A counterexample to the strong subadditivity of extremal plurisubharmonic functions

A simple example is given which shows that one way have $$h_E (z^0 ) + h_F (z^0 ) > h_{E \cup F} (z^0 ) + h_{E \cap F} (z^0 )$$ for some pointz ⁰∈ω, where $$h_E (z) = \sup \{ u(z):u \in PSH (\Omega ),u \leqslant 0 on E,u \leqslant 1 in \Omega \} ,z \in \Omega ,$$ is the extremal function often studied in complex analysis.     

Problems similar to the additive divisor problem

For multiplicative functions ?(n), let the following conditions be satisfied: ?(n)≥0 ?(p ^r)≤A ^r,A>0, and for anyε>0 there exist constants $A_\varepsilon$ ,α>0 such that $f(n) \leqslant A_\varepsilon n^\varepsilon$ and Σ _p≤x ?(p) lnp≥αx. For such functions, the following relation is proved: $$\sum\limits_{n \leqslant x} {f(n)} \tau (n - 1) = C(f)\sum\limits_{n \leqslant x} {f(n)lnx(1 + 0(1))}$$ . Hereτ(n) is the number of divisors ofn andC(?) is a constant.     

Bell Numbers, Log-Concavity, and Log-Convexity

Let {b _k (n)} _n=0 ^∞ be the Bell numbers of order k. It is proved that the sequence {b _k (n)/n!} _n=0 ^∞ is log-concave and the sequence {b _k (n)} _n=0 ^∞ is log-convex, or equivalently, the following inequalities hold for all n?0, $$1 \leqslant \frac{{b_k (n + 2)b_k (n)}}{{b_k (n + 1)^2 }} \leqslant \frac{{n + 2}}{{n + 1}}$$ . Let {α(n)} _n=0 ^∞ be a sequence of positive numbers with α(0)=1. We show that if {α(n)} _n=0 ^∞ is log-convex, then α(n)α(m)?α(n+m), ?n,m?0. On the other hand, if {α(n)/n!} _n=0 ^∞ is log-concave, then $$\alpha (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)\alpha (n)\alpha (m),{\text{ }}\forall n,m \geqslant 0$$ . In particular, we have the following inequalities for the Bell numbers $$b_k (n)b_k (m) \leqslant b_k (n + m) \leqslant \left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right)b_k (n)b_k (m),{\text{ }}\forall n,m \geqslant 0$$ . Then we apply these results to characterization theorems for CKS-space in white noise distribution theory.     

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 $$\left| {1 + \alpha } \right|\left| {a_3 - \mu a_2^2 } \right| \leqslant ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})\max \left[ {1,\left| {1 + 2\alpha - {3 \mathord{\left/ {\vphantom {3 {2\mu }}} \right. \kern-\nulldelimiterspace} {2\mu }}(1 + \alpha )} \right|} \right].$$ .     

Über arithmetische Eigenschaften von Lückenreihen mit algebraischen Koeffizienten und algebraischem Argument

Let \(f(z) = \sum\limits_{h = 0}^\infty {f_h z^h } \) be a power series with positive radius of convergenceR _f ≤1,f _h algebraic and lacunary in the following sense: Let {r _n }, {s _n } be two infinite sequences of integers, satisfying $$0 = s_0 \leqslant r_1< s_1 \leqslant r_2< s_2 \leqslant r_3< s_3 \leqslant ..., \mathop {lim}\limits_{n \to \infty } (s_n /F(n)) = \infty $$ such that $$f_h = 0 if r_n< h< s_n ,f_{r_n } \ne 0,f_{s_n } \ne 0 for n = 1,2,3,...;$$ F(n) denotes a certain function ofn, dependent onr _n and \(f_0 ,f_1 ,f_2 , \ldots f_{r_n } \) . Using ideas from a note ofK. Mahler, among other results the following main theorem is proved: The function valuef(α) (with α algebraic, 0<|α|<R _f ) is algebraic if and only if there exists a positive integerN=N(α) such that $$P_n (\alpha ): = \sum\limits_{h = s_n }^{r_{n + 1} } {f_h \alpha ^h = 0 for all n \geqslant N.} $$      

On a multiplicative function on the set of shifted primes

It is proved that if ?(n) is a multiplicative function taking a valueζ on the set of primes such thatζ ³ = 1,ζ ≠ 1 and? ³(p ^r)=1 forr≥2, then there exists aθ ∈ (0, 1), for which $$|\sum\limits_{p \leqslant x} f (p + 1)| \leqslant \theta \pi (x)$$ , where $$\pi (x) = \sum\limits_{p \leqslant x} 1$$ .     

On the radius of α-convexity of certain classes of starlike functions

LetM (α) denote the class of α-convex functions, α real, that is the class of analytic functions? (z) =z + Σ _n=2/^∞ a _n z ⁿ 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 + Σ _n=0/^∞ b _n z ⁿ 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.     

Some properties of elliptic Riesz means at critical index on Hp(TH)

Suppose f∈H^p(Tⁿ), 0 _r ^δ , δ=n/p?(n+1)/2. In this paper we eastablish the following inequality $$\mathop {\sup }\limits_{R > 1} \left\{ {\frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta } \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqslant C_{R,p} \left\| f \right\|_{H^p (T^R )} $$ It implies that $$\mathop {\lim }\limits_{R \to \infty } \frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta - f} \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} = 0$$ Moreover we obtain the same conclusion when p=1 and n=1.     

The variety of solutions of the singular generalized Cauchy-Riemann System

We prove that the equation $$2\bar z\partial _{\bar z} \bar w = 0_1 z \in G,$$ in whichB(z) ∈C ^∞(G),B ₀(z)=O(|z})α),α>0,z → 0, and $$b(\varphi ) = \sum\limits_{k = - m_o }^m {b_k e^{ik\varphi } } $$ does not have nontrivial solutions in the classC ^∞(G).     

Bemerkungen zu einem Satz von Fejér

Последовательность {ita_k} _(n) ^k =1/∞ вещественных ч исел называется дважды мо нотонной, еслиa _k -2a _k+1 +a _k+2 ≧0 дляk≧1. В работе доказываютс я следующие утвержде ния, являющиеся обобщени ем двух теорем Фейера:

1. Если {ita_k — дважды моно тонная последовател ьность, то для ¦z¦<1 $$\operatorname{Re} \sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } /\sum\limits_{\kappa = 1}^n {a_\kappa z^\kappa } > 1/2$$ дляи≧ 1.
2. Если О≦β<1 и последова тельность (k+1-2β)ak} дважд ы монотонна, то для ¦z¦<1 $$\operatorname{Re} \sum\limits_{\kappa = 1}^\infty {ka_\kappa z^\kappa } /\sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } > \beta $$ , то есть $$\sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } \varepsilon S_\beta ^\kappa $$ . При помощи 2) получены о бобщения и уточнения теорем из работы [1] о линейных комбинациях некотор ых однолистных функц ий.

     

Limit ultraspherical series and their approximation properties

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x ₂)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S _n ^?1 (f) = S _n ^?1 (f, x) is the projection onto the subspace of algebraic polynomials p _n = p _n (x) of degree at most n, i.e., S _n (p _n ) = p _n ; in addition, S _n ^?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S _n ^?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ _n (x) of the partial sums S _n ^?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

Ranges of certain functionals in classes of regular functions

LetP _κ,n (λ,β) be the class of functions \(g(z) = 1 + \sum\nolimits_{v = n}^\infty {c_\gamma z^v }\) , regular in ¦z¦<1 and satisfying the condition $$\int_0^{2\pi } {\left| {\operatorname{Re} \left[ {e^{i\lambda } g(z) - \beta \cos \lambda } \right]} \right|} /\left( {1 - \beta } \right)\cos \lambda \left| {d\theta \leqslant \kappa \pi ,} \right.z = re^{i\theta } ,$$ , 0 < r < 1 (κ?2,n?1, 0?Β<1, -π<λ<π/2;M _κ,n (λ,β,α),n?2, is the class of functions \(f(z) = z + \sum\nolimits_{v = n}^\infty {a_v z^v }\) , regular in¦z¦<1 and such thatF _α(z)∈P _κ,n?1(λ,β), where \(F_\alpha (z) = (1 - \alpha )\frac{{zf'(z)}}{{f(z)}} + \alpha (1 + \frac{{zf'(z)}}{{f'(z)}})\) (0?α?1). Onr considers the problem regarding the range of the system {g ^(v?1)(z_?)/(v?1)!}, ?=1,2,...,m,v=1,2,...,N _?, on the classP _κ,1(λ,β). On the classesP _κ,n (λ,β),M _κ,n (λ,β,α) one finds the ranges of C_v, v?n, a_m, n?m?2n-2, and ofg(?),F _?(?), 0<¦ξ¦<1, ξ is fixed.     

On the Fourth Moment of Coefficients of Symmetric Square L-Function

Let f(z) be a holomorphic Hecke eigencuspform of weight k for the full modular group. Let ?? _f (n) be the nth normalized Fourier coefficient of f(z). Suppose that L(sym² f, s) is the symmetric square L-function associated with f(z), and $ \lambda _{sym^2 f} (n) $ (n) denotes the nth coefficient L(sym² f, s). In this paper, it is proved that $$ \sum\limits_{n \leqslant x} {\lambda _{sym^2 f}^4 (n)} = xP2(\log x) + O(x^{\frac{{79}} {{81}} + \varepsilon } ), $$ , where P ₂(t) is a polynomial in t of degree 2. Similarly, it is obtained that $$ \sum\limits_{n \leqslant x} {\lambda _f^4 (n^2 )} = x\tilde P2(\log x) + O(x^{\frac{{79}} {{81}} + \varepsilon } ), $$ , where $ \tilde P_2 (t) $ is a polynomial in t of degree 2.     

Estimates of maximal functions measuring local smoothness

Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. Letf ∈L(I ⁿ ) (I≡[0,1]. Set $${\mathcal{N}}_\eta f(x) = \sup \frac{1}{{\left| Q \right|\eta (\left| Q \right|^{1/n} )}} \smallint _Q \left| {f(t) - f(x)} \right|dt,$$ , where the supremum is taken over all cubes containing the pointx. Forη=t ^α (0<α≤1) this definition was given by A.Calderón. In the paper we prove estimates of the maximal functions ${\mathcal{N}}_\eta f$ , along with some embedding theorems. In particular, we prove the following Sobolev type inequality: if $$1 \leqslant p< q< \infty , \theta \equiv n(1/p - 1/q)< 1, and \eta (t) \leqslant t^\theta \sigma (t),$$ , then $$\parallel {\mathcal{N}}_\sigma {f} {\parallel_{q,p}} \leqslant c \parallel {\mathcal{N}}_\eta {f} {\parallel_p} .$$ . Furthermore, we obtain estimates of ${\mathcal{N}}_\eta f$ in terms of theL ^p -modulus of continuity off. We find sharp conditions for ${\mathcal{N}}_\eta f$ to belong toL ^p (I ⁿ ) and the Orlicz class?(L), too.     

Approximation of integrable functions by linear methods almost everywhere

It is shown that 2π periodic functions whose (r-1)-th derivatives have bounded variation (r > 0) can be approximated by de La Vallée-Poussin sums σ _n,m (an ?m =m (n) ?An,0 <a<A<1) at almost all points with a rate o(n^?r). For functions belonging to the class Lip (α, L) (0 <α < 1), any natural N, and a positive ?, we have almost everywhere $$|f(x) - \sigma _{n,m} (f;x)| \leqslant c(f,x)n^{ - \alpha } lnn \ldots ln_N^{1 + \varepsilon } n,$$ where \(ln_k x = \underbrace {ln \ldots ln x}_k(k = 1, 2, \ldots )\) . For any triangular method of summation T with bounded coefficients we construct functions belonging to Lip (α, L) (0 < α < 1) and such that almost everywhere, $$\mathop {\overline {\lim } }\limits_{n \to \infty } |f(x) - \tau _n (f;x)|n^a (ln n \ldots ln_N n)^{ - a} = \infty $$ where the τ_n(f; x) are the means of the method T.     

Estimates of capacities associated with Besov spaces

Definition

Let A??ⁿ, 0<β≤∞. Define $$h_{\varphi ,\beta } (A) = \inf \left( {\sum\limits_{i = 0}^{ + \infty } {\left( {m_j \varphi (2^{ - i} } \right)^\beta } } \right)^{1/\beta } $$ where the infinum is taken over all coverings of A by a countable number of balls, whose radii r_j do not exceed 1, while m_i is the number of balls from this covering whose radii r_j belong to the set (2^?i?1, 2^?i], i∈N₀.

Theorem 1

Let p≤1, θ=∞, and let the function ?(t)t^lp?n increase. Then the following conditions are 2quivalent;

1. for any compact set K, K??ⁿ, if $\overline {cap} (K, X) = 0$ , then h_?,∞(K)=0;

Theorem 2

Let θ<1. Then for any set A the inequalities $c_1 \overline {cap} (A,X) \leqslant h_{t^{n - lp} ,\theta /p} (A) \leqslant c_2 \overline {cap} (A,X)$ hold. Bibliography:6 titles.     

On the best polynomial approximation of entire transcendental functions in banach spaces. I

We study the behavior of the best approximationsE _n (?) _p of entire transcendental functions ?(z) of the order ρ=∞ by polynomials of at mostn th degree in the metric of the Banach space E′_p(Ω) of functions /tf(z) analytic in a bounded simply connected domain Ω with rectifiable Jordan boundary and such that $$\left\| f \right\|_{E'_p } = \left\{ {\iint_\Omega {\left| {f\left( z \right)} \right|^p }dxdy} \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}}< \infty $$ . In particular, we describe the relationship between the best approximationsE _n (?)_p and theq-order andq-type of the function ?(z).     

The maximal variation of a bounded martingale

Let \(\chi _0^n = \left\{ {X_t } \right\}_0^n \) be a martingale such that 0≦X_i≦1;i=0, …,n. For 0≦p≦1 denote by ? _p ⁿ the set of all such martingales satisfying alsoE(X₀)=p. Thevariation of a martingale χ ₀ ⁿ is denoted byV ₀ ⁿ and defined by \(V(\chi _0^n ) = E\left( {\sum {_{l = 0}^{n - 1} } \left| {X_{l + 1} - X_l } \right|} \right)\) . It is proved that $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\mathop {Sup}\limits_{x_0^n \in \mathcal{M}_p^n } \left[ {\frac{1}{{\sqrt n }}V(\chi _0^n )} \right]} \right\} = \phi (p)$$ , where ?(p) is the well known normal density evaluated at itsp-quantile, i.e. $$\phi (p) = \frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi _p^2 ) where \int_{ - \alpha }^{x_p } {\frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi ^2 )} dx = p$$ . A sequence of martingales χ ₀ ⁿ ,n=1,2, … is constructed so as to satisfy \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .