Series of univalent functions

We are considering a class S of functions F(z), F(0) = 0, F′(0) = 1 that are univalent and regular in the circle ¦z¦ < 1, and its subclasses s _h ^* and K of starlike functions of order h and of convex functions respectively. Among others, we establish the following results: If F(z)εs and 0 < α < 1, then IfF (z) ε s (0 < a < 1) and $$\begin{gathered} 1 + \operatorname{Re} {{z_1 F^n \left( {z_1 } \right)} \mathord{\left/ {\vphantom {{z_1 F^n \left( {z_1 } \right)} {F'\left( {z_1 } \right)}}} \right. \kern-\nulldelimiterspace} {F'\left( {z_1 } \right)}} = \operatorname{Re} {{\alpha z_1 F''\left( {\alpha z_1 } \right)} \mathord{\left/ {\vphantom {{\alpha z_1 F''\left( {\alpha z_1 } \right)} {F'\left( {\alpha z_1 } \right)}}} \right. \kern-\nulldelimiterspace} {F'\left( {\alpha z_1 } \right)}} \hfill \\ \left( {2 - \sqrt 3< \left| {z_1 } \right| = r< 1} \right) \hfill \\ \end{gathered} $$ then we obtain the domain of values of the point αz₁.     

On the finite-increment theorem for complex polynomials

For an arbitrary polynomial P of degree at most n and any points z ₁ and z ₂ on the complex plane, we establish estimates of the form $$ \left| {P(z_1 ) - P(z_2 )} \right| \geqslant d_n \left| {P'(z_1 )} \right|\left| {z_1 - \zeta } \right| $$ , where ζ is one of the roots of the equation P(z) = P(z ₂), and d _n is a positive constant depending only on the number n.     

On the analogue of a de la Vallée Poussin's problem for arcs in the complex plane

пУстьL — кОНЕЧНАь жОР ДАНОВА ДУгА, И тОЧкАz ₀?L (НЕ сОВпАДАУЩАь НИ с ОДНИ М Иж кОНцОВL) ДЕлИтL НА ДВЕ ЧАстИL′ ИL″'. пРИ НЕкОт ОРых ОгРАНИЧЕНИьх НА гЕОМ ЕтРИУ ДУгИ НАИДЕН пОРьДОк НАИлУ ЧшИх пРИБлИжЕНИИ МНО гОЧлЕНАМИ Дль ФУНкцИИ $$f(z) = f(z,z_0 ) = \left\{ {\begin{array}{*{20}c} {z - z_0 ,z \in L';} \\ {z_0 - z,z \in L''.} \\ \end{array} } \right.$$      

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

On estimates of lengths of lemniscates

For any natural number n and any C > 0, we obtain an integral formula for calculating the lengths |L(P _n , C)| of the lemniscates $$L\left( {P_n ,C} \right): = \left\{ {z:\left| {P_n \left( z \right)} \right| = C} \right\}$$ of algebraic polynomials P _n (z):= z ⁿ + c _n?1 z ^n?1 + ... + c ₀ in the complex variable z with complex coefficients c _j , j = 0, ..., n ? 1, and establish the upper bound for the quantities $$\lambda _n : = \sup \left\{ {\left| {L\left( {P_n ,1} \right)} \right|:P_n (z)} \right\},$$ which is currently best for 3 ≤ n ≤ 10¹⁴. We also study the properties of the derivative S′(C) of the area function S(C) of the set {z: |P _n (z)| ≤ C}.     

Sections of the Taylor expansion of exp(g(z)) whereg(z) is entire and admissible in the sense of Hayman

The author considers $$f(z) = \exp (g(z)) = \sum\limits_{j = 0}^\infty {a_j z^j ,}$$ , whereg(z) is a real, entire, transcendental function admissible in the sense of W. K. Hayman [(1956): Reine Angew. Math.,196:67-95]. The aim of the paper is to study, asm→+∞, the distribution of the zeros of the partial sums $$s_m (z) = \sum\limits_{j = 0}^m {a_j z^j .}$$ The results are stated in terms of Hayman's auxiliary functions Ifr>0 is large enough, botha(r) andb(r) are positive,a(r) is strictly increasing, and $$a(r) \to + \infty ,b(r) \to + \infty (r \to + \infty ).$$ Define the sequence (R _m ) (m>m ₀) by the relationsa(R _m )=m. From the following proposition, typical of those stated in the paper, it is easy to deduce accurate information regarding those zeros ofs _m (z) that lie near the positive axis: Letζ be an auxiliary complex variable; then asm→+∞, and forR=R _m , the functions $$\left\{ {1 + \zeta \left( {\frac{2}{{b(R)}}} \right)^{1/2} } \right\}^{ - m} \{ f(R)\} ^{ - 1} s_m \left( {R\left( {1 + \zeta \left( {\frac{2}{{b(R)}}} \right)^{1/2} } \right)} \right)$$ tend to $$\frac{1}{2}e^{\zeta ^2 } \left( {1 - \frac{2}{{\sqrt \pi }}\int_0^\zeta {e^{ - \sigma ^2 } d\sigma } } \right)$$ uniformly on every compact subset of theζ-plane. There are similar, equally precise, results covering those zeros ofs _m (z) that lie near any rayte ^i?(0<t<+∞,?≠0).     

On optimal extrapolation and interpolation of fuzzy analytic functions

Для класса ? аналитич еских в единичном кру ге функций, ограниченны х по модулю единицей, погрешност ью наилучшего прибли жения в точкеz ₀ по значениям в точкахz ₁,..., z_n, заданным с погрешнос тьюδ, называется вели чинаr(z ₀, z₁ z..., z_n, α)=inf sup sup ¦f(z₀)-S(f₁, ...f_n)¦, где нижняя грань бере тся по всевозможным ф ункциям S: Сⁿ→С. ДляE～((?1,1) иz ₀∈ ∈(-1,1)Е рассматривается задача о нахождении п орядка информативности мно жестваЕ, т.е. минимальногоп, на котором достигается нижняя грань в равенстве $$R(z_0 ,\delta ,E) = \mathop {\inf }\limits_n {\text{ }}\mathop {\inf }\limits_{z_1 , \ldots ,z_n \in E} {\text{ }}r(z_0 ,z_1 , \ldots ,z_n ,\delta ).$$ Кроме того, приδ, близ ких к 1, решена задача о нахождении величины $$r_n (\delta ,E) = \mathop {\inf }\limits_{z_1 , \ldots ,z_n \in Ez_0 \in E} \sup r(z_0 ,z_1 , \ldots ,z_n ,\delta )$$ и найдены узлы, на кото рых достигается нижн яя грань.     

A mean value theorem for the Dedekind zeta-function of a quadratic number field

Let \(K = \mathbb{Q}(\sqrt d )\) be any quadratic number field with discriminantd. ζ _K (s) denotes the Dedekind zeta-function. The purpose of this note is to prove the following asymptotic formula: $$\int\limits_0^T {|\zeta _K ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + it)|^2 dt = ({6 \mathord{\left/ {\vphantom {6 {\pi ^2 }}} \right. \kern-\nulldelimiterspace} {\pi ^2 }})} \prod\limits_{p/d} {(1 + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p})^{ - 1} \cdot R_K^2 \cdot T \cdot \log ^2 T + O_\varepsilon \left\{ {\left| d \right|1 + \varepsilon \cdot T \cdot \log T} \right\},} $$ where the implied constant depends only on ε. HereR _K, denotes the residue of ζ _K (s) ats=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.     

On the monotonicity of some functionals in the family of univalent functions

LetS denote the class of regular and univalent functions in |z|<1 with the normalizationf(0)=0,f′(0)=1. Denoted _f=inf _f∈s {|α||f(z)≠ α, |z|<1} and letS(d)={f¦f∈S,d _f=d, 1/4≦d≦1}. The analytic functionf(z) is univalent in |z|<1 if and only if $$log\frac{{f(z) - f(\zeta )}}{{z - \zeta }} = \sum\limits_{m,n = 0}^\infty {d_{mn} z^m \zeta ^n } $$ converges in the bicylinder |z|<1, |ξ|<1. LetC _mn =√mnd _mn andC _nn (d)= Max _fεS(d){Re(C _nn )}. The paper deals with the monotonicity ofc _nn(d) and related functionals.     

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).     

Small Divisor Problem in Dynamical Systems and Analytic Solutions of a q-Difference Equation with a Singularity at the Origin

In this paper, we consider a q-difference equation $$\sum_{j=0}^{k}\sum_{t=1}^{\infty}C_{t,j}(z)(y(q^jz))^{t}=G(z)$$ in the complex field ${\mathbb C,}$ where C _t,j (z) and G(z) have a h ₁ order pole and a h ₂ order pole at z = 0, respectively. Under the case 0 < |q| < 1 or |q| = 1, we give the existence of local analytic solutions for the above equation by using small divisor theory in dynamical systems.     

Numerical computations of integrals over paths on Riemann surfaces of genusN

This paper is a continuation of work by Forest and Lee [1,2]. In [1,2] it was proved that the function theory of periodic soliton solutions occurs on the Riemann surfaces ? of genusN, where the integrals over paths on ? play the most fundamental role. In this paper a numerical method is developed to evaluate these integrals. Predisely, the aim is to develop a computational code for integrals of the form $$\int\limits_\gamma {f(z)\frac{{dz}}{{R(z)}}, or} \int\limits_\gamma {f(z)R(z)dz,} $$ wheref(z) is any single-valued analytic function on the complex planeC, andR(z) is a two-valued function onC of the form $$R^2 (z) = \prod\limits_{k = 1}^{2N + \delta } {(z - z_0 (k)), \delta = 0 or 1,} $$ where {z ₀(k),1≤k≤2N+δ} are distinct complex numbers which play the role of the branch points of the Riemann surface ? = {(z, R(z))} of genusN?1+δ. The integral path γ is continuous on ?. The numerical code is developed in “Mathematica” [3].     

L p inequalities for polynomials with restricted zeros

LetP(z) be a polynomial of degreen which does not vanish in the disk |z|<k. It has been proved that for eachp>0 andk≥1, $$\begin{gathered} \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P^{(s)} (e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} \leqslant n(n - 1) \cdots (n - s + 1) B_p \hfill \\ \times \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P(e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} , \hfill \\ \end{gathered} $$ where $B_p = \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {k^s + e^{i\alpha } } \right|^p d\alpha } } \right\}^{ - 1/p} $ andP ^(s)(z) is thesth derivative ofP(z). This result generalizes well-known inequality due to De Bruijn. Asp→∞, it gives an inequality due to Govil and Rahman which as a special case gives a result conjectured by Erdös and first proved by Lax.     

Lower bounds for the modulus of the logarithmic derivative of a polynomial

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 ? x_n) is a polynomial with real zeros. Then, for any δ > 0, on any intervala ?x ?b, containing all of the x_k (k=1, 2, ..., n), outside an exceptional set E_δ?[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.     

Обобщение регулярны х в круге функций, близ ких к звездным, на случай многих комп лексных переменных И. И. Баврин

LetD denote a bounded complete bicircular domain (centered at (0, 0)) in the space C of two complex variablesz ₁ andz ₁. Extremal problems are treated in the classP _D which is a generalization, to the two complex variables case, of the class of close-to-starlike functions regular in a disc. Our considerations include estimates of the basic functionals, in particular, those specific for the functions of several complex variables; exactness of estimates is considered, as well. For example, we obtain estimates of the functionals $$A_k (D) = \mathop {\sup }\limits_{(z_1 ,z_2 ) \in D} \mathop \sum \limits_{l = 0}^k |a_{k - l,l} |^2 |z_1 |^{2(k - l)} |z_2 |^{2l} ,B_k (D) = \mathop {\sup }\limits_{(z_1 ,z_2 ) \in D} \left| {\mathop \sum \limits_{l = 0}^k a_{k - l,l} z_1^{k - l} z_2^l } \right|,k = 1,2 \ldots .$$ It is also proved that the Carathéodory class is a subclass of the classP _D.     

On rational approximations of functions of complex variable integrable over plane domains

пУстьE — ИжМЕРИМОЕ пО лЕБЕгУ ОгРАНИЧЕННОЕ МНОжЕстВО пОлОжИтЕльНОИ плОЩА ДИ mes₂ E кОМплЕксНОИ плОск ОстИ с. кАк ОБыЧНО, пРИp≧1 ОБОжНАЧИМ ЧЕРЕжL ^p (E) БА НАхОВО пРОстРАНстВО ИжМЕРИ Мых пО лЕБЕгУ НАE кОМплЕксНОжНАЧНых Ф УНкцИИf с сУММИРУЕМО Иp—стЕпЕНьУ Их МОДУль И ОБыЧНОИ НОРМОИ \(\left\| \cdot \right\|_p = \left\| \cdot \right\|_{L_p (E)}\) . ЧЕР ЕжL ^p R _n (f,E) ОБОжНАЧИМ НАИМЕН ьшЕЕ УклОНЕНИЕf?L ^p (E) От РАц ИОНАльНых ФУНкцИИ ст ЕпЕНИ ≦n кОМплЕксНОгО пЕРЕМЕ ННОгОz пО НОРМЕ ∥ · ∥. пОлОжИМf(z)=0 Дльz?¯CE,E _δ —δ-ОкРЕстНОсть МНО жЕстВАE (δ>0), И $$\omega _p (\delta ,f) = \mathop {\sup {\mathbf{ }}}\limits_{\left| h \right|< \delta } \{ \int\limits_{E_\sigma } {\int {{\mathbf{ }}|f(z + h) - f(z)|^p } d\sigma } \} ^{1/p} .$$ тЕОРЕМА.пУсть 1≦p<2,f?L _p (E),n≧4.тОгДА $$\begin{array}{*{20}c} {L^p R_n (f,E) \leqq 12\omega _p \left( {\frac{{\delta + \ln n}}{{\sqrt n }},f} \right){\mathbf{ }}npu{\mathbf{ }}p = 1,} \\ {L^p R_n (f,E) \leqq \frac{{24}}{{(p - 1)(2 - p)}}\omega _p (n^{(p - 2)/2p} ,f){\mathbf{ }}npu{\mathbf{ }}1< p< 2,} \\ {L^1 R_n (\bar z,[0,1] \times [0,1]) \geqq \frac{1}{{32\sqrt n }}.} \\ \end{array} $$ .     

Classes of functions subharmonic in ℝm and bounded on certain sets

Let Z_j be the Euclidean space of vectors \((z_{j,1,...,} z_{j_{j \cdot n_j + 1} } ), Z = \mathop \oplus \limits_{j = 1}^P Z_j\) . The function u: Z → ?₊, u ?0, is said to be logarithmically p-subharmonic if log u(z) is upper semicontinuous with respect to the totality of the variables and subharmonic or identically equal to ?∞ with respect to each z_j when the remaining ones are fixed. For such functions, with the growth estimate $$log u(z) \leqslant \delta \mathop \Pi \limits_{j = 1}^P (1 + |z_{j,n_j + 1} |) + N(\mathop {\sum\limits_{\mathop {1 \leqslant j \leqslant p}\limits_{} } {z_{j,k}^2 } }\limits_{1 \leqslant k \leqslant n_j } )^{1/2} + C; \delta ,N \geqslant 0, C \in \mathbb{R}$$ one proves theorems on equivalence of^∞) (L_q)-norms of their restrictions to \(X = \mathop \oplus \limits_{j = 1}^P (Z_{j,1} ,...,z_{j,n_j } )\) and to a relatively dense subset of it, generalizing the known Cartwright and Plancherel-Pólya results.     

Rows and diagonals of the Walsh array for entire functions with smooth Maclaurin series coefficients

Let \(f(z): = \sum\nolimits_{j = 0}^\infty {a_j z^J } \) be entire, witha _j≠0,j large enough, \(\lim _{J \to \infty } a_{j + 1} /a_J = 0\) , and, for someq∈C, \(q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q\) asj→∞. LetE _mn(f; r) denote the error in best rational approximation off in the uniform norm on |z‖≤r, by rational functions of type (m, n). We study the behavior ofE _mn(f; r) asm and/orn→∞. For example, whenq above is not a root of unity, or whenq is a root of unity, butq _m has a certain asymptotic expansion asm→∞, then we show that, for each fixed positive integern, ,m→∞. In particular, this applies to the Mittag-Leffler functions \(f(z): = \sum\nolimits_{j = 0}^\infty {z^j /\Gamma (1 + j/\lambda )} \) and to \(f(z): = \sum\nolimits_{j = 0}^\infty {z^j /(j!)^{I/\lambda } } \) , λ>0. When |q‖<1, we also handle the diagonal case, showing, for example, that ,n→∞. Under mild additional conditions, we show that we can replace 1+0(1) ⁿ by 1+0(1). In all cases we show that the poles of the best approximants approach ∞ asm→∞.