The following theorem is proved: there is a functionf(z) analytic in |z|<1 and having the natural boundary |z|=1 such that for an infinite sequence of rational functions of degreen, rn(z)=Pn(z)/qn(z), the inequality1 $$\left| {f(z) - r_n (z)} \right|< \varepsilon _n $$ holds in the closed unit circle |z|≦1. Here?1,?2,...,?n is any sequence of positive numbers, tending to zero asn approaches infinity. This theorem is a refinement of a theorem of Aharonov and Walsh, who showed the existence of anf(z) satisfying (*) in |z|≦1 (with an infinite sequence {rn(z)}) but having the natural boundary |z|=3. 相似文献
The functionf(z), analytic in the unit disc, is inAp if \(\int {\int {_{\left| z \right|< 1} \left| {f(z)} \right|^p dxdy< \infty } } \) . A necessary condition on the moduli of the zeros ofAp functions is shown to be best possible. The functionf(z) belongs toBp if \(\int {\int {_{\left| z \right|< 1} \log ^ + \left| {f(z)} \right|)^p } } \) . Let {zn } be the zero set of aBp function. A necessary condition on |zn | is obtained, which, in particular, implies that Σ(1?|zn |)1+(1/p)+g<∞ for all ε>0 (p≧1). A condition on the Taylor coefficients off is obtained, which is sufficient for inclusion off inBp. This in turn shows that the necessary condition on |zn | is essentially the best possible. Another consequence is that, forq≧1,p<q, there exists aBp zero set which is not aBq zero set. 相似文献
LetF(b, M) (b ≠ 0 complex,M>1/2) denote the class of functionsf(z) =z + Σn=2∞anzn analytic in U={z:|z|<1} which satisfy for fixedM, f(z)/z ≠ 0 inU and \(\left| {\frac{{b - 1 + \left[ {zf'{{\left( z \right)} \mathord{\left/ {\vphantom {{\left( z \right)} {f\left( z \right)}}} \right. \kern-0em} {f\left( z \right)}}} \right]}}{b} - M} \right|< M, z \in U\) . In this note we obtain various representations for functions inF(b, M). We maximize |a3=μa22| over the classF(b, M). Also sharp coefficient bounds are established for functions inF(b, M). We also obtain the sharp radius of starlikeness of the classF(b, M). 相似文献
Letf be an entire function that is real on the real axis and not a polynomial. Let 1<α<4/3. A condition $$\log \left| {f(z)} \right| = O((\log \left| z \right|)^a )$$ is known to guarantee uniform distribution of the sequencef(n) (n=1,2...). However, we show here by an example that no quantitative version of the uniform distribution can be deduced from (1). 相似文献
Letf∈Aρ(ρ>1), whereAρ denotes the class of functions analytic in ¦z¦ <ρ but not in ¦z¦≤ρ. For any positive integerl, the quantity Δl,n?1(f; z) (see (2.3)) has been studied extensively. Recently, V. Totik has obtained some quantitative estimates for \(\overline {\lim _{n \to \infty } } \max _{\left| z \right| = R} \left| {\Delta _{l,n - 1}^ - \left( {f;z} \right)} \right|^{1/n} \) . Here we investigate the order of pointwise convergence (or divergence) of Δl,n?1(f; z), i.e., we study \(B_1 \left( {f;z} \right) = \overline {\lim _{n \to \infty } } \left| {\Delta _{l,n - 1} \left( {f;z} \right)} \right|^{1/n} \) . We also study some problems arising from the results of Totik. 相似文献
Sunto Si considerano le serie di potenze f(z) = ∑anzn convergenti (almeno) per | z |<1 e si studia l'andamento della funzione maggiorante ∑ | an | ru, ponendolo in relazione con quello della media quadratica del modulo { ∑ | an |2r2n}1/2 (che in ogni caso è minore del massimo modulo
. Si stabiliscono le ? migliori costanti ? soltanto per qualcuno dei quesiti. Alla fine si esamina il caso in cui {∑ | an |2r2n} è convergente con ipotesi che tengono conto delle funzioni ? lentamente oscillanti ?.
A Mauro Picone nel suo 70mo compleanno. 相似文献
设 H 是一个Hilbert空间. B(H) 表示所有H 到 H 的有界线性算子构成的Banach空间. 设 T= {f(z): f(z)=zI-∑∞n=2 znAn 在单位圆盘|z|<1上解析, 其中系数An是 H 到 H 的紧正Hermitian算子, I 表示 H 上的恒等算子, ∑∞n=2 n(An x, x) ≤1 对所有x ∈H, ∣|x∣∣=1 成立. 该文研究了函数族 T 的极值点. 相似文献
Letf be an entire function (in Cn) of exponential type for whichf(x)=0(?(x)) on the real subspace \(\mathbb{R}^w (\phi \geqslant 1,{\mathbf{ }}\mathop {\lim }\limits_{\left| x \right| \to \infty } \phi (x) = \infty )\) and ?δ>0?Cδ>0 $$\left| {f(z)} \right| \leqslant C_\delta \exp \left\{ {h_s (y) + S\left| z \right|} \right\},z = x + iy$$ where h, (x)=sup〈3, x〉, S being a convex set in ?n. Then for any ?, ?>0, the functionf can be approximated with any degree of accuracy in the form p→ \(\mathop {\sup }\limits_{x \in \mathbb{R}^w } \frac{{\left| {P(x)} \right|}}{{\varphi (x)}}\) by linear combinations of functions x→expi〈λx〉 with frequenciesX belonging to an ?-neighborhood of the set S. 相似文献
LetM (α) denote the class of α-convex functions, α real, that is the class of analytic functions? (z) =z + Σn=2/∞anzn in the unit discD = {z: |z | < 1} which satisfies inD the condition ?′ (z) ?(z)/z ≠ 0 and $$\operatorname{Re} \left\{ {(1 - a) \frac{{z f'(z)}}{{f (z)}} + a \left( {1 + \frac{{z f''(z)}}{{f' (z)}}} \right)} \right\} > 0. Let W (a) $$ denote the class of meromorphic α-convex functions. α real, that is the class of analytic functions ? (z) =z?1 + Σn=0/∞bnzn inD* = {z: 0 < |z | < 1} which satisfies inD* the conditionsz?′(z)/?(z) ≠ 0 and $$\operatorname{Re} \left\{ {(1 - a) \frac{{z\phi ' (z)}}{{\phi (z)}} + a \left( {1 + \frac{{z\phi ''(z)}}{{\phi ' (z)}}} \right)} \right\}< 0. $$ In this paper we obtain the relation betweenM (a) and W(α). The radius of α-convexity for certain classes of starlike functions is also obtained. 相似文献
We study the subclass Wσ(A) of the class of entire transcendental functions f(z)of exponential type with index not greater than σ satisfying the condition $$\int_{ - \infty }^\infty {\left| {f(x)} \right|^2 dx \leqslant A^2 .}$$ We find the set of values of the quantities f(z), f′(z), etc. when z is fixed and f runs through the subclass Wσ(A). We study extremal values of functionals of the type Φ(f(z), f ′(z)). In particular, we obtain upper bounds on the quantities ¦f(z +β/2) ± f(z?β/2) ¦ and ¦af '(z) + bof(z)¦. 相似文献
We consider functions defined by regrouped power series \(f(z) = \sum\nolimits_{n = 0}^\infty z ^{\lambda _{n_{P_{k_n } } } } (z)\) in the disk |z|<1 and also in some domain D outside of this disk. We obtain conditions under whichf(z) is analytically continuable outside of the disk |z|<1, the analytic continuation being effected with the help of the given series. We also consider the analytic continuability of functionsf(z, w). 相似文献
If $P(z) = \sum\limits_{\nu = 0}^n {c_\nu z^\nu } $ is a polynomial of degree n, then for |β| ≤ 1, it was proved in [4] that $\left| {zP'(z) + n\frac{\beta } {2}P(z)} \right| \leqslant n\left| {1 + \frac{\beta } {2}} \right|\mathop {\max }\limits_{|z| = 1} |P(z)|,|z| = 1 $ In this paper, first we generalize the above result for the sth derivative of polynomials and next we improve the above inequality for polynomials with restricted zeros. 相似文献
Gehring and Pommerenke have shown that if the Schwarzian derivativeSf of an analytic functionf in the unit diskD satisfies |Sf(z)|≤, 2(1 - |z|2)–2 thenf(D) is a Jordan domain except whenf(D) is the image under a Möbius transformation of an infinite parallel strip. The condition |Sf(z)|≤ 2(1 - |z|2)–2 is the classical sufficient condition for univalence of Nehari. In this paper we show that the same type of phenomenon established by Gehring and Pommerenke holds for a wider class of univalence criteria of the form|Sf(z)|≤p(|z|) also introduced by Nehari. These include|Sf((z)|≤π2/2 and|Sf((z)|≤4(1-|z|2)–1. We also obtain results on Hölder continuity and quasiconformal extensions. 相似文献
In this paper, we use a simpler argument and solve the following more general function equation: $$\left| {f\left( {z + w} \right)} \right| + \left| {g\left( {z - w} \right)} \right| = \left| {h\left( {z + \bar w} \right)} \right| + \left| {k\left( {z - \bar w} \right)} \right|$$ where f, g, h, k are unknown entire functions and z, w are complex variables. 相似文献
LetK be a compact point set in the complex plane having positive logarithmic capacity and connected complement. For anyf continuous onK and analytic in the interior ofK we investigate the distribution of the extreme points for the error in best uniform approximation tof onK by polynomials. More precisely, if $$A_n (f): = \{ z \in K:|f(z) - p_n^* (f;z)| = \parallel f - p_n^* (f)\parallel _K \} ,$$ wherepn*(f) is the polynomial of degree≤n of best uniform approximation tof onK, we show that there is a subsequencenk with the property that the sequence of (nk+2)-point Fekete subsets of \(A_{n_k }\) has limiting distribution (ask→∞) equal to the equilibrium distribution forK. Analogues for weighted approximation are also given. 相似文献
in the space of analytic functions on the unit polydisk Un in the complex vector space ?n. We show that the operator is bounded in the mixed norm space
, with p, q ∈ [1, ∞) and α = (α1, …, αn), such that αj > ?1, for every j = 1, …, n, if and only if \(\sup _{z \in U^n } \prod\nolimits_{j = 1}^n {\left( {1 - \left| {z_j } \right|} \right)} \left| {g(z)} \right| < \infty \). Also, we prove that the operator is compact if and only if \(\lim _{z \to \partial U^n } \prod\nolimits_{j = 1}^n {\left( {1 - \left| {z_j } \right|} \right)} \left| {g(z)} \right| = 0\).