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1.
The asymptotic behavior of quadratic Hermite–Padé polynomials
associated with the exponential function is studied for n→∞. These polynomials are defined by the relation (*) where O(·) denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in (*), and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are defined with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic Hermite–Padé approximants, which will be done in a follow-up paper. 相似文献
pn(z)+qn(z)ez+rn(z)e2z=O(z3n+2) as z→0,
2.
3.
Ryszard Mazur 《Mathematische Nachrichten》1980,99(1):355-361
Let Q(D) be a class of functions q, q(0) = 0, |q(z)| < 1 holomorphic in the Reinhardt domain D ? C n, a and b — arbitrary fixed numbers satisfying the condition — 1 ≤ b < a ≤ 1. ??(a, b; D) — the class of functions p such that p ? ??(a, b; D) iff for some q ? Q(D) and every z ? D. S*(a, b; D) — the class of functions f such that f ? S*(a, g; D) iff Sc(a, b; D) — the class of functions q such that q ? Sc(a, b; D) iff , where p ε ??(a, b; D) and K is an operator of the form for z=z1,z2,…zn. The author obtains sharp bounds on |p(z)|, f(z)| g(z)| as well as sharp coefficient inequalities for functions in ??(a, b; D), S*(a, b; D) and Sc(a, b; D). 相似文献
4.
James R. Holub 《Israel Journal of Mathematics》1985,52(3):231-238
LetW(D) denote the set of functionsf(z)=Σ
n=0
∞
A
n
Z
n
a
nzn for which Σn=0
∞|a
n
|<+∞. Given any finite set lcub;f
i
(z)rcub;
i=1
n
inW(D) the following are equivalent: (i) The generalized shift sequence lcub;f
1(z)z
kn
,f
2(z)z
kn+1, …,f
n
(z)z
(k+1)n−1rcub;
k=0
∞
is a basis forW(D) which is equivalent to the basis lcub;z
m
rcub;
m=0
∞
. (ii) The generalized shift sequence is complete inW(D), (iii) The function
has no zero in |z|≦1, wherew=e
2πiti
/n. 相似文献
5.
Let F(z) = Re(P(z)) + h.o.t be such that M = (F = 0) defines a germ of real analytic Levi-flat at 0 ∈ ℂ
n
, n ≥ 2, where P (z) is a homogeneous polynomial of degree k with an isolated singularity at 0 ∈ ℂ
n
and Milnor number μ. We prove that there exists a holomorphic change of coordinate ϕ such that ϕ(M) = (Re(h) = 0), where h(z) is a polynomial of degree μ + 1 and j
0
k
(h) = P. 相似文献
6.
A polynomial projector Π of degree d on is said to preserve homogeneous partial differential equations (HPDE) of degree k if for every and every homogeneous polynomial of degree k, q(z)=∑|α|=kaαzα, there holds the implication: q(D)f=0q(D)Π(f)=0. We prove that a polynomial projector Π preserves HPDE of degree if and only if there are analytic functionals with such that Π is represented in the following form with some , where uα(z)zα/α!. Moreover, we give an example of polynomial projectors preserving HPDE of degree k (k1) without preserving HPDE of smaller degree. We also give a characterization of Abel–Gontcharoff projectors as the only Birkhoff polynomial projectors that preserve all HPDE. 相似文献
7.
For two nonisomorphic orientations D and D′ of a graph G, the orientation distance do(D,D′) between D and D′ is the minimum number of arcs of D whose directions must be reversed to produce an orientation isomorphic to D′. The orientation distance graph 𝒟o(G) of G has the set 𝒪(G) of pairwise nonisomorphic orientations of G as its vertex set and two vertices D and D′ of 𝒟0(G) are adjacent if and only if do(D,D′) = 1. For a nonempty subset S of 𝒪(G), the orientation distance graph 𝒟0(S) of S is the induced subgraph 〈S〉 of 𝒟o(G). A graph H is an orientation distance graph if there exists a graph G and a set S⊆ 𝒪(G) such that 𝒟o(S) is isomorphic to H. In this case, H is said to be an orientation distance graph with respect to G. This paper deals primarily with orientation distance graphs with respect to paths. For every integer n ≥ 4, it is shown that 𝒟o(Pn) is Hamiltonian if and only if n is even. Also, the orientation distance graph of a path of odd order is bipartite. Furthermore, every tree is an orientation distance graph with respect to some path, as is every cycle, and for n ≥ 3 the clique number of 𝒟o(Pn) is 2 if n is odd and is 3 otherwise. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 230–241, 2001 相似文献
8.
Zhigang Peng 《Journal of Mathematical Analysis and Applications》2008,340(1):209-218
Let B denote the set of functions ?(z) that are analytic in the unit disk D and satisfy |?(z)|?1(|z|<1). Let P denote the set of functions p(z) that are analytic in D and satisfy p(0)=1 and Rep(z)>0(|z|<1). Let T denote the set of functions f(z) that are analytic in D, normalized by f(0)=0 and f′(0)=1 and satisfy that f(z) is real if and only if z is real (|z|<1). In this article we investigate the support points of the subclasses of B, P and T of functions with fixed coefficients. 相似文献
9.
F. M. Al-Oboudi 《Complex Analysis and Operator Theory》2011,5(3):647-658
Let A denote the class of analytic functions f, in the open unit disk E = {z : |z| < 1}, normalized by f(0) = f′(0) − 1 = 0. In this paper, we introduce and study the class STn,al,m(h){ST^{n,\alpha}_{\lambda,m}(h)} of functions f ? A{f\in A}, with
\fracDn,al fm(z)z 1 0{\frac{D^{n,\alpha}_\lambda f_m(z)}{z}\neq 0}, satisfying
\fracz(Dn,al f(z))¢Dn,al fm(z)\prec h(z), z ? E,\frac{z\left(D^{n,\alpha}_\lambda f(z)\right)'}{D^{n,\alpha}_\lambda f_m(z)}\prec h(z),\quad z\in E, 相似文献
10.
V. Mityushev 《Aequationes Mathematicae》1999,57(1):37-44
Summary. Local solutions of the functional equation¶¶zk f( z) = ?k=1nGk( z) f( skz ) +g( z) z{^\kappa} \phi \left( z\right) =\sum_{k=1}^nG_k\left( z\right) \phi \left( s_kz \right) +g\left( z\right) ¶with k > 0 \kappa > 0 and | sk| \gt 1 \left| s_k\right| \gt 1 are considered. We prove that the equation is solvable if and only if a certain system of k \kappa conditions on Gk (k = 1, 2, ... , n) and g is fulfilled. 相似文献
11.
Let k be a positive integer, let M be a positive number, let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity at least k, and let h be a holomorphic function in D, h ≢ 0. If, for every f ∈ F, f and f
(k) share 0, and |f(z)| ≥ M whenever f
(k)(z) = h(z), then F is normal in D. The condition that f and f
(k) share 0 cannot be weakened, and the condition that |f(z)| ≥ M whenever f
(k)(z) = h(z) cannot be replaced by the condition that |f(z)| ≥ 0 whenever f
(k)(z) = h(z). This improves some results due to Fang and Zalcman [2] etc. 相似文献
12.
A criterion of normality based on a single holomorphic function 总被引:1,自引:0,他引:1
Let F be a family of functions holomorphic on a domain D ⊂ ℂ Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k −1, such that h(z) has no common zeros with any f ∈ F. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ f′(z) = h(z); and (b) f′(z) = h(z) ⇒ |f
(k)(z)| ≤ c, where c is a constant. Then F is normal on D. 相似文献
13.
V. N. Margaryan H. G. Ghazaryan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2018,53(1):6-15
A linear differential operator P(x, D) = P(x1,... x n , D1,..., D n ) = ∑αγα(x)Dα with coefficients γα(x) defined in E n is called formally almost hypoelliptic in E n if all the derivatives DνξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in En. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions. 相似文献
14.
Horst Alzer 《Proceedings Mathematical Sciences》2010,120(2):131-137
Let n ≥ 1 be an integer and let P
n
be the class of polynomials P of degree at most n satisfying z
n
P(1/z) = P(z) for all z ∈ C. Moreover, let r be an integer with 1 ≤ r ≤ n. Then we have for all P ∈ P
n
:
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