The Fourier expansion of $$\varvec{\eta (z)\eta (2z)\eta (3z)/\eta (6z)}$$

We compute the Fourier coefficients of the weight one modular form \(\eta (z)\eta (2z)\eta (3z)/\eta (6z)\) in terms of the number of representations of an integer as a sum of two squares. We deduce a relation between this modular form and translates of the modular form \(\eta (z)^4/\eta (2z)^2\). In the last section we use our main result to give an elementary proof of an identity by Victor Kac.     

Sufficient Conditions for Starlikeness and Strong-starlikeness of a Given Order

We find sharp upper bounds for $ \vert \,f^{\prime \prime }(z)/f^{\prime }(z) \vert $ such that the function f be starlike or strongly-starlike of a given order, where f is holomorphic in the unit disc U , with $ f(0)=f^{\prime }(0)-1=0 $ . These bounds are obtained by using the differential subordination technique. As a useful ingredient we obtain the radius of convexity for the function $ F(z)= z /(\exp (z)-1) $     

On the Diophantine system $$f(z)\,{=}\,f(x)f(y)\,{=}\,f(u) f(v)$$

We show that the Diophantine system

$$\begin{aligned} f(z)=f(x)f(y)=f(u)f(v) \end{aligned}$$

has infinitely many nontrivial positive integer solutions for \(f(X)=X^2-1\), and infinitely many nontrivial rational solutions for \(f(X)=X^2+b\) with nonzero integer b.

     

微分方程f″+e^{az}f′+Q(z)f=F(z) 的复振荡

该文研究了线性微分方程f″+e^{az}f′+Q(z)f=F(z)的复振荡问题，其中Q(z)、F(z )( 0)是整函数，且σ(Q)=1,σ(F)<+∞,Q(z)=h(z)e^{bz},h(z)是多项式，b≠-1是复常数，那么上述线性微分方程的所有解f(z)满足~λ(f)=λ(f)=σ(f)=∞,~λ_2(f)=λ_2(f)=σ_2(f)=1.至多除去两个例外复数a及一个可能的有穷级例外解f_0(z)。     

The Region of Values of the System {f(z 1),..., f(z n )} in the Class of Typically Real Functions

The paper studies the region of values of the system {f(z₁), f(z₂),... , f(z_n)} in the class T of functions f(z) = z + a₂z² + ⋯ regular in the unit disk and satisfying the condition Im f(z) Im z > 0 for Im z ≠ 0. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 41–51.     

Lower bounds on the canonical height associated to the morphism $${\phi(z)= z^d+c}$$

A lower bound is obtained of the canonical height associated to the morphism evaluated at wandering points α. The lower bound is of the form Ch(c), for some constant C depending on the the number of primes of bad reduction for , and the degree of the number field . The author was supported by a PDF grant from NSERC.     

关于φ（z）f（z）f＇（z）的值分布

本文研究有穷圆内的值分布，其中ｆ（２）为非常数亚纯函数，为非零亚纯函数，并对平面上这样的函数，若，得到     

A note on log (f(z)/z) forf inS

We prove that forf a normalized schlicht function in the disk the boundary values of log |f(z)/z| satisfy a growth condition on subarcs of the unit circle .     

On the exponential Diophantine equation $$(3pm^2-1)^x+(p(p-3)m^2+1)^y=(pm)^z$$

Let m be a positive integer, and let p be a prime with \(p \equiv 1~(\mathrm{mod}~4).\) Then we show that the exponential Diophantine equation \((3pm^2-1)^x+(p(p-3)m^2+1)^y=(pm)^z\) has only the positive integer solution \((x, y, z)=(1, 1, 2)\) under some conditions. As a corollary, we derive that the exponential Diophantine equation \((15m^2-1)^x+(10m^2+1)^y=(5m)^z\) has only the positive integer solution \((x, y, z)=(1, 1, 2).\) The proof is based on elementary methods and Baker’s method.     

On Univalence of the Power Deformation z（f（z）/z）c

The authors mainly concern the set U _f of c ?? ? such that the power deformation $ z(\frac{{f(z)}} {z})^c $ is univalent in the unit disk |z| < 1 for a given analytic univalent function f(z) = z + a ₂ z ² + ?? in the unit disk. It is shown that U _f is a compact, polynomially convex subset of the complex plane ? unless f is the identity function. In particular, the interior of U _f is simply connected. This fact enables us to apply various versions of the ??-lemma for the holomorphic family $ z(\frac{{f(z)}} {z})^c $ of injections parametrized over the interior of U _f . The necessary or sufficient conditions for U _f to contain 0 or 1 as an interior point are also given.     

Zeros of Differential Polynomial f（z）f（k）（z） -- a and Its Normality

Suppose that function f(z) is transcendental and meromorphic in the plane. The aim of this work is to investigate the conditions in which differential monomials f(z)f(k)(z) takes any non-zero finite complex number infinitely times and to consider the normality relation to differential monomials f(z)f(k)(z).     

On the continued fraction \frac{1 |}{| z } +\frac{1 |}{| 1 } + \frac{2 |}{| z } +\frac{3 |}{| 1 } + \frac{4 |}{| z} + \cdots

The article on hand deals with the continued fraction $$\frac{1 |}{| z } +\frac{1 |}{| 1 } + \frac{2 |}{| z } +\frac{3 |}{| 1 } + \frac{4 |}{| z} + \cdots.$$ The famous Indian mathematician Srinivasa Ramanujan has given a pre-presentation by a power series, but he however concealed a proof. Subsequently a proof has been established, but a direct verification is intricate. Here we give a quick and direct approach with comparitively little effort.     

THE ZEROS AND ORDER OF MEROMORPHIC SOLUTIONS OF f^（k）＋Bf=H（z）

1.IntroductionandResultsConsidernon-homogeneouslineardifferentialequationsoftheform1.Lameprovedin[7]TheoremA.LetB(z),PO(z),PI(z)*06epolynomialssuchthatdegB=n21,degPO=p<(n k)/kandH=PI(z)epo('),then(a)IfdegPI相似文献   

The exponential Diophantine equation x2 + (3n 2 + 1) y = (4n 2 + 1) z

Let n be a positive integer. In this paper, using the results on the existence of primitive divisors of Lucas numbers and some properties of quadratic and exponential diophantine equations, we prove that if n ≡ 3 (mod 6), then the equation x ² + (3n ² + 1) ^y = (4n ² + 1) ^z has only the positive integer solutions (x, y, z) = (n, 1, 1) and (8n ³ + 3n, 1, 3).     

Behavior of the roots of the equationw′1(z)+бw1(z)=0

Some properties of the roots of the equationw′₁(z)+бw₁(z)=0 when the parameter σ is real are studied.     

Il sistema diofanteo N+1=x2, 3N+1=y2, 8N+1=z2

Sunto Nel 1969 A. Baker e H. Davenport hanno dimostrato che il sistema3x ² −2=y ² ,8x ² −7=z ² ha soltanto le soluzioni |x|=|y|=|z|=1; |x|=11, |y|=19, |z|=31. Una seconda dimostrazione di P. Kanagasabapathy — Tharmambikai Ponmudurai è apparsa nell'ottobre 1975. L'A. dà una terza dimostrazione che consente, con l'uso dei metodi classici dell'aritmetica, di controllare i ragionamenti in ogni punto. A Dario Graffi nel suo 70° compleanno Entrata in Redazione il 26 novembre 1975.     

Inequalities for entire functions of exponential type satisfying f(z) \equiv e^{i\gamma } e^{i\tau z} \overline {f(\bar z)}

Let f be an entire function of exponential type satisfying the condition $ f(z) \equiv e^{i\gamma } e^{i\tau z} \overline {f(\bar z)} $ for some real γ. Lower and upper estimates for ∫ _?∞ ^∞ |f′(x)| ^p dx in terms of ∫ _?∞ ^∞ |f(x)| ^p dx, for such a function f belonging to L ^p (R), have been known in the case where p ? [1, ∞) and γ = 0. In this paper, these estimates are shown to hold for any p ? (0, ∞) and any real γ.     

微分多项式f kQ [ f ]+P [ f ] 的零点分布——纪念李国平院士吴新谋教授诞辰100周年

该文讨论了亚纯函数及其微分多项式f ^kQ[ f ]+P[ f ]例外集理论的产生, 发展和最新进展,并且为下一步研究提出了建议.