泛函方程组的解析解

§ 1　IntroductionThe Feigenbaum functional equation plays an importantrole in the theory concerninguniversal properties of one-parameter families of maps of the interval that has the formf2 (λx) +λf(x) =0 ,0 <λ=-f(1 ) <1 ,f(0 ) =1 ,(1 .1 )where f is a map ofthe interval[-1 ,1 ] into itself.Lanford[1 ] exhibited a computer-assist-ed proof for the existence of an even analytic solution to Eq.(1 .1 ) .It was shown in[2 ]that Eq.(1 .1 ) does not have an entire solution.Si[3] discussed the it…     

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

On a linear combination of some expressions in the theory of the univalent functions

LetH(α) denote the class of regular functionsf(z) normalized so thatf(0)=0 andf′(0)=1 and satisfying in the unit discE the condition $$\operatorname{Re} \left\{ {(1 - \alpha )f'(z) + \alpha (1 + zf''(z)/f'(z))} \right\} > 0$$ for fixed α. It is known thatH(0) is a particular class NW of close-to-convex univalent functions. The authors show the following results:Theorem 1. Letf(z)∈H(α). Thenf(z)∈NW if α≤0 andz∈E.Theorem 2. Letf(z)∈NW. Thenf(z)∈H(α) in |z|=r<r _α where i) \(r_\alpha = (1 + \sqrt {2\alpha } )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}\) , α≥0 and ii) \(r_\alpha = \sqrt {\frac{{1 - \alpha - \sqrt {\alpha (\alpha - 1)} }}{{1 - \alpha }}}\) , α<0. All results are sharp.Theorem 3. Iff(z)=z+a ₂ z ²+a ₃ z ³+... is inH(α) and if μ is an arbitrary complex number, then $$\left| {1 + \alpha } \right|\left| {a_3 - \mu a_2^2 } \right| \leqslant ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})\max \left[ {1,\left| {1 + 2\alpha - {3 \mathord{\left/ {\vphantom {3 {2\mu }}} \right. \kern-\nulldelimiterspace} {2\mu }}(1 + \alpha )} \right|} \right].$$ .     

Bloch-Bergman Pullbacks with Logarithmic Weights

We characterize the composition operators mapping Blochs boundedly into the weighted Bergman spaces of logarithmic weight. For 0 < p < ∞, 1 < α < ∞, let A^{p, log α} denote the space of holomorphic functions F in the unit disc D for which

On α-close-to-convex functions

LetP(α) denote the class of functionsf analytic in the unit discE, withf(0)=0,f(z)≠0 (0<|z|<1) andf′(z)≠0 inE, satisfying the condition $$\int\limits_{\theta _1 }^{\theta _2 } {\operatorname{Re} } \left\{ {a\left( {1 + \frac{{zf''\left( z \right)}}{{f'\left( z \right)}}} \right) + \left( {1 - a} \right)\frac{{zf'\left( z \right)}}{{f\left( z \right)}}} \right\}d\theta > - \pi $$ whenever 0≤θ₁≤θ₂≤θ₁+2π,z=re^iθ r<1 and α is any positive real number. The functions inP(α) unify the classes of close-to-starlike (α=0) and close-to-convex (α=1) functions. We callf∈P(α) and α-close-to-convex function. In this paper we investigate certain properties of the classP(α).     

Estimates for sums of moduli of blocks in trigonometric Fourier series

We consider the following two problems. Problem 1: what conditions on a sequence of finite subsets A _k ? ? and a sequence of functions λ _k : A _k → ? provide the existence of a number C such that any function f ∈ L ₁ satisfies the inequality ‖U _A,Λ(f)‖ _p ≤ C‖f‖₁ and what is the exact constant in this inequality? Here, \(U_{\mathcal{A},\Lambda } \left( f \right)\left( x \right) = \sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {\lambda _k \left( m \right)c_m \left( f \right)e^{imx} } } \right|}\) and c _m (f) are Fourier coefficients of the function f ∈ L ₁. Problem 2: what conditions on a sequence of finite subsets A _k ? ? guarantee that the function \(\sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {c_m \left( h \right)e^{imx} } } \right|}\) belongs to L _p for every function h of bounded variation?     

The value regions of initial coefficients in a certain class of meromorphic functions

Let M_k,λ(0≤λ≤1, k≥2) be the class of functions f(z)=1/z+a_o+a₁z+... that are regular and locally univalent for 0<⩛z⩛<1 and satisfy the condition where J_λ(z)=λ(1+zf″(z)/f'(z))+(1-λ)zf'(z)/f(z). In the class M_k,λ we consider sorne coefficient problems and problems concerning distortion theorems. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 91–96. Translated by N. Yu. Netsvetaev.     

Zeros of analytic functions and norms of inverse matrices

Letk _n be the smallest constant such that for anyn-dimensional normed spaceX and any invertible linear operatorT ∈L(X) we have $|\det (T)| \cdot ||T^{ - 1} || \le k_n |||T|^{n - 1} $ . LetA ₊ be the Banach space of all analytic functionsf(z)=Σ _k≥0 a _kz^k on the unit diskD with absolutely convergent Taylor series, and let ‖f‖A ₊=σ _k≥0 |ακ|; define ? _n on $\overline D ^n $ by $ \begin{array}{l} \varphi _n \left( {\lambda _1 ,...,\lambda _n } \right) \\ = inf\left\{ {\left\| f \right\|_{A + } - \left| {f\left( 0 \right)} \right|; f\left( z \right) = g\left( z \right)\prod\limits_{i = 1}^n {\left( {\lambda _1 - z} \right), } g \in A_ + , g\left( 0 \right) = 1 } \right\} \\ \end{array} $ . We show thatk _n=sup {? _n(λ₁,…, λ _n ); (λ₁,…, λ _n )∈ $\overline D ^n $ }. Moreover, ifS is the left shift operator on the space ?∞:S(x ₀,x ₁, …,x _p, …)=(x ₁,…,x _p,…) and if J_n(S) denotes the set of allS-invariantn-dimensional subspaces of ?∞ on whichS is invertible, we have $k_n = \sup \{ |\det (S|_E )|||(S|_E )^{ - 1} ||E \in J_n (S)\} $ . J. J. Schäffer (1970) proved thatk _n≤√en and conjectured thatk _n=2, forn≥2. In factk ₃>2 and using the preceding results, we show that, up to a logarithmic factor,k _n is of the order of √n whenn→+∞.     

On higher heine-stieltjes polynomials

Take a linear ordinary differential operator $\mathfrak{d}\left( z \right) = \sum\nolimits_{i = 1}^k {Q_i \left( z \right)\frac{{d^i }} {{dz^i }}}$\mathfrak{d}\left( z \right) = \sum\nolimits_{i = 1}^k {Q_i \left( z \right)\frac{{d^i }} {{dz^i }}} with polynomial coefficients and set r = max _i=1,…,k(deg Q _i (z) − i). If d(z) satisfies the conditions: (i) r ≥ 0 and (ii) deg Q _k (z) = k + r, we call it a non-degenerate higher Lamé operator. Following the classical examples of E. Heine and T. Stieltjes we initiated in [13] the study of the following multiparameter spectral problem: for each positive integer n find polynomials V (z) of degree at most r such that the equation

On the dependence of the growth rate on the length of the defining relator

Let { } be a sequence of finitely presented groups with generating setA={a1, …, am}, and letRk be the symmetrized set of words over the alphabetA∪A−1 obtained from the defining words and their inverses by all cyclic shifts. We shall assume that the words inRk are cyclically irreducible, and their lengths tend to ∞ ask increases. In the paper, it is proved that ifRk satisfies the small cancellation conditionC'(1/6) and the number of relators increases not very rapidly with increasingk, then the growth rate ψ(Gk) tends to 2m−1 ask→∞. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 611–617, April, 1999.     

Precise estimate of total deficiency of meromorphic derivatives

Let f(z) be a transcendental meromorphic function in the finite plane andk be a positive integer. Then we have . Moreover, if the order of f(z) is finite, then we also have , where δ(a, f^(k)) denotes the deficiency of the valuea with respect to f^(k) and θ(∞,f) is the ramification index of ∞ with respect tof.     

Systems with nonextendable convergence of quasipolynomials

The system , where Λ={λ _n } is the set of zeros (of multiplicitiesm _n ) of the Fourier transform

Multiple Wiener-Ito integrals possessing a continuous extension

LetF(W) be a Wiener functional defined byF(W)=I _n(f) whereI _n(f) denotes the multiple Wiener-Ito integral of ordern of the symmetricL ²([0, 1] ⁿ ) kernelf. We show that a necessary and sufficient condition for the existence of a continuous extension ofF, i.e. the existence of a function ø(·) from the continuous functions on [0, 1] which are zero at zero to which is continuous in the supremum norms and for which ø(W)=F(W) a.s, is that there exists a multimeasure (dt ₁,...,dt _n ) on [0, 1] ⁿ such thatf(t ₁, ...,t _n ) = ((t ₁, 1]), ..., (t _n , 1]) a.e. Lebesgue on [0, 1] ⁿ . Recall that a multimeasure (A ₁,...,A _n ) is for every fixedi and every fixedA _i,...,A_i-1, A_i+1,...,A_n a signed measure inA _i and there exists multimeasures which are not measures. It is, furthermore, shown that iff(t ₁,t ₂, ...,t _n ) = ((t ₁, 1], ..., (t _n , 1]) then all the tracesf ^(k), off exist, eachf(^k) induces ann–2k multimeasure denoted by ^(k), the following relation holds

A zero-one law for random subgroups of some totally disconnected groups

Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Γ = Aut(F _k ) on the set of marked, k-generated, dense subgroups $ {D_{k,A}}: = \left\{ {\eta \in {\text{Hom}}\left( {{F_k},A} \right)\left| {\overline {\left\langle {\phi \left( {{F_k}} \right)} \right\rangle } = A} \right.} \right\} Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Γ = Aut(F _k ) on the set of marked, k-generated, dense subgroups D_k,A: = { h ? \textHom( F_k,A )| [`( á f( F_k ) ñ )] = A } {D_{k,A}}: = \left\{ {\eta \in {\text{Hom}}\left( {{F_k},A} \right)\left| {\overline {\left\langle {\phi \left( {{F_k}} \right)} \right\rangle } = A} \right.} \right\} . We prove the ergodicity of this action for the following two families of simple, totally disconnected, locally compact groups:

•	A = PSL2(K) where K is a non-Archimedean local field (of characteristic ≠ 2);
•	A = Aut0(T q+1)—the group of orientation-preserving automorphisms of a q + 1 regular tree, for q \geqslant 2.q \geqslant 2.

In contrast, a recent result of Minsky’s shows that the same action fails to be ergodic for A = PSL₂(C) and, when k is even, also for A = PSL₂(R). Therefore, if k \geqslant 4 k \geqslant 4 is even and K is a local field (with char(K) ≠ 2), the action of Aut(F _k ) on D_{k,\textPS\textL2(K)} {D_{k,{\text{PS}}{{\text{L}}_2}(K)}} is ergodic if and only if K is non-Archimedean. Ergodicity implies that every “measurable property” either holds or fails to hold for almost every k-generated dense subgroup of A.     

Some inequalities about mixed volumes

We prove inequalities about the quermassintegralsV _k (K) of a convex bodyK in ℝ ⁿ (here,V _k (K) is the mixed volumeV((K, k), (B _n ,n − k)) whereB _n is the Euclidean unit ball). (i) The inequality

Some remarks on the identity between a variational integral and its relaxed functional

Summary Letf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] be measurable inx and convex inz. It is proved, by an example, that even iff verifies a condition as|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) with 1<p<q,a∈L _loc ^s (R ⁿ),s>1, the functional that isL ¹(Ω)-lower semicontinuous onW ^1,1(Ω), does not agree onW ^1,1(Ω) with its relaxed functional in the topologyL ¹(Ω) given by inf

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Riassunto Siaf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] misurabile inx e convessa inz. Si mostra con un esempio che anche sef verifica una condizione del tipo|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) con 1<p<q,a∈L _loc ^s (R ⁿ),s>1, il funzionale , che èL ¹(Ω)-semicontinuo inferiormente suW ^1,1(Ω), non coincide suW ^1,1(Ω) con il suo funzionale rilassato nella topologiaL ¹(Ω) definito da inf

     

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

全纯函数的加权积分

§ 1　 IntroductionLet D be the unit disc in the complex plane C,dm be the Lebesgue area measure onD.We denote H (D) the setof all holomorphic functions on D.For 0 相似文献   

More quantitative results on walsh equiconvergence: I. Lagrange Case

Letf∈A _ρ (ρ>1), whereA _ρ denotes the class of functions analytic in ¦z¦ <ρ but not in ¦z¦≤ρ. For any positive integerl, the quantity Δ _l,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 Δ _l,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.