154.     

Structural formulas and value regions of functionals in certain classes of regular functions

We study the structural properties of the class M_k,λ,b(k≥2, 0≤λ≤1, b∈ℂ\{0}) of functions f(z)=z+ ... which are regular in |z|<1 and satisfy the conditions f(z)f′(z)z⁻¹≠0 and , where J(z)=λ(1+b⁻¹zf″(z)/f′(z)+(1−λ)(b⁻¹zf′(z)/f(z)+1−b⁻¹). The value regions of some functionals on this class are found. The case λ=1 was considered in our previous paper. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 55–60. Translated by O. A. Ivanov.     

具有四个有穷的IM公共小函数的整函数

本文证明了两个非常数整函数若具有四个有穷的ＩＭ公共小函数，则它们必恒等．从而在整函数的情形下，Ｎｅｖａｎｌｉｎｎａ五值定理中的五个ＩＭ公共值能否更换为五个ＩＭ公共小函数（含∞）的问题的回答是肯定的．     

On a conjecture of Hayman

Letf(z) be a transcendental meromorphic function in the plane. Hayman conjectured thatf f′ assumes all finite values except possibly zero infinitely many times. In this paper, we solve this conjecture partly. The Project Supported by National Natural Science Foundation of China     

On a region of values in the class of typically real functions

The paper studies the regions of values of the systems {f(z₁), f(r₁), f(r₂),…, f(r_n)} and {f(r₁), f(r₂),…, f (r_n)}, where n ⁥ 2; z₁ is an arbitrary fixed point of the disk U = {z: |z| < 1} with Im z₁ ≠ 0; r_j are fixed numbers, 0 < r_j < 1, j = 1, 2,…, n; f ∈ T, and the class T consists of the functions f(z), f(0) = 0, f′(0) = 1, regular in the disk U and satisfying the condition Im f(z) · Imz > 0 for Im z ≠ 0. As an implication, the region of values of f(z₁) in the subclass of functions f ∈ T with prescribed values f(r_j) (j = 1, 2,…, n) is determined. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 5–16.     

Picard values of <Emphasis Type="Italic">p</Emphasis>-adic meromorphic functions

We investigate Picard-Hayman behavior of derivatives of meromorphic functions on an algebraically closed field K, complete with respect to a non-trivial ultrametric absolute value. We present an analogue of the well-known Hayman’s alternative theorem both in K and in any open disk. Here the main hypothesis is based on the behaviour of |f|(r) when r tends to +∞ on properties of special values and quasi-exceptional values.We apply this study to give some sufficient conditions on meromorphic functions so that they satisfy Hayman’s conjectures for n = 1and for n = 2. Given a meromorphic transcendental function f, at least one of the two functions f′f and f′f ² assumes all non-zero values infinitely often. Further, we establish that if the sequence of residues at simple poles of a meromorphic transcendental function on K admits no infinite stationary subsequence, then either f′ + af ² has infinitely many zeros that are not zeros of f for every a ∈ K* or both f′ + bf ³ and f′ + bf ⁴ have infinitely many zeros that are not zeros of f for all b ∈ K*. Most of results have a similar version for unbounded meromorphic functions inside an open disk.     

涉及导数与差分的亚纯函数唯一性

设f,g是两个非常数亚纯函数,a是一个非零有穷复数,n≥5是一个正整数.若[f(z)]~n与[g(z)]~n CM分担a,f(z)与g(z) CM分担∞,且N_(1))(r,f)=S(r,f),则或者f(z)三tg(z),其中t~n=1;或者f(z)g(z)≡t,其中t~n=a~2.由此改进了涉及导数与差分的一些亚纯函数唯一性的结果.     

Escaping points of meromorphic functions with a finite number of poles

We establish several new properties of the escaping setI(f)={z∶f ⁿ (z)→∞ andf ⁿ(z)⇑∞ for eachn∈N} of a transcendental meromorphic functionf with a finite number of poles. By considering a subset ofI(f) where the iterates escape about as fast as possible, we show thatI(f) always contains at least one unbounded component. Also, iff has no Baker wandering domains, then the setI(f)⊂J(f), whereJ(f) is the Julia set off, has at least one unbounded component. These results are false for transcendental meromorphic functions with infinitely many poles.     

单级零点较少的亚纯函数的一个界囿不等式

本文证明了:若.f(z)为复平面上的非线性亚纯函数,满足N1(r,1/f)=S(r,f),则对任何非零复数c,成立T(r,f)<11N(r,f)+11N(r,1/f'-c)+S(r,f).这里N1(r,1/f)表示f(z)的单级零点的计数函数.     

A FUNDAMENTAL INEQUALITY AND ITS APPLICATION

Let f(z) be meromorphie in |z|k+4+[2/k].In this note,a fundamental inequality is established such that thecharacreristic function T(r,f)can be limibd by N(r,1/f)and _(τ-1)(r,1/(f~(k)-1).As anapplication,the following criterion for normality is also proved:Let be a family ofmeromorphic functions in a region D.If for every f(z)∈ ,f(z)≠0 and all the zeros off~(k)(z)-1 are of multiplicity >k+4+[2/k]in D,then is normal there.     

The growth of solutions of <Emphasis Type="Italic">f</Emphasis>″ + e<Superscript>-z</Superscript><Emphasis Type="Italic">f</Emphasis>′+ Q(z)<Emphasis Type="Italic">f</Emphasis> = 0 where the order (Q) = 1

This paper investigates the growth of solutions of the equation f″ + e^-z f′ + Q(z)f = 0 where the order (Q) = 1. When Q(z) = h(z)e^bz, h(z) is nonzero polynomial, b ≠ - 1 is a complex constant, every solution of the above equation has infinite order and the hyper-order 1. We improve the results of M. Frei, M. Ozawa, G. Gundersen and J. K. Langley.     

Normal families of holomorphic functions and shared values

LetF be a family of holomorphic functions in a domainD, and leta, b be two distinct finite complex numbers. If, for anyf ∈F,f andf′ sharea IM, andf′(z)=b wheneverf(z)=b, thenF is normal inD. This improves results due to Pang, Pang and Zalcman, Xu, etc. Supported by the NNSF of China (Grant No. 10071038) and “Qing Lan Project” of the Educational Department of Jiangsu Province.     

The boundary distortion and extremal problems in certain classes of univalent functions

We consider the class S₁(τ), 0<τ<1, of functions f(z)=rz+a₂z²+... that are regular and univalent in the unit disk U and have |f(z)|<1. We obtain sharp estimates for the 1-measure of the sets {θ: |f(e^iθ)|=1}. As a corollary, for the familiar class S we find Kolmogorov-type estimates for the sets {θ: |f(e^iθ)|>M}, M>1, and prove inequalities for the harmonic measure, which are similar to those by Carleman-Milloux and Baernstein. We also consider problems on distortion of fixed systems of boundary arcs in the classes of functions that are regular (or meromorphic) and univalent in the disk or circular annulus. Bibliography: 23 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 115–142. Translated by A. Yu. solynin.     

On the derivative of a polynomial

For an entire function f(z), let . If p(z) is a polynomial of degree n, then, in general, it is difficult to obtain a lower bound for M(p′, 1). But if the zeros of the polynomial are close to the origin, then various lower bounds for M(p′, 1) have been obtained in the past. In this paper, we have considered polynomials having all their zeros in ⋎z⋎≤k(k≥1), with a possible zero of order m(m≥0) at the origin and have obtained a lower bound for M(p′, 1), which is better than most of the known lower bounds. Our bound is sharp for m=0.     

Caratheodory inequality for analytic operator function

Abstract. Suppose H is a complex Hilbert space, AH (△) denotes the set of all analytic operator functions on     

The region of values of a system of functionals in the class of typically real functions

Let T_R be the class of functions that are regular and typically real in the disk E={z:⋱z⋱<1}. For this class, the region of values of the system {f(z₀), f(r)} for z₀ ∈ ℝ, r∈(-1,1) is studied. The sets D_r={f(z₀):f∈T_R, f(r)=a} for −1≤r≤1 and Δ_r={(c₂, c₃): f ∈ T_R, −f(−r)=a} for 0<r≤1 are found, where aε(r(1+r)⁻², r(1−r)⁻²) is an arbitrary fixed number. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 69–79.     

Sharp forms of nevanlinna’s error terms

Let f(z) be a meromorphic function in the plane. If ψ(t)/t andp(t) are two positive, continuous and non-decreasing functions on [1,∞) with ∫ ₁ ^∞ dt/ψ(t) = ∞ and ∫ ₁ ^∞ dt/p(t) = ∞, then asr → ∞ outside a small exceptional set, provided that the divergence of the integral ∫ ₁ ^r dt/ψ(t) is slow enough. The same forms for the logarithmic derivative and for the ramification term are obtained. It is shown by example that the estimates are best possible. Author supported by Max-Planck-Gesellschaft Z.F.D.W and by NSFC.     

Weierstrass factorizations in compact Riemann surfaces

Letν′ be the complementary in a compact Riemann surface ν of a point (or a finite set). In this paper are characterized the subfields, of the field of meromorphic functions inν′, containing sufficient functions to verify a factorization property, similar to that of the classical Weierstrass theorem. It is also seen that the field generated by the Baker functions is not of this type, and the problem is solved of determining the divisors, inν′, of the holomorphic functions admiting Weierstrass factorizations with Baker functions as factors. As an application, a theorem is obtained characterizing the infinite products, of meromorphic functions in ν with bounded degree, which converge normally inν′. The first-named author is partially supported by PB91-0188.     

与例外函数和分担函数相关的亚纯函数的正规族

设F为区域D内的一族亚纯函数,对于每个f∈F,f的所有极点重数至少是2,a(z)和b(z)为两个在D内满足a(z)■b(z)的全纯函数.若对于每个f∈F,f(z)≠a(z)和f(z)≠b(z),则F在D内正规.这个结果改进了经典的Montel定则.此外,我们也讨论了亚纯函数族中每个函数与其导函数分担两个全纯函数的正规性.