共查询到20条相似文献,搜索用时 31 毫秒
1.
Bappaditya Bhowmik Saminathan Ponnusamy Karl-Joachim Wirths 《Monatshefte für Mathematik》2010,29(4):59-75
Let Co(α) denote the class of concave univalent functions in the unit disk
\mathbbD{\mathbb{D}}. Each function f ? Co(a){f\in Co(\alpha)} maps the unit disk
\mathbbD{\mathbb{D}} onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional (1-|z|2)( f¢¢(z)/f¢(z)), f ? Co(a){(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next
we obtain the set of variability of the functional (1-|z|2)(f¢¢(z)/f¢(z)), f ? Co(a){(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)} whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp
coefficient inequalities, we prove that functions in Co(α) belong to the H
p
space for p < 1/α. 相似文献
2.
Let
W í \Bbb C\Omega \subseteq {\Bbb C}
be a simply connected domain in
\Bbb C{\Bbb C}
, such that
{¥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}]
is connected. If g is holomorphic in Ω and every derivative of g extends continuously on
[`(W)]\bar{\Omega}
, then we write g ∈ A∞ (Ω). For g ∈ A∞ (Ω) and
z ? [`(W)]\zeta \in \bar{\Omega}
we denote
SN (g,z)(z) = ?Nl=0\fracg(l) (z)l ! (z-z)lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l
. We prove the existence of a function f ∈ A∞(Ω), such that the following hold:
相似文献
i) | There exists a strictly increasing sequence μn ∈ {0, 1, 2, …}, n = 1, 2, …, such that, for every pair of compact sets Γ, Δ ⊂ [`(W)]\bar{\Omega} and every l ∈ {0, 1, 2, …} we have supz ? G supw ? D \frac?l?wl Smn ( f,z) (w)-f(l)(w) ? 0, as n ? + ¥ and\sup_{\zeta \in \Gamma} \sup_{w \in \Delta} \frac{\partial^l}{\partial w^l} S_{\mu_ n} (\,f,\zeta) (w)-f^{(l)}(w) \rightarrow 0, \hskip 7.8pt {\rm as}\,n \rightarrow + \infty \quad {\rm and} |
ii) | For every compact set K ì \Bbb CK \subset {\Bbb C} with K?[`(W)] = ?K\cap \bar{\Omega} =\emptyset and Kc connected and every function h: K? \Bbb Ch: K\rightarrow {\Bbb C} continuous on K and holomorphic in K0, there exists a subsequence { m¢n }¥n=1\{ \mu^\prime _n \}^{\infty}_{n=1} of {mn }¥n=1\{\mu_n \}^{\infty}_{n=1} , such that, for every compact set L ì [`(W)]L \subset \bar{\Omega} we have supz ? L supz ? K Sm¢n ( f,z)(z)-h(z) ? 0, as n? + ¥.\sup_{\zeta \in L} \sup_{z\in K} S_{\mu^\prime _n} (\,f,\zeta )(z)-h(z) \rightarrow 0, \hskip 7.8pt {\rm as} \, n\rightarrow + \infty . |
3.
Here we give a quantitative Voronovskaya formula for a class of Mellin convolution operators of type
(Twf)(s) = ò0+¥ Kw(zs-1)f(z)\fracdzz.({T_w}f)(s) = {\int_0^{+\infty}} {K_w}(zs^{-1})f(z)\frac{dz}{z}. 相似文献
4.
Nadia J. Gal James E. Jamison Aristomenis G. Siskakis 《Complex Analysis and Operator Theory》2010,4(2):245-255
We are interested in the isometric equivalence problem for the Cesàro operator
C(f) (z) = \frac1z ò0zf(x) \frac11-xd x{C(f) (z) =\frac{1}{z} \int_{0}^{z}f(\xi) \frac{1}{1-\xi}d \xi} and an operator
Tg(f)(z)=\frac1zò0zf(x) g¢(x) d x{T_{g}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\xi) g^{\prime}(\xi) d \xi}, where g is an analytic function on the disc, on the Hardy and Bergman spaces. Then we generalize this to the isometric equivalence
problem of two operators Tg1{T_{g_{1}}} and Tg2{T_{g_{2}}} on the Hardy space and Bergman space. We show that the operators Tg1{T_{g_{1}}} and Tg2{T_{g_{2}}} satisfy Tg1U1=U2Tg2{T_{g_{1}}U_{1}=U_{2}T_{g_{2}}} on H
p
, 1 ≤ p < ∞, p ≠ 2 if and only if g2(z) = lg1(eiqz){g_{2}(z) =\lambda g_{1}(e^{i\theta}z) }, where λ is a modulus one constant and U
i
, i = 1, 2 are surjective isometries of the Hardy Space. This is analogous to the Campbell-Wright result on isometrically equivalence
of composition operators on the Hardy space. 相似文献
5.
Petros Galanopoulos Daniel Girela Rodrigo Hernández 《Journal of Geometric Analysis》2011,21(3):665-682
This paper is concerned mainly with the logarithmic Bloch space ℬlog which consists of those functions f which are analytic in the unit disc
\mathbbD{\mathbb{D}} and satisfy
sup|z| < 1(1-|z|)log\frac11-|z||f¢(z)| < ¥\sup_{\vert z\vert <1}(1-\vert z\vert )\log\frac{1}{1-\vert z\vert}\vert f^{\prime}(z)\vert <\infty , and the analytic Besov spaces B
p
, 1≤p<∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We
give explicit examples of:
|