Convex Univalent Research Articles

Univalent Functions, VMOA and Related Spaces

This paper is concerned mainly with the logarithmic Bloch space ℬlog which consists of those functions f which are analytic in the unit disc ${\mathbb{D}}$ and satisfy $\sup_{\vert z\vert <1}(1-\vert z\vert )\log\frac{1}{1-\vert z\vert}\vert f^{\prime}(z)\vert <\infty $ , and the analytic Besov spaces B p , 1≤p<∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We give explicit examples of: We also prove that the situation changes for certain subclasses of univalent functions. Namely, we prove that the convex univalent functions in ${\mathbb{D}}$ which belong to any of the spaces ℬ0, VMOA, B p (1≤p<∞), ℬlog , or some other related spaces are the same, the bounded ones. We also consider the question of when the logarithm of the derivative, log g′, of a univalent function g belongs to Besov spaces. We prove that no condition on the growth of the Schwarzian derivative Sg of g guarantees log g′∈B p . On the other hand, we prove that the condition $\int _{{\mathbb{D}}}(1-\vert z\vert^{2})^{2p-2}\vert \mathit{Sg}(z)\vert ^{p}\,dA(z)<\infty $ implies that log g′∈B p and that this condition is sharp. We also study the question of finding geometric conditions on the image domain $g({\mathbb{D}})$ which imply that log g′ lies in B p . First, we observe that the condition of $g({\mathbb{D}})$ being a convex Jordan domain does not imply this. On the other hand, we extend results of Pommerenke and Warschawski, obtaining for every p∈(1,∞), a sharp condition on the smoothness of a Jordan curve Γ which implies that if g is a conformal mapping from ${\mathbb{D}}$ onto the inner domain of Γ, then log g′∈B p .

Region of Variability for Exponentially Convex Univalent Functions

For $\alpha\in\IC\setminus \{0\}$ let $\mathcal{E}(\alpha)$ denote the class of all univalent functions $f$ in the unit disk $\mathbb{D}$ and is given by $f(z)=z+a_2z^2+a_3z^3+\cdots$, satisfying $$ {\rm Re\,} \left (1+ \frac{zf''(z)}{f'(z)}+\alpha zf'(z)\right)>0 \quad {in ${\mathbb D}$}. $$ For any fixed $z_0$ in the unit disk $\mathbb{D}$ and $\lambda\in\overline{\mathbb{D}}$, we determine the region of variability $V(z_0,\lambda)$ for $\log f'(z_0)+\alpha f(z_0)$ when $f$ ranges over the class $$\mathcal{F}_{\alpha}(\lambda)=\left\{f\in\mathcal{E}(\alpha) \colon f''(0)=2\lambda-\alpha %\quad{and} f'''(0)=2[(1-|\lambda|^2)a+ %(\lambda-\alpha)^2 -\lambda\alpha] \right\}. $$ We geometrically illustrate the region of variability $V(z_0,\lambda)$ for several sets of parameters using Mathematica. In the final section of this article we propose some open problems.

Variability Regions for Families of Convex Functions

Let Ω be a simply connected domain contained in the right half plane, i.e. Ω ⊂ {w ∈ ℂ: Rew > 0}, and satisfying 1 ∈ Ω. Let P Ω be the conformal mapping of the unit disk \(\mathbb{D}\) onto Ω with P Ω(0) = 1 and P′Ω(0) > 0. Define \(\mathcal{C}V_\Omega\) to be the class of analytic functions f in \(\mathbb{D}\) such that $$1 + \frac{{zf''(z)}} {{f'(z)}} \in \Omega for all z \in \mathbb{D} and f(0) = f'(0) - 1 = 0.$$ Then \(\mathcal{C}V_\Omega\) is a subclass of the normalized convex univalent functions in \(\mathbb{D}\). If Ω is starlike with respect to 1 and $$Re\left( {P\Omega (z) - 1 + \frac{{zP'_\Omega (z)}} {{P_\Omega (z) - 1}}} \right) > 0$$ in \(\mathbb{D}\), then we can determine the variability region \(\left\{ {f\left( {z_0 } \right):f \in \mathcal{C}\mathcal{V}_\Omega } \right\}\). As an application we shall show a subordination result and determine variability regions for uniformly convex functions.

New results related to the starlikeness of Bernardi integral operator

In Lewandowski, Miller and Zlotkiewicz, Generating functions for some classes of univalent functions, Proc. Amer. Math. Soc., 56 (1976), pp. 111–117 and Pascu, Alpha-close-to-convex functions, Romanian Finish Seminar on Complex Analysis, Springer, Berlin, 1979, pp. 331–335 it has been proved that the integral operator defined by Bernardi in Convex and starlike univalent functions, Trans. Amer. Math. Soc., 135 (1969), pp. 429–446 given by preserves certain classes of univalent functions such as the class of starlike functions, the class of convex functions and the class of close-to-convex functions. In this article, we determine conditions that a function f ∈ A needs to satisfy such that the function F given by (1) is starlike.

Region of variability for close-to-convex functions-II

For a complex number α with Re α > 0 let K ϕ ( α ) be the class of analytic functions f in the unit disk D with f ( 0 ) = 0 satisfying Re ( f ′ ( z ) / ϕ ′ ( z ) ) > 0 in D , f ′ ( 0 ) / ϕ ′ ( 0 ) = α , for some convex univalent function ϕ in D . For any fixed z 0 ∈ D , and λ ∈ D ¯ we shall determine the region of variability V ϕ ( z 0 , α , λ ) for f ( z 0 ) when f ranges over the class K ϕ ( α , λ ) = f ∈ K ϕ ( α ) : d dz f ′ ( z ) ϕ ′ ( z ) z = 0 = 2 λ ( Re α ) . In the final section we graphically illustrate the region of variability for several sets of parameters z 0 and α .

Weak subordination for convex univalent harmonic functions

For two complex-valued harmonic functions f and F defined in the open unit disk Δ with f ( 0 ) = F ( 0 ) = 0 , we say f is weakly subordinate to F if f ( Δ ) ⊂ F ( Δ ) . Furthermore, if we let E be a possibly infinite interval, a function f : Δ × E → C with f ( ⋅ , t ) harmonic in Δ and f ( 0 , t ) = 0 for each t ∈ E is said to be a weak subordination chain if f ( Δ , t 1 ) ⊂ f ( Δ , t 2 ) whenever t 1 , t 2 ∈ E and t 1 < t 2 . In this paper, we construct a weak subordination chain of convex univalent harmonic functions using a harmonic de la Vallée Poussin mean and a modified form of Pommerenke's criterion for a subordination chain of analytic functions.

Region of variability for close-to-convex functions

For a complex number α with Re α > 0 let 𝒦 φ (α) be the class of analytic functions f in the unit disk 𝔻 with f (0) = 0 satisfying Re(f ′(z)/φ′(z)) > 0 in 𝔻, f ′(0)/φ′(0) = α, for some convex univalent function φ in 𝔻. For any fixed z 0 ∈ 𝔻 we shall determine the region of variability for f (z 0) when f ranges over the class 𝒦 φ (α). As corollaries, we present a number of its consequences.

Norm estimates for the Bernardi integral transforms of functions defined by subordination

In this paper, we obtain sharp norm estimates for the Bernardi integral transform of functions belonging to the class ${\mathcal K}(A,B)$, $-1\le B<A\le 1$, which is a subclass of the well-known class of convex univalent functions. As a consequence, a number of open questions arise naturally, concerning ${\mathcal S}^*(A,B)$--a subclass of the well-known class of starlike univalent functions, and many other classes.

Inclusion relations and convolution properties of a certain class of analytic functions associated with the Ruscheweyh derivatives

Let A denote the class of functions f ( z ) with f ( 0 ) = f ′ ( 0 ) − 1 = 0 , which are analytic in the open unit disk U . By means of the Ruscheweyh derivatives, we introduce and investigate the various properties and characteristics of a certain two-parameter subclass T ( α , λ ; h ) of A , where α ≧ 0 , λ > − 1 , and h ( z ) is analytic and convex univalent in U with h ( 0 ) = 1 . In particular, some inclusion relations and convolution properties for the function class T ( α , λ ; h ) are presented here.

Horocyclically convex univalent functions

(i) C is a maximal curve of zero curvature; (ii) C is a noncompact maximal curve of constant curvature. Now we turn to the hyperbolic metric ds = (1 − |z|2)−1|dz| in the unit disk D. Let C : w(s), s ∈ I, be a curve of class C 2 in D that is parameterized by the hyperbolic arc-length s; that is, |w ′| = 1− |w|2, w ′′ is continuous in I. (1.1) Then the hyperbolic (= geodesic) curvature κ satisfies the differential equation w ′′ w ′ + 2ww ′ 1− |w|2 = iκ. (1.2)

Convolution and subordination in the convex hull of convex mappings

In this work we give an extension of the Ruscheweyh and Stankiewicz theorem [S. Ruscheweyh, J. Stankiewicz, Subordination under convex univalent functions, Bull. Polish Acad. Sci. Math. 33 (1985) 499–502] on the subordination under convex functions in the unit disc Δ = { z : | z | < 1 } . We prove that if f ≺ F ∈ c o ¯ K and g ≺ G ∈ c o ¯ K , then f ⋆ g ≺ F ⋆ G where ≺ denotes the subordination, ⋆ denotes the Hadamard product and c o ¯ K is the closed convex hull of the class of convex functions.

On Convex Univalent Functions with Convex Univalent Derivatives

On Convex Univalent Functions with Convex Univalent Derivatives

Minimal graphs in ℝ³ over convex domains

Krust established that all conjugate and associate surfaces of a minimal graph over a convex domain are also graphs. Using a convolution theorem from the theory of harmonic univalent mappings, we generalize Krust’s theorem to include the family of convolution surfaces which are generated by taking the Hadamard product or convolution of mappings. Since this convolution involves convex univalent analytic mappings, this family of convolution surfaces is much larger than just the family of associated surfaces. Also, this generalization guarantees that all the resulting surfaces are over close-to-convex domains. In particular, all the associate surfaces and certain Goursat transformation surfaces of a minimal graph over a convex domain are over close-to-convex domains.

On the univalence of an integral on a subclass of meromorphic convex univalent functions

A nonlinear integral operator is studied on the class of convex meromorphic functions in the exterior of the unit disk. In this paper, we improve a sufficient condition for univalency of the operator obtained earlier by the first author.

Certain convex harmonic functions

We define and investigate a family of complex‐valued harmonic convex univalent functions related to uniformly convex analytic functions. We obtain coefficient bounds, extreme points, distortion theorems, convolution and convex combinations for this family.

On hyperbolically convex functions

Let\(\mathbb{D}\) be the unit disk of the complex plane. A conformai map of\(\mathbb{D}\) into itself is called hyperbolically convex if the non-Euclidean segment between any two points of\(f(\mathbb{D})\) also belongs to\(f(\mathbb{D})\). In this paper we prove several inequalities that are analogous to inequalities about (Euclidean) convex univalent functions. We show that if ƒ (0) = 0, then Re zf′/f > 1/2. This inequality is the key for the results of this paper. In particular we deduce a three-variable inequality corresponding to that of Ruscheweyh and Sheil-Small. The sharp bound for the Schwarzian derivative remains open.

On spherically convex univalent functions.

On spherically convex univalent functions.

Exponentially convex univalent functions

In this article we introduce the tamily ε(α), α ∈ C. of α-exponentially convex functions f as normalized univalent functions in the unit disk D such that e αf is convex univalent. Out study was suggested by a formal and concrete relationship between ε(α) and the class of convex functions of bounded type, defined by Professor A. W. Goodman. We also exhibit the connections to the class C(α) of normalized convex univalent functions which omit the balue -1/α. C-C(0) and to the class CH(α) of normalized convex univalent functions whose ranges lie mside a horizontal strip of width φ/α. Beside the description of some basic geometrical and analytic properties of the functions in ε(α). we are mainly interested in the determination of various Koebe domains or Koebe radii. We solve (up to certain constrained minimizations the Koebe radius and Koebe domain problem for C∩ε(α) and CH(α), and the Koebe domain problems for C(α) and ε(α) Because of technical difficulties this cannot be used to obtain the sharp (explicit) Koebe radius for ε(α). We have, however, complete numerical results. At least we can show that the Koebe radius for C∩ε(α) is larger than the one for ε(α), for small α

Two-Point Distortion Theorems for Strongly Close-to-Convex Functions

A holomorphic function f defined on the unit disk is called strongly close-to-convex of order α > 0 if there is a convex univalent function φ defined on such that For α ε (0.1) this condition f is univalent in . By convention a convex univalent function is called close-to-convex of order 0. If f is close-to-convex of order α ε [0, 1 ] and , then where and is the hyperbolic distance between a and b. Equality holds for distinct if and only if where S is a conformal automorphism of automorphism , T is a conformal automorphism of ; and .

Coefficient Inequalities for Strongly Close-to-Convex Functions

Letf(z)=z+a2z2+a3z3+··· be a normalized strongly close-to-convex function of order α>0 defined on the unit disk D . This means that there is a normalized convex univalent function ϕ and β∈ R such that[formula]forz∈ D . Then|a3−a22|+13|ax|2≤13(1+4α+2α2)and|a3−23a22+23|a2|2≤13(3+4α+2α2)with equality if and only iffis a rotation of[formula]