A bound of the exterior arcs for a univalent mapping

In this paper we consider the intersection of the circle ¦w|=x with the image of the disc ¦z|&#x2264;r, 0f(z)=z+c₂z²+... which is univalent analytic in ¦z|<1. Earlier I. E. Bazilevich proved that for xc^/er the measure of the above intersection does not exceed the measure of the intersection produced by the functionf ^*(z)=z/(1–z)², ¦=1. In this paper I. E. Bazilevich's ideas are used to strengthen some of his results.Translated from Matematicheskie Zametki, Vol. 18, No. 3, pp. 367–378, September, 1975.     

Об одной теореме И. И. П ривалова

Let G be a finite Jordan domain,f(z)=u(z)+iv(z) an analytic function inG. Connections between smoothness of the functionu(z) on dG and smoothness of the functionf(z) on ¯G are obtained. In these results the regionG is assumed to satisfy the condition

The imaginary zeros of a mixed Bessel function

In the present work we study the existence and monotonicity properties of the imaginary zeros of the mixed Bessel functionM _v(z)=(z²+)J_v(z)+zJ_v(z). Such a function includes as particular cases the functionsJ _v(z)(==0), J_v(z)(=–v²,=1)x andH _v(z)=J_v(z)+zJ_v(z), whereJ _v(z) is the Bessel function of the first kind and of orderv>–1 andJ _v(z), J_v(z) are the first two derivatives ofJ _v(z). Upper and lower bounds found for the imaginary zeros of the functionsJ _v(z), J_v(z) andH _v(z) improve previously known bounds.

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Zusammenfassung Dieser Artikel betrifft die Existenz und Monotonie von Eigenschaften imaginärer Nullen der gemischten BesselfunktionM _v(z)=(z²+)J_v(z)+zJ_v(z). Eine solche Funktion enthält als Spezialfall die FunktionenJ _v(z)(==0), J_v(z)(=–v²,=1) undH _v(z)=J_v(z)+zJ_v(z), woJ _v(z)die Besselfunktion von erster Art und Ordnungv>–1 andJ _v(z), J_v(z) sind die erste und zweite Ableitung vonJ _v(z). Untere und obere Schranken, die für die imaginären Nullen der FunktionenJ _v(z), J_v(z) undH _v(z) gefunden wurden, verbessern früher bekannte Resultate.

     

On the existence of entire functions of boundedl-index andl-regular growth

We prove that, under certain conditions on a positive functionl continuous on [0, +∞], there exists an entire transcendental functionf of boundedl-index such that lnlnM _f(r)lnL(r),r→∞, whereM _f (r)=max {|f(z)|: |z|=r} andL(r)=∫ ₀ ^r l(t)dt. Ifl(r)=r ^p-1 forr≥1, 0<ρ<∞, then there exists an entire functionf of boundedl-index such thatM _f (r)≈r ^p . Lvov University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1166–1182, September, 1996.     

An introduction to second degree forms

We are dealing with orthogonal sequences with respect to forms verifying a second degreee equation, i.e. that its formal Stieltjes functionS(u)(z) satisfies a quadratic equation of the formB(z)S ²(u)(z)+C(z)S(u)(z)+D(z)=0, whereB, C, D are polynomials. Various algebraic properties are given, especially those concerning the quasi-orthogonality of associated sequences. A classification is outlined. Some examples are quoted. In particular, we give the representation of Tchebychev co-recursive forms for any complex value of the parameter.     

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.     

Differential sobordination and Bazilevič functions

LetM(z)=z ⁿ +…,N(z)=z ⁿ +… be analytic in the unit disc Δ and let λ(z)=N(z)/zN′(z). The classical result of Sakaguchi-Libera shows that Re(M′(z)/N′(z))<0 implies Re(M(z)/N(z))>0 in Δ whenever Re(λ(z))>0 in Δ. This can be expressed in terms of differential subordination as follows: for anyp analytic in Δ, withp(0)=1,p(z)+λ(z)zp′(z)<1+z/1−z impliesp(z)<1+z/1−z, for Reλ(z)>0,z∈Δ. In this paper we determine different type of general conditions on λ(z),h(z) and ϕ(z) for which one hasp(z)+λ(z)zp′(z)<h(z) impliesp(z)<ϕ(z)<h(z) z∈Δ. Then we apply the above implication to obtain new theorems for some classes of normalized analytic funotions. In particular we give a sufficient condition for an analytic function to be starlike in Δ.     

Structural formulas and value regions of functionals in certain classes of regular functions

We study the structural properties of the class M_k,λ,b(k≥2, 0≤λ≤1, b∈ℂ\{0}) of functions f(z)=z+ ... which are regular in |z|<1 and satisfy the conditions f(z)f′(z)z⁻¹≠0 and , where J(z)=λ(1+b⁻¹zf″(z)/f′(z)+(1−λ)(b⁻¹zf′(z)/f(z)+1−b⁻¹). The value regions of some functionals on this class are found. The case λ=1 was considered in our previous paper. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 55–60. Translated by O. A. Ivanov.     

Geometric conditions for interpolation in weighted spaces of entire functions

We use L² estimates for the equation to find geometric conditions on discrete interpolating varieties for weighted spaces A_p(ℂ) of entire functions such that |f(z)|≤Ae^Bp(z) for some A, B>0. In particular, we give a characterization when p(z)=e^|z| and more generally, when In p(e^r) is convex andIn p(r) is concave. Acknowledgements and Notes. The author wishes to thank X. Massaneda for useful talks and remarks.     

Zeros ofA p functions and related classes of analytic functions

The functionf(z), analytic in the unit disc, is inA ^p if \(\int {\int {_{\left| z \right|< 1} \left| {f(z)} \right|^p dxdy< \infty } } \) . A necessary condition on the moduli of the zeros ofA ^p functions is shown to be best possible. The functionf(z) belongs toB ^p if \(\int {\int {_{\left| z \right|< 1} \log ^ + \left| {f(z)} \right|)^p } } \) . Let {z _n } be the zero set of aB ^p function. A necessary condition on |z _n | is obtained, which, in particular, implies that Σ(1?|z _n |)^1+(1/p)+g <∞ for all ε>0 (p≧1). A condition on the Taylor coefficients off is obtained, which is sufficient for inclusion off inB ^p. This in turn shows that the necessary condition on |z _n | is essentially the best possible. Another consequence is that, forq≧1,p<q, there exists aB ^p zero set which is not aB ^q zero set.     

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}.      13. An introduction to second degree forms      P. Maroni 《Advances in Computational Mathematics》1995,3(1-2):59-88 We are dealing with orthogonal sequences with respect to forms verifying a second degreee equation, i.e. that its formal Stieltjes functionS(u)(z) satisfies a quadratic equation of the formB(z)S 2(u)(z)+C(z)S(u)(z)+D(z)=0, whereB, C, D are polynomials. Various algebraic properties are given, especially those concerning the quasi-orthogonality of associated sequences. A classification is outlined. Some examples are quoted. In particular, we give the representation of Tchebychev co-recursive forms for any complex value of the parameter.      14. Analytic functions which are regular in a disc and smooth on its boundary      B. I. Korenblyum  V. C. Korolevich 《Mathematical Notes》1970,7(2):100-104 A theorem is established, asserting that the norm of the derivativef (n)(z) in the space H2 for a functionf(z) regular in the disc is not increased if we replacef by the ratiof (z)/G(z), where G(z) is any interior function dividingf(z) whose singular part is of a particular form.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 165–172, February, 1970.      15. Fourier sums for the banach indicatrix      K. I. Oskolkov 《Mathematical Notes》1974,15(4):309-312 We prove the existence of a functionf(t), which is continuous on the interval [0, 1], is of bounded variation, minf(t)=0, maxf(t)=1, for which the integral $$I(x) = \frac{1}{\pi }\int_0^\infty {\left[ {\int_0^1 {cos} y(f(l) - x)\left| {df(l)} \right|} \right]} dy$$ diverges for almost all X ∈ [0, 1]. This result gives a negative answer to a question posed by Z. Ciesielski.      16. The composition operators on weighted Bloch space   总被引：9，自引：0，他引：9   R. Yoneda 《Archiv der Mathematik》2002,78(4):310-317 We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B log = { f ? H(D): supz ? D (1-| z|2) ( log\frac21-| z|2 )| f￠(z)| B_{ \log }= \{ f \in H(D): \sup_{z \in D } (1-\left| z\right|^2) \left( \log \frac{2}{1-\left| z\right|^2} \right)\left| f'(z)\right| < +￥} +\infty \} , where H(D) be the class of all analytic functions on D.      17. Automorphism groups of rational elliptic surfaces with section and constant J-map      Tolga Karayayla 《Central European Journal of Mathematics》2014,12(12):1772-1795 In this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have constant J-maps are classified. The ground field is ?. The automorphism group of such a surface β: B → ?1, denoted by Au t(B), consists of all biholomorphic maps on the complex manifold B. The group Au t(B) is isomorphic to the semi-direct product MW(B) ? Aut σ (B) of the Mordell-Weil groupMW(B) (the group of sections of B), and the subgroup Aut σ (B) of the automorphisms preserving a fixed section σ of B which is called the zero section on B. The Mordell-Weil group MW(B) is determined by the configuration of singular fibers on the elliptic surface B due to Oguiso and Shioda [9]. In this work, the subgroup Aut σ (B) is determined with respect to the configuration of singular fibers of B. Together with a previous paper [4] where the case with non-constant J-maps was considered, this completes the classification of automorphism groups of relatively minimal rational elliptic surfaces with section.      18. Direct and inverse spectral problems for generalized strings      Heinz Langer  Henrik Winkler 《Integral Equations and Operator Theory》1998,30(4):409-431 Let the functionQ be holomorphic in he upper half plane + and such that ImQ(z 0 and ImzQ(z) 0 ifz +. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf + fdm = 0; f(0–) = 0. In this note we show that the set of functionsQ which are holomorphic in + and such that the kernel has negative squares of + and ImzQ(z) 0 ifz + is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf +f dm + 2 fdD = 0 on [0,L),f(0–) = 0. Here is the number of pointsx whereD increases or 0 >m(x + 0) –m(x – 0) –; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.To the memory of M.G. Krein with deep gratitude and affection.This author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 09832      19. A note on regularity for a class of quasilinear elliptic equations      LiJunjie BianBaojun 《高校应用数学学报(英文版)》2000,15(3):273-280 The following regularity of weak solutions of a class of elliptic equations of the form are investigated.      20. A theorem of the Wiener—Tauberian type forL 1(H n)      Rama Rawat 《Proceedings Mathematical Sciences》1996,106(4):369-377 The Heisenberg motion groupHM(n), which is a semi-direct product of the Heisenberg group Hn and the unitary group U(n), acts on Hn in a natural way. Here we prove a Wiener-Tauberian theorem for L1 (Hn) with this HM(n)-action on Hn i.e. we give conditions on the “group theoretic” Fourier transform of a functionf in L1 (Hn) in order that the linear span ofgf : g∈HM(n) is dense in L1(Hn), wheregf(z, t) =f(g·(z, t)), forg ∈ HM(n), (z,t)∈Hn.

An introduction to second degree forms

We are dealing with orthogonal sequences with respect to forms verifying a second degreee equation, i.e. that its formal Stieltjes functionS(u)(z) satisfies a quadratic equation of the formB(z)S ²(u)(z)+C(z)S(u)(z)+D(z)=0, whereB, C, D are polynomials. Various algebraic properties are given, especially those concerning the quasi-orthogonality of associated sequences. A classification is outlined. Some examples are quoted. In particular, we give the representation of Tchebychev co-recursive forms for any complex value of the parameter.     

Analytic functions which are regular in a disc and smooth on its boundary

A theorem is established, asserting that the norm of the derivativef ⁽ⁿ⁾(z) in the space H² for a functionf(z) regular in the disc is not increased if we replacef by the ratiof (z)/G(z), where G(z) is any interior function dividingf(z) whose singular part is of a particular form.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 165–172, February, 1970.     

Fourier sums for the banach indicatrix

We prove the existence of a functionf(t), which is continuous on the interval [0, 1], is of bounded variation, minf(t)=0, maxf(t)=1, for which the integral $$I(x) = \frac{1}{\pi }\int_0^\infty {\left[ {\int_0^1 {cos} y(f(l) - x)\left| {df(l)} \right|} \right]} dy$$ diverges for almost all X ∈ [0, 1]. This result gives a negative answer to a question posed by Z. Ciesielski.     

The composition operators on weighted Bloch space

We will characterize the boundedness and compactness of the composition operators on weighted Bloch space B _log = { f ? H(D): sup_{z ? D} (1-| z|²) ( log\frac21-| z|² )| f￠(z)| B_{ \log }= \{ f \in H(D): \sup_{z \in D } (1-\left| z\right|^2) \left( \log \frac{2}{1-\left| z\right|^2} \right)\left| f'(z)\right| < +￥} +\infty \} , where H(D) be the class of all analytic functions on D.     

Automorphism groups of rational elliptic surfaces with section and constant J-map

In this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have constant J-maps are classified. The ground field is ?. The automorphism group of such a surface β: B → ?¹, denoted by Au t(B), consists of all biholomorphic maps on the complex manifold B. The group Au t(B) is isomorphic to the semi-direct product MW(B) ? Aut _σ (B) of the Mordell-Weil groupMW(B) (the group of sections of B), and the subgroup Aut _σ (B) of the automorphisms preserving a fixed section σ of B which is called the zero section on B. The Mordell-Weil group MW(B) is determined by the configuration of singular fibers on the elliptic surface B due to Oguiso and Shioda [9]. In this work, the subgroup Aut _σ (B) is determined with respect to the configuration of singular fibers of B. Together with a previous paper [4] where the case with non-constant J-maps was considered, this completes the classification of automorphism groups of relatively minimal rational elliptic surfaces with section.     

Direct and inverse spectral problems for generalized strings

Let the functionQ be holomorphic in he upper half plane ⁺ and such that ImQ(z 0 and ImzQ(z) 0 ifz ⁺. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf + fdm = 0; f(0–) = 0. In this note we show that the set of functionsQ which are holomorphic in ⁺ and such that the kernel has negative squares of ⁺ and ImzQ(z) 0 ifz ⁺ is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf +f dm + ² fdD = 0 on [0,L),f(0–) = 0. Here is the number of pointsx whereD increases or 0 >m(x + 0) –m(x – 0) –; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.To the memory of M.G. Krein with deep gratitude and affection.This author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 09832     

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.     

A theorem of the Wiener—Tauberian type forL 1(H n)

The Heisenberg motion groupHM(n), which is a semi-direct product of the Heisenberg group Hⁿ and the unitary group U(n), acts on Hⁿ in a natural way. Here we prove a Wiener-Tauberian theorem for L¹ (Hⁿ) with this HM(n)-action on Hⁿ i.e. we give conditions on the “group theoretic” Fourier transform of a functionf in L¹ (Hⁿ) in order that the linear span of^gf : g∈HM(n) is dense in L¹(Hⁿ), where^gf(z, t) =f(g·(z, t)), forg ∈ HM(n), (z,t)∈Hⁿ.