On the fourth and fifth coefficients of strongly starlike functions

For 0 < α ≤ 1, analytic functions f(z) = z + a₂z² + a₃z³ + … in the unit disk U are strongly starlike of order α if ¦arg {zf′ (z)/f(z)}¦ < πα / 2, z ∈ U. We find sharp estimates on the fourth and fifth coefficients of functions in this class.     

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.     

A constructive description of Hölder classes on closed Jordan curves

Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, leta ε int Γ, and let R_n(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the pointa. Let Λ^α(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γ_n(z) > 0 on Γ such that f∈Λ^α(Γ) if and only if for any nonnegative integer n there exists a function R_n, R_n ε R_n(a) such that ¦f(z) ? R_n(z)¦ ≤ c_f·γ_n(z), z ε Γ. It is proved that the weights γ_n cannot be expressed simply in terms of ρ ₁ ⁺ /n^(z) and ρ ₁ ^- /n^(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on \(\mathbb{C}\backslash \mathbb{D}\) .     

The Hardy classes for functions in the class MV[α, k]

Suppose that f(z) = z + a₂z² + ··· + a_nz_n + ··· is regular in the unit disc $\text{D}\text{with}\text{[f(z) f′}\text{(z)}\text{z] ≠ 0}\text{in}\text{D}$, and further let α ? 0 and k ? 2. If $\text{∝}{}^{\mathrm{2\pi }}{}_{\mathrm{o}}\text{¦}\text{Re}\text{{(1 ? α)z[}\text{f′(z)}\text{f(z)}\text{] + α(1 + z[}\text{f″(z)}\text{f′(z)}\text{])}¦ dθ ? kπ}\text{for}\text{z ? D}$, then f(z) is said to belong to the class MV[α, k]. This class contains many of the special classes of regular and univalent functions. The authors determine the Hardy classes of which f(z), f′(z) and f″(z) belong and obtain growth estimates of a_n.     

A Functional Equation Of J.A. Baker Involving Integral Means

The functional equation $$f(x)={1\over 2}\int^{x+1}_{x-1}f(t)\ dt\ \ \ {\rm for}\ \ \ x\ \in\ {\rm R}$$ has the linear functions ?(x) = a + bx (a, b ∈ ?) as trivial solutions. It is shown that there are two kinds of nontrivial solutions, (i) ?(x) = eλ_i ^x (i = 1, 2, …), where the λ_i∈ ? are the fixed points of the map z ? sinh z, and (ii) C^∞-solutions ? for which the values in the interval [?1,1] can be prescribed arbitrarily, but with the provision that ?^(j)(? 1) = ?^(j)(0) = ?^(j)(1) = 0 for all j = 0, 1, 2 …     

一类高阶整函数系数微分方程解的增长级、下级与超级

该文研究了一类高阶整函数系数微分方程解的增长性,对方程f~(k)+A_(k-1)(z)e~(ak-1z).f~(k-1)+…+A_0(z)e~(a0z)f=0与方程f~(k)+(A_(k-1)(z)e~(ak-1z)+D_(k-1)(z))f~(k-1)+…+(A_0(z)e~(a0z)+D_0(z))f=0中a_j(0≤j≤k-1)幅角主值不全相等的情形,得到了解的增长级、下级与超级的精确估计.     

Extremal problems in a subclass of entire functions of finite power

We study the subclass W_σ(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 W_σ(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)¦.     

On certain estimates for orthogonal polynomials depending on parameters

Рассматривается сис тема ортогональных м ногочленов {P _n (z)} ₀ ^∞ , удовлетворяющ их условиям $$\frac{1}{{2\pi }}\int\limits_0^{2\pi } {P_m (z)\overline {P_n (z)} d\sigma (\theta ) = \left\{ {\begin{array}{*{20}c} {0,m \ne n,P_n (z) = z^n + ...,z = \exp (i\theta ),} \\ {h_n > 0,m = n(n = 0,1,...),} \\ \end{array} } \right.} $$ где σ (θ) — ограниченная неу бывающая на отрезке [0,2π] функция с бесчисленным множе ством точек роста. Вводится последовательность параметров {а_{n 0} ^∞ , независимых дру г от друга и подчиненных единств енному ограничению { ¦а_n¦<1} ₀ ^∞ ; все многочлены {Р _n (z)} 0/∞ можно найти по формуле $$P_0 = 1,P_{k + 1(z)} = zP_k (z) - a_k P_k^ * (z),P_k^ * (z) = z^k \bar P_k \left( {\frac{1}{z}} \right)(k = 0,1,...)$$ . Многие свойства и оце нки для {P _n (z)} ₀ ^∞ и (θ) можн о найти в зависимости от этих параметров; например, условие \(\mathop \Sigma \limits_{n = 0}^\infty \left| {a_n } \right|^2< \infty \) , бо лее общее, чем условие Г. Cerë, необходимо и достато чно для справедливости а симптотической форм улы в области ¦z¦>1. Пользуясь этим ме тодом, можно найти также реш ение задачи В. А. Стекло ва.     

The domain of regularity of the limit function of a sequence of analytic functions

Letf (z) be an entire function λ_n(n=0,1,2,...) complex numbers, such that the system f(λ_{n n=0} ^∞ is not complete in the circle ¦z¦n(z) have the form \(\sum\nolimits_{k = 0}^{p_n } {\alpha _{nk} } f(\lambda _k \cdot z)\) . We study the properties of the limit function of the sequence Q_n(z) in the case when $$f(z) = 1 + \sum\nolimits_{n = 1}^\infty {\frac{{z^n }}{{P(1)P(2)...P(n)}}} ,$$ . where P(z) is a polynomial having at least one negative integral root.     

Picardsche Ausnahmewerte bei Lösungen linearer Differentialgleichungen

Every solution w(z)0 of the linear differential equation L_n(w)=w⁽ⁿ⁾+a_n–1(z)w^(n–1)+...+a_o(z)w=0 with polynomial coefficients a_j(z), a_o(z)0, assumes all values a0, infinitely often. Strictly speaking, the only deficient values are zero and infinity. In this paper we study differential equations L_n(w)=0, which have a fundamental system with the following property: Every function of this fundamental system takes the value zero only a finite number of times. It is shown, that such a fundamental system exists if and only if the transformation w(z)=exp(q(z))u(z), where q(z) is a suitable polynomial, transforms the differential equation into one with constant coefficients for u(z).     

The completeness of systems of functions of the Mittag-Leffler type for weighted uniform approximation in a complex domain

For a givenρ(1/2 <ρ < + ∞) let us set L _ρ = {z: |arg z| = π/(2ρ)} and assume that a real valued measurable function ?(t) such that ?(t) ≥ 1(t ∈ L _ρ ) and \(\mathop {\lim }\limits_{|t| \to + \infty } \varphi (t) = + \infty (t \in L_\rho )\) is defined on L _ρ . Let C _? (L _ρ ) denote the space of continuous functionsf(t) on L _ρ such that \(\lim \tfrac{{f(t)}}{{\varphi (t)}} = 0\) , where the norm of an elementf is defined as: \(\parallel f\parallel = \mathop {\sup }\limits_{t \in L_\rho } \tfrac{{|f(t)|}}{{\varphi (t)}}\) . In this note we pose the question about the completeness of the system of functions of the Mittag-Leffler type {Eρ(ut; μ)} (μ ≥ 1, 0 ≤ u ≤a) or, what is the same thing, of the system of functions \(p(t) = \int_0^a {E_\rho (ut;\mu )d\sigma (u)} \) in C _? (L _ρ ). The following theorem is proved: The system of functions of the Mittag-Leffler type is complete in C _? (L _ρ ) if and only if sup |p(z)| ≡ +∞, z ∈ L _ρ , where the supremum is taken over the set of functions p(t) such that ∥p(t) (t + 1)^?1 ∥ ≤ 1.     

A rotation theorem in the class of bounded univalent functions

Let S_M, M > 1, be the class of functionsf(z) which are regular and univalent in the disk ¦z¦ < 1 and satisfy the conditionsf(0) = 0,f'(0) = 1, and ¦f(z)¦ < M. In the present note we will obtain an exact estimate for the argument of the derivative of a function of the class S_M.     

A gap in the energy spectrum of the one-dimensional Dirac operator

One considers the one-dimensional Dirac operator with a slowly oscillating potential (1) $$H = \left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right)\frac{d}{{dx}} + q\left( {\begin{array}{*{20}c} {\cos z(x)} & {\sin z(x)} \\ {\sin z(x)} & { - \cos z(x)} \\ \end{array} } \right)_, x \in ( - \infty ,\infty ),q - const,$$ where . The following statement holds. The double absolutely continuous spectrum of the operator (1) fills the intervals (?∞,?¦q¦), (¦q¦, ∞). The interval (?¦q¦, ¦q¦) is free from spectrum. The operator has a simple eigenvalue only for singn C₊=sign C_?, situated either at the point (under the condition C₊>0) or at the point λ=?¦q¦ (under the condition). The proof is based on the investigation of the coordinate asytnptotics of the corresponding equation.     

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.     

Region of Variability for Exponentially Convex Univalent Functions

For ${\alpha\in\mathbb C{\setminus}\{0\}}For a ? \mathbb C\{0}{\alpha\in\mathbb C{\setminus}\{0\}} let E(a){\mathcal{E}(\alpha)} denote the class of all univalent functions f in the unit disk \mathbbD{\mathbb{D}} and is given by f(z)=z+a₂z²+a₃z³+?{f(z)=z+a_2z^2+a_3z^3+\cdots}, satisfying

${\rm Re}\left (1+ \frac{zf'(z)}{f'(z)}+\alpha zf'(z)\right ) > 0 \quad {\rm in }\,{\mathbb D}.${\rm Re}\left (1+ \frac{zf'(z)}{f'(z)}+\alpha zf'(z)\right ) > 0 \quad {\rm in }\,{\mathbb D}.      16. Oscillation theory at a finite singularity      Don B Hinton  Roger T Lewis 《Journal of Differential Equations》1978,30(2):235-247 It is well known that every weak solution (with boundary values 0) of a semilinear equation $\text{Au + ?(x, u) = g}$ is a regular solution if ? fulfils the growth condition $\text{(1) ¦?(x, u)¦? c ¦u¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; ?\; ?</mtext>}}$. Here 2m is the order of A. In this paper we weaken this condition to $\text{c ¦u ¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1</mtext>}}\text{? ?(x, u)u ? ?c ¦u ?}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1\; ?\; ?</mtext>}}$. This requires a technique completely different from that which may be applied in case (1).      17. The value regions of initial coefficients in a certain class of meromorphic functions      E. G. Goluzina 《Journal of Mathematical Sciences》1997,83(6):745-749 Let Mk,λ(0≤λ≤1, k≥2) be the class of functions f(z)=1/z+ao+a1z+... that are regular and locally univalent for 0<⩛z⩛<1 and satisfy the condition where Jλ(z)=λ(1+zf″(z)/f'(z))+(1-λ)zf'(z)/f(z). In the class Mk,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.      18. Базисы из экспонент в пространствахL p      А. М. Седлецкий 《Analysis Mathematica》1982,8(3):215-232 Let the rootsλ n of an entire functionL(z) be separated and lie in some horizontal strip ¦Im z¦ ≦h, and suppose that $$0< c \leqq |L(z)|(1 + |z|)^{ - b} \exp ( - a|\operatorname{Im} z|) \leqq C< \infty$$ for ¦Imz¦≧H>h. If 1<p<2 and - 1/pq (1/q+1/p=1), then the system {exp (iλ n x)} n=0 ∞ constitutes a basis нn the spaceL p (-a,a). In the caseb=1/q orb=?1/p the theorem fails, Equivalence of the following two statements is also proved: {exp (iλ n x)} n=0 ∞ is an extendable convergence system inL p from the interval (-a, a). {exp (iλ n x)} n=0 ∞ is a continuable basis inL p (-a,a).       19. On a linear combination of some expressions in the theory of the univalent functions      Hassoon S. Al-Amiri  Maxwell O. Reade 《Monatshefte für Mathematik》1975,80(4):257-264 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 $$\left| {1 + \alpha } \right|\left| {a_3 - \mu a_2^2 } \right| \leqslant ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})\max \left[ {1,\left| {1 + 2\alpha - {3 \mathord{\left/ {\vphantom {3 {2\mu }}} \right. \kern-\nulldelimiterspace} {2\mu }}(1 + \alpha )} \right|} \right].$$ .      20. Subordination properties of certain integrals      S. Ponnusamy 《Proceedings Mathematical Sciences》1991,101(3):219-226 Let B1(μ,β) denote the class of functions f(z)= z + a 2 z 2 + h+ a n z m +… that are analytic in the unit disc Δ and satisfy the condition Ref′(z)(f(z)/z)?-1 > β, zεΔ, for some ?>0 and β< 1. Denote by S*(0)for B1(0,0). For μ andc such thatc > -μ, letF =I gm,c (f) be defined by $$F(z) = \left[ {\frac{{\mu + c}}{{Z^c }}\int_0^z {f^\mu (t)} t^{c - 1} dt} \right]^{1/\mu } ,z \in \Delta .$$ The author considers the following two types of problems: (i) To find conditions on ?,c and ρ > 0 so thatfεB 1 (μ -ρ) implies Iμ,c(f<εS*(0); (ii) To determine the range of μ and δ > 0 so that fεB1 (μ -δ) impliesI μο (f)εS*(0); We also prove that if / satisfies Re(f′(z) +zf′’(z)) > 0 in Δ then the nth partial sumf n off satisfiesfn(z)/z? -1 -(2/z)log(l -z)in Δ. Here, ? denotes the subordination of analytic functions with univalent analytic functions. As applications we also give few examples.

Oscillation theory at a finite singularity

It is well known that every weak solution (with boundary values 0) of a semilinear equation $\text{Au + ?(x, u) = g}$ is a regular solution if ? fulfils the growth condition $\text{(1) ¦?(x, u)¦? c ¦u¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; ?\; ?</mtext>}}$. Here 2m is the order of A. In this paper we weaken this condition to $\text{c ¦u ¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1</mtext>}}\text{? ?(x, u)u ? ?c ¦u ?}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1\; ?\; ?</mtext>}}$. This requires a technique completely different from that which may be applied in case (1).     

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.     

Базисы из экспонент в пространствахL p

Let the rootsλ _n of an entire functionL(z) be separated and lie in some horizontal strip ¦Im z¦ ≦h, and suppose that $$0< c \leqq |L(z)|(1 + |z|)^{ - b} \exp ( - a|\operatorname{Im} z|) \leqq C< \infty$$ for ¦Imz¦≧H>h. If 1<p<2 and - 1/pq (1/q+1/p=1), then the system {exp (iλ _n x)} _n=0 ^∞ constitutes a basis нn the spaceL ^p (-a,a). In the caseb=1/q orb=?1/p the theorem fails, Equivalence of the following two statements is also proved:

1. {exp (iλ _n x)} _n=0 ^∞ is an extendable convergence system inL ^p from the interval (-a, a).
2. {exp (iλ _n x)} _n=0 ^∞ is a continuable basis inL ^p (-a,a).

     

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

Subordination properties of certain integrals

Let B₁(μ,β) denote the class of functions f(z)= z + a ₂ z ² + h+ a _n z ^m +… that are analytic in the unit disc Δ and satisfy the condition Ref′(z)(f(z)/z)^?-1 > β, zεΔ, for some ?>0 and β< 1. Denote by S*(0)for B₁(0,0). For μ andc such thatc > -μ, letF =I _gm,c (f) be defined by $$F(z) = \left[ {\frac{{\mu + c}}{{Z^c }}\int_0^z {f^\mu (t)} t^{c - 1} dt} \right]^{1/\mu } ,z \in \Delta .$$ The author considers the following two types of problems: (i) To find conditions on ?,c and ρ > 0 so thatfεB ₁ (μ -ρ) implies I_μ,c(f<εS*(0); (ii) To determine the range of μ and δ > 0 so that fεB₁ (μ -δ) impliesI _μο (f)εS*(0); We also prove that if / satisfies Re(f′(z) +zf′’(z)) > 0 in Δ then the nth partial sumf _n off satisfiesf_n(z)/z? -1 -(2/z)log(l -z)in Δ. Here, ? denotes the subordination of analytic functions with univalent analytic functions. As applications we also give few examples.