On integral means of star-like functions

We study univalent holomorphic functions in the unit diskU = {z: |z| < 1} of the formf(z)=z+∑ _n=2 ^∞ a _n z ⁿ that satisfy the condition Re zf’(z)/f(z) > α with α∈ [0, 1) inU. Some integral means of such funcions are estimated.     

On generalized shift bases for the wiener disc algebra

LetW(D) denote the set of functionsf(z)=Σ _n=0 ^∞ A _n Z ⁿ a _nzⁿ for which Σ_n=0 ^∞|a _n |<+∞. Given any finite set lcub;f _i (z)rcub; _i=1 ⁿ inW(D) the following are equivalent: (i) The generalized shift sequence lcub;f ₁(z)z ^kn ,f ₂(z)z ^kn+1, …,f _n (z)z ^(k+1)n−1rcub; _k=0 ^∞ is a basis forW(D) which is equivalent to the basis lcub;z ^m rcub; _m=0 ^∞ . (ii) The generalized shift sequence is complete inW(D), (iii) The function has no zero in |z|≦1, wherew=e ^2πiti /n.     

A unified approach for univalent functions with negative coefficients using the Hadamard product

For given analytic functions ϕ(z) = z + Σ _n=2^∞ λ _n z ⁿ , Ψ(z) = z + Σ _n=2^∞ μ with λ _n ≥ 0, μ _n ≥ 0, and λ _n ≥ μ _n and for α, β (0≤α<1, 0<β≤1), let E(φ,ψ; α, β) be of analytic functions ƒ(z) = z + Σ _n=2^∞ a _n z ⁿ in U such that f(z)*ψ(z)≠0 and

The region of values of the system {ƒ(z1), …, ƒ(zn)} in the class of typically real functions. II

The paper studies the region of values D_m,1(T) of the system {ƒ(z₁), ƒ(z₂), …, ƒ(zm), ƒ(r)}, m e 1, where z_j (j = 1, 2, …,m) are arbitrary fixed points of the disk U = {z: |z| < 1} with Im zj ≠ 0 (j = 1, 2, …,m), and r, 0 < r < 1, is fixed, in the class T of functions ƒ(z) = z+a₂z²+ ⋯ regular in the disk U and satisfying in the latter the condition Im ƒ(z) Imz > 0 for Im z ≠ 0. An algebraic characterization of the set D_m,1(T) in terms of nonnegative-definite Hermitian forms is given, and all the boundary functions are described. As an implication, the region of values of ƒ(z_m) in the subclass of functions from the class T with prescribed values ƒ(z_k) (k = 1, 2, …,m − 1) and ƒ(r) is determined. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 24–33. Original article submitted June 13, 2005.     

Characterization and the pre-Schwarzian norm estimate for concave univalent functions

Let Co(α) denote the class of concave univalent functions in the unit disk \mathbbD{\mathbb{D}}. Each function f ? Co(a){f\in Co(\alpha)} maps the unit disk \mathbbD{\mathbb{D}} onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional (1-|z|²)( f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional (1-|z|²)(f^￠￠(z)/f^￠(z)), f ? Co(a){(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)} whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in Co(α) belong to the H ^p space for p < 1/α.     

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.     

The boundary distortion and extremal problems in certain classes of univalent functions

We consider the class S₁(τ), 0<τ<1, of functions f(z)=rz+a₂z²+... that are regular and univalent in the unit disk U and have |f(z)|<1. We obtain sharp estimates for the 1-measure of the sets {θ: |f(e^iθ)|=1}. As a corollary, for the familiar class S we find Kolmogorov-type estimates for the sets {θ: |f(e^iθ)|>M}, M>1, and prove inequalities for the harmonic measure, which are similar to those by Carleman-Milloux and Baernstein. We also consider problems on distortion of fixed systems of boundary arcs in the classes of functions that are regular (or meromorphic) and univalent in the disk or circular annulus. Bibliography: 23 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 115–142. Translated by A. Yu. solynin.     

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.     

The sum of coefficients of bounded univalent functions

We solve the maximal value problem for the functional in the class of functionsf(z)=z+a ₂z²+… that are holomorphic and univalent in the unit disk and satisfy the inequality |f(z)|<M. We prove that the Pick functions are extremal for this problem for sufficiently largeM whenever the set of indicesk ₁,…,k_m contains an even number. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 728–733, May, 1997. Translated by S. S. Anisov     

Differential subordinations concerning starlike functions

Denote byS ^* (⌕), (0≤⌕<1), the family consisting of functionsf(z)=z+a ₂z²+...+a_nzⁿ+... that are analytic and starlike of order ⌕, in the unit disc ⋎z⋎<1. In the present article among other things, with very simple conditions on μ, ⌕ andh(z) we prove the f’(z) (f(z)/z)^μ−1<h(z) implies f∈S^*(⌕). Our results in this direction then admit new applications in the study of univalent functions. In many cases these results considerably extend the earlier works of Miller and Mocanu [6] and others.     

The region of values of the system { f ( z 1),..., f ( z n )} in the class of typically real functions. III

The paper studies the region of values D_m,n(T) of the system {f(z₁), f(z₂),..., f(z_m), f(r₁), f(r₂),..., f(r_n)}, where m ≥ 1; n > 1; z_j, j = 1, ... m, are arbitrary fixed points of the disk U = {z: |z| < 1} with Im z_j ≠ 0, j = 1, 2, ..., m; r_j, 0 < r_j < 1, j = 1, 2, ..., n, are fixed; f ∈ T, and the class T consists of functions f(z) = z + c₂z² + ... regular in the disk U and satisfying the condition Im f(z) · Im z > 0 for Im z ≠= 0, z ∈ U. An algebraic characterization of the set D_m,n(T) in terms of nonnegative-definite Hermitian forms is provided, and all the boundary functions are described. As an implication, the region of values of f(z₁) in the subclass of functions f ∈ T with prescribed values f(r_j) (j = 1, 2, 3) is determined. Bibliography: 12 titles. Dedicated to the 100th anniversary of my father’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 23–34.     

Remarks on the descriptive metric characterization of singular sets of analytic functions

This work presents two remarks on the structure of singular boundary sets of functions analytic in the unit diskD: |z|<1. The first remark concerns the conversion of the Plessner theorem. We prove that three pairwise disjoint subsetsE ₁,E ₂, andE ₃ of the unit circle Γ: |z|=1, = Γ, are the setsI(ƒ) of all Plessner points,F(ƒ) of all Fatou points, andE(ƒ) of all exceptional boundary points, respectively, for a function ƒ holomorphic inD if and only ifE ₁ is aG _δ-set andE ₃ is a -set of linear measure zero. In the second part of the paper it is shown that for any -subsetE of the unit circle Γ with a zero logarithmic capacity there exists a one-sheeted function onD whose angular limits do not exist at the points ofE and do exist at all the other points of Γ. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 56–61, January, 1998.     

Coefficients of nonvanishing functions inH ∞

LesB denote the class of functions analytic in the unit disc ofC which satisfy 0<|f(z)|<1. It is proved that there exists a numberc<1 such that iff∈B and iff(z)=Σ _n=0 ^∞ a _n z ⁿ , then |a _n |<c forn>=1.     

On Classes of Functions Related to Starlike Functions with Respect to Symmetric Conjugate Points Defined by a Fractional Differential Operator

Let A denote the class of analytic functions f, in the open unit disk E = {z : |z| < 1}, normalized by f(0) = f′(0) − 1 = 0. In this paper, we introduce and study the class ST^n,a_l,m(h){ST^{n,\alpha}_{\lambda,m}(h)} of functions f ? A{f\in A}, with \fracD^n,a_l f_m(z)z 1 0{\frac{D^{n,\alpha}_\lambda f_m(z)}{z}\neq 0}, satisfying

On Bernstein–Walsh-type lemmas in regions of the complex plane

Let G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to [`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A _p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition

Bounded starlike functions of complex order

LetF(b, M) (b ≠ 0 complex,M>1/2) denote the class of functionsf(z) =z + Σ _n=2 ^∞ a _n z ⁿ analytic in U={z:|z|<1} which satisfy for fixedM, f(z)/z ≠ 0 inU and \(\left| {\frac{{b - 1 + \left[ {zf'{{\left( z \right)} \mathord{\left/ {\vphantom {{\left( z \right)} {f\left( z \right)}}} \right. \kern-0em} {f\left( z \right)}}} \right]}}{b} - M} \right|< M, z \in U\) . In this note we obtain various representations for functions inF(b, M). We maximize |a₃=μa ₂ ² | over the classF(b, M). Also sharp coefficient bounds are established for functions inF(b, M). We also obtain the sharp radius of starlikeness of the classF(b, M).     

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}.      18. Sull'andamento delle funzioni maggioranti delle serie di potenze      Giovanni Ricci 《Annali di Matematica Pura ed Applicata》1955,40(1):285-306 Sunto Si considerano le serie di potenze f(z) = ∑a n zn convergenti (almeno) per | z |<1 e si studia l'andamento della funzione maggiorante ∑ | an | ru, ponendolo in relazione con quello della media quadratica del modulo { ∑ | an |2r2n}1/2 (che in ogni caso è minore del massimo modulo . Si stabiliscono le ? migliori costanti ? soltanto per qualcuno dei quesiti. Alla fine si esamina il caso in cui {∑ | an |2r2n} è convergente con ipotesi che tengono conto delle funzioni ? lentamente oscillanti ?. A Mauro Picone nel suo 70mo compleanno.      19. Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration      Miran Černe 《Arkiv f?r Matematik》2002,40(1):27-45 LetX(-ϱB m ×C n be a compact set over the unit sphere ϱB m such that for eachz∈ϱB m the fiberX z ={ω∈C n ;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain inC n . The polynomial hull ofX is described in terms of the Perron-Bremermann function for the homogeneous defining function ofX. Moreover, for each point (z 0,w 0)∈Int there exists a smooth up to the boundary analytic discF:Δ→B m ×C n with the boundary inX such thatF(0)=(z 0,w 0). This work was supported in part by a grant from the Ministry of Science of the Republic of Slovenia.      20. On power deformations of univalent functions      Yong Chan Kim  Toshiyuki Sugawa 《Monatshefte für Mathematik》2012,23(10):231-240 For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) − 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f c (z) = z(f (z)/z) c for a complex number c. We determine those values c for which the operator maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.

Sull'andamento delle funzioni maggioranti delle serie di potenze

Sunto Si considerano le serie di potenze f(z) = ∑a _n zⁿ convergenti (almeno) per | z |<1 e si studia l'andamento della funzione maggiorante ∑ | a_n | r^u, ponendolo in relazione con quello della media quadratica del modulo { ∑ | a_n |²r²ⁿ}^1/2 (che in ogni caso è minore del massimo modulo . Si stabiliscono le ? migliori costanti ? soltanto per qualcuno dei quesiti. Alla fine si esamina il caso in cui {∑ | a_n |²r²ⁿ} è convergente con ipotesi che tengono conto delle funzioni ? lentamente oscillanti ?. A Mauro Picone nel suo 70^mo compleanno.     

Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration

LetX(-ϱB ^m ×C ⁿ be a compact set over the unit sphere ϱB ^m such that for eachz∈ϱB ^m the fiberX _z ={ω∈C ⁿ ;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain inC ⁿ . The polynomial hull ofX is described in terms of the Perron-Bremermann function for the homogeneous defining function ofX. Moreover, for each point (z ₀,w ₀)∈Int there exists a smooth up to the boundary analytic discF:Δ→B ^m ×C ⁿ with the boundary inX such thatF(0)=(z ₀,w ₀). This work was supported in part by a grant from the Ministry of Science of the Republic of Slovenia.     

On power deformations of univalent functions

For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) − 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f _c (z) = z(f (z)/z) ^c for a complex number c. We determine those values c for which the operator maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.