Bohr phenomenon for subordinating families of certain univalent functions

On the Fekete–Szegö problem for concave univalent functions

Let $Co(\alpha)$ denote the class of concave univalent functions in the unit disk $\ID$. Each function $f\in Co(\alpha)$ maps the unit disk $\ID$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional $(1-|z|^2)\left ( f''(z)/f'(z)\right)$, $f\in Co(\alpha)$. 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 $(1-|z|^2)\left(f''(z)/f'(z)\right)$, $f\in Co(\alpha)$ 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 $Co(\alpha)$ belong to the $H^p$ space for $p<1/\alpha$.

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 ${\mathbb{D}}$. Each function ${f\in Co(\alpha)}$ maps the unit disk ${\mathbb{D}}$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional ${(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}$. 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 ${(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}$ 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 Hp space for p < 1/α.

We consider functions that map the open unit disc conformally onto the complement of an unbounded convex set with opening angle pa, a ? (1, 2], at infinity. In this paper, we show that every such function is close-to-convex of order (a - 1) and is included in the set of univalent functions of bounded boundary rotation. Many interesting consequences of this result are obtained. We also determine the extreme points of the set of concave functions with respect to the linear structure of the Hornich space.

On concave univalent functions

Conformal maps of the unit disk onto convex domains are a classical topic. Recently Avkhadiev and Wirths discovered that conformal maps onto concave domains (the complements of convex closed sets) have some novel properties, namely that there are non-trivial lower estimates, a rare thing for univalent functions (Avkhadiev, F.G. and Wirths, K.-J., 2002, Convex holes produce lower bounds for coefficients, Complex Variables, Theory and Application, 47, 556–563). We prove an inequality involving two variables and introduce a lower order for concave functions. §Dedicated to Professor Peter L. Duren on the occasion of his 70th birthday.

Lorena Cruz

Papers

On concave univalent functions

Strongly hyperbolically convex functions