共查询到20条相似文献,搜索用时 31 毫秒
1.
J. Gwoździewicz 《Commentarii Mathematici Helvetici》1999,74(3):364-375
Let f be a real analytic function defined in a neighborhood of
0 ? \Bbb Rn 0 \in {\Bbb R}^n such that f-1(0)={0} f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: |f(x)| 3 c|x|a |f(x)|\geq c|x|^{\alpha} , |grad f(x)| 3 c|x|b |{\rm grad}\,f(x)|\geq c|x|^{\beta} , |grad f(x)| 3 c|f(x)|q |{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c > 0 c > 0 . We prove that a = b+1 \alpha=\beta+1, q = b/a\theta=\beta/\alpha . Moreover b = N+a/b \beta=N+a/b where $ 0 h a < b h N^{n-1} $ 0 h a < b h N^{n-1} . If f is a polynomial then |f(x)| 3 c|x|(degf-1)n+1 |f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero. 相似文献
2.
The composition operators on weighted Bloch space 总被引:9,自引:0,他引:9
R. Yoneda 《Archiv der Mathematik》2002,78(4):310-317
We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B log = { f ? H(D): supz ? D (1-| z|2) ( log\frac21-| z|2 )| f¢(z)| B_{ \log }= \{ f \in H(D): \sup_{z \in D } (1-\left| z\right|^2) \left( \log \frac{2}{1-\left| z\right|^2} \right)\left| f'(z)\right| < +¥} +\infty \} , where H(D) be the class of all analytic functions on D. 相似文献
3.
А. А. Долгобородов 《Analysis Mathematica》2013,39(2):123-134
We consider functions f(z) (f(0) = 1, |z| < 1) from spaces endowed with mixed norm; in particular, the Bergmann-Dzharbashian space, with zeroes z k (f) (|z 1(f)| ≤ |z 2(f)| ≤ ...). We construct examples of such functions f that the products $$\pi n(f) = \left( {\left| {z_1 \left( f \right)} \right|} \right) \cdots \left( {\left| {z_n \left( f \right)} \right|} \right)^{ - 1}$$ have a well defined order of magnitude as n→∞ with respect to certain subsequences of n. We also establish necessary and sufficient conditions for the existence of such subsequences. These results are applied to study a number of spaces under consideration. 相似文献
4.
J. Brzdek 《Aequationes Mathematicae》2000,59(3):248-254
Summary. Let \Bbb K {\Bbb K} be either the field of reals or the field of complex numbers, X be an F-space (i.e. a Fréchet space) over \Bbb K {\Bbb K} n be a positive integer, and f : X ? \Bbb K f : X \to {\Bbb K} be a solution of the functional equation¶¶f(x + f(x)n y) = f(x) f(y) f(x + f(x)^n y) = f(x) f(y) .¶We prove that, if there is a real positive a such that the set { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} contains a subset of second category and with the Baire property, then f is continuous or { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} for every x ? X x \in X . As a consequence of this we obtain the following fact: Every Baire measurable solution f : X ? \Bbb K f : X \to {\Bbb K} of the equation is continuous or equal zero almost everywhere (i.e., there is a first category set A ì X A \subset X with f(X \A) = { 0 }) f(X \backslash A) = \{ 0 \}) . 相似文献
5.
For any fixed finite interval [a, b] on the real line, an arbitrary natural numberr and σ>0, we describe the extremal function to the problem $\left\| {f^{(k)} } \right\|L_p \left[ {a,b} \right]^{ \to \sup } \left( {1 \leqslant k \leqslant r - 1, 1 \leqslant p< \infty } \right)$ over all functionsf ∈W ∞ r such that |f (r)(x)| ≤σ, |f(x)|≤1 on (?∞, ∞). Similarly, we solve the problem, raised by Paul Erdös, of characterizing the trigonometric polynomial of fixed uniform norm whose graph has maximal arc length over [a, b]. 相似文献
6.
Binyamin Schwarz 《Israel Journal of Mathematics》1965,3(1):29-38
Letx
1,...,x
m be points in the solid unit sphere ofE
n and letx belong to the convex hull ofx
1,...,x
m. Then
. This implies that all such products are bounded by (2/m)
m
(m −1)
m−1. Bounds are also given for other normed linear spaces. As an application a bound is obtained for |p(z
0)| where
andp′(z
0)=0. 相似文献
7.
Estimates are given for the measure of a section of an arbitrary straight line of the set $$E_\delta = \left\{ {z:\left| {P' {{\left( z \right)} \mathord{\left/ {\vphantom {{\left( z \right)} {\left( {nP \left( z \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {nP \left( z \right)} \right)}} \leqslant \delta } \right|} \right\} \left( {\delta > 0} \right)$$ where P (z) is a polynomial of degree n. THEOREM. Suppose P (x) = (x ? x1) ... (x ? xn) is a polynomial with real zeros. Then, for any δ > 0, on any intervala ?x ?b, containing all of the xk (k=1, 2, ..., n), outside an exceptional set Eδ?[a,b] such that $$mes E_\delta \leqslant \left( {\sqrt {1 + \delta ^2 \left( {b - a} \right)^2 } - 1} \right)/\delta $$ , we have the inequality $$\left| {P' {{\left( x \right)} \mathord{\left/ {\vphantom {{\left( x \right)} {\left( {nP \left( x \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {nP \left( x \right)} \right)}}} \right| > \delta $$ . A similar estimate is given for polynomials whose roots lie either in Imz ? 0 or in Imz ? 0. 相似文献
8.
Let f be a transcendental entire function of order less than 1/2. Denote the maximum and minimum modulus of f by M(r, f) = max{|f(z)|: |z| = r} and m(r, f) = min{|f(z)|: |z| = r}. We obtain a minimum modulus condition satisfied by many f of order zero that implies all Fatou components are bounded. A special case of our result is that if
$
\log \log M(r,f) = O(\log r/(\log \log r)^K )
$
\log \log M(r,f) = O(\log r/(\log \log r)^K )
相似文献
9.
Consider j = f +[`(g)]\varphi = f + \overline {g}, where f and g are polynomials, and let TjT_{\varphi} be the Toeplitz operators with the symbol j\varphi. It is known that if TjT_{\varphi} is hyponormal then |f¢(z)|2 3 |g¢(z)|2|f'(z)|^{2} \geq |g'(z)|^{2} on the unit circle in the complex plane. In this paper, we show that it is also a necessary and sufficient condition under
certain assumptions. Furthermore, we present some necessary conditions for the hyponormality of TjT_{\varphi} on the weighted Bergman space, which generalize the results of I. S. Hwang and J. Lee. 相似文献
10.
We study functions f(z) holomorphic in having the property f(z) ≠ 0 for 0 < Im z < 1 and we obtain lower bounds for |f(z)| for 0 < Im z < 1. In our analysis we deal with scalar functions f(z) as well as with operator valued holomorphic functions I + A(z) assuming that A(z) is a trace class operator for and I + A(z) is invertible for 0 < Im z < 1 and is unitary for .
A. Borichev was partially supported by the ANR project DYNOP. 相似文献
11.
O. D. Jones 《Aequationes Mathematicae》2002,63(3):251-265
Summary. We give conditions for the multivariate Böttcher equation b(f (x)) = b(x)l \beta(f (x)) = \beta(x)^{\lambda} to have a solution, in the case where f : \mathbbRd ? \mathbbRd f : \mathbb{R}^d \rightarrow \mathbb{R}^d is a polynomial with non-negative coefficients. The solution is constructed from the limit of the functional iterates -l-n logfn(x) -\lambda^{-n} \log f^{n}(x) . 相似文献
12.
A. L. Grigoryan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(1):8-13
It is well-known that if f(x) belongs to the spaceW
r
of 2π-periodic functions, the r-th derivatives of which are continuous and satisfy the inequality |f
(r)(x)| ≤ 1, then the modulus of the remainder term in the Fourier series of f(x) has independent of x finite upper bound. Besides, for any natural r A. N. Kolmogorov has proved the asymptotic formula
|