Let ${\mathcal{P}}_{{\mathcal{H}}}^{0}(M)$ denote the class of normalised harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ satisfying $\text{Re}\,(zh^{\prime \prime }(z))>-M+|zg^{\prime \prime }(z)|$, where $h^{\prime }(0)-1=0=g^{\prime }(0)$ and $M>0$. Let ${\mathcal{B}}_{{\mathcal{H}}}^{0}(M)$ denote the class of sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ satisfying $|zh^{\prime \prime }(z)|\leq M-|zg^{\prime \prime }(z)|$, where $M>0$. We discuss the coefficient bound problem, the growth theorem for functions in the class ${\mathcal{P}}_{{\mathcal{H}}}^{0}(M)$ and a two-point distortion property for functions in the class ${\mathcal{B}}_{{\mathcal{H}}}^{0}(M)$.

Dorff et al. [‘Convolutions of harmonic convex mappings’, Complex Var. Elliptic Equ. 57(5) (2012), 489–503] formulated a question concerning the convolution of two right half-plane mappings, where the normalisation of the functions was considered incorrectly. In this paper, we reformulate the problem correctly and provide a solution to it in a more general form. We also obtain two new theorems which correct and improve related results.

We apply the Schwarz lemma to find general formulas for the third coefficient of Carathéodory functions dependent on a parameter in the closed unit polydisk. Next we find sharp estimates of the Hankel determinant $H_{2,2}$ and Zalcman functional $J_{2,3}$ over the class ${\mathcal{C}}{\mathcal{V}}$ of analytic functions $f$ normalised such that $\text{Re}\{(1-z^{2})f^{\prime }(z)\}>0$ for $z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, that is, the subclass of the class of functions convex in the direction of the imaginary axis.

Let $f$ be analytic in $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ and given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$. We give sharp bounds for the initial coefficients of the Taylor expansion of such functions in the class of strongly Ozaki close-to-convex functions, and of the initial coefficients of the inverse function, together with some growth estimates.

We prove the sharp inequality $|H_{3,1}(f)|\leq 4/135$ for convex functions, that is, for analytic functions $f$ with $a_{n}:=f^{(n)}(0)/n!,~n\in \mathbb{N}$, such that

$$\begin{eqnarray}Re\bigg\{1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg\}>0\quad \text{for}~z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\},\end{eqnarray}$$

$H_{3,1}(f)$

$$\begin{eqnarray}H_{3,1}(f):=\left|\begin{array}{@{}ccc@{}}a_{1} & a_{2} & a_{3}\\ a_{2} & a_{3} & a_{4}\\ a_{3} & a_{4} & a_{5}\end{array}\right|.\end{eqnarray}$$

Let ${\mathcal{S}}$ denote the class of analytic and univalent functions in $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ which are of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$. We determine sharp estimates for the Toeplitz determinants whose elements are the Taylor coefficients of functions in ${\mathcal{S}}$ and certain of its subclasses. We also discuss similar problems for typically real functions.

By considering the norm of a locally univalent function given by

we obtain such norm estimates for an operator of functions involving a convolution structure of convex univalent functions with a subclass of convex functions (defined by subordination). We also obtain some inequalities concerning this norm for functions under a certain fractional integral operator. Some implications of our results are briefly pointed out in the concluding section.

Let the function $f$ be analytic in $\mathbb{D}=\{z:|z|<1\}$ and given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$. For $0<\unicode[STIX]{x1D6FD}\leq 1$, denote by ${\mathcal{C}}(\unicode[STIX]{x1D6FD})$ the class of strongly convex functions. We give sharp bounds for the initial coefficients of the inverse function of $f\in {\mathcal{C}}(\unicode[STIX]{x1D6FD})$, showing that these estimates are the same as those for functions in ${\mathcal{C}}(\unicode[STIX]{x1D6FD})$, thus extending a classical result for convex functions. We also give invariance results for the second Hankel determinant $H_{2}=|a_{2}a_{4}-a_{3}^{2}|$, the first three coefficients of $\log (f(z)/z)$ and Fekete–Szegö theorems.

The logarithmic coefficients $\unicode[STIX]{x1D6FE}_{n}$ of an analytic and univalent function $f$ in the unit disc $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ with the normalisation $f(0)=0=f^{\prime }(0)-1$ are defined by $\log (f(z)/z)=2\sum _{n=1}^{\infty }\unicode[STIX]{x1D6FE}_{n}z^{n}$. In the present paper, we consider close-to-convex functions (with argument 0) with respect to odd starlike functions and determine the sharp upper bound of $|\unicode[STIX]{x1D6FE}_{n}|$, $n=1,2,3$, for such functions $f$.

For any real number ${\it\beta}$ with ${\it\beta}>1$, let ${\mathcal{M}}(\,{\it\beta})$ (${\mathcal{N}}(\,{\it\beta})$ respectively) denote the class of analytic functions $f$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ and satisfying $\text{Re}\,P_{f}<{\it\beta}$ ($\text{Re}\,Q_{f}<{\it\beta}$ respectively) in $\mathbb{D}$, where $P_{f}=zf^{\prime }(z)/f(z)$ and $Q_{f}=1+zf^{\prime \prime }(z)/f^{\prime }(z)$. Also, for ${\it\beta}>1$, let ${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$ (${\mathcal{N}}{\rm\Sigma}(\,{\it\beta})$ respectively) denote the class of analytic functions $g$ of the form $g(z)=z(1+\sum _{n=1}^{\infty }b_{n}z^{-n})$ and satisfying $\text{Re}\,P_{g}<{\it\beta}$ ($\text{Re}\,Q_{g}<{\it\beta}$ respectively) for $z\in {\rm\Delta}=\{z\in \mathbb{C}:1<|z|<\infty \}$. In this paper, we shall determine the coefficient bounds, inverse coefficient bounds, the growth and distortion theorem and the upper bounds for the Fekete–Szegő functional ${\rm\Lambda}_{{\it\lambda}}(f)=a_{3}-{\it\lambda}a_{2}^{2}$ for functions $f$ in the classes ${\mathcal{M}}(\,{\it\beta})$ and ${\mathcal{N}}(\,{\it\beta})$. Further, we shall solve the maximal area problem for functions of the type $z/f(z)$ when $f\in {\mathcal{M}}(\,{\it\beta})$, which is Yamashita’s conjecture for the class ${\mathcal{M}}(\,{\it\beta})$. We shall obtain the radius of convexity for the class ${\mathcal{N}}(\,{\it\beta})$. We shall also determine the coefficient bounds for functions $g$ in the classes ${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$ and ${\mathcal{N}}{\rm\Sigma}(\,{\it\beta})$ and the inverse coefficient bounds for functions $g$ in the class ${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$. All the results are sharp.

For a normalized analytic function $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, the estimate of the integral means

$$\begin{eqnarray}L_{1}(r,f):=\frac{r^{2}}{2{\it\pi}}\int _{-{\it\pi}}^{{\it\pi}}\frac{d{\it\theta}}{|f(re^{i{\it\theta}})|^{2}}\end{eqnarray}$$

$f(z)$

$\mathbb{D}\setminus \{0\}$

${\rm\Delta}(r,f)$

$\mathbb{D}_{r}:=\{z\in \mathbb{C}:|z|<r\}$

$f$

$0<r\leq 1$

$L_{1}(r,f)$

${\rm\Delta}(r,z/f)$

$r$

$f$

$m$

We consider a recent work of Pascu and Pascu [‘Neighbourhoods of univalent functions’, Bull. Aust. Math. Soc. 83(2) (2011), 210–219] and rectify an error that appears in their work. In addition, we study certain analogous results for sense-preserving harmonic mappings in the unit disc $\vert z\vert \lt 1$. As a corollary to this result, we derive a coefficient condition for a sense-preserving harmonic mapping to be univalent in $\vert z\vert \lt 1$.

Extreme points of compact, convex integral families of analytic functions are investigated. Knowledge about extreme points provides a valuable tool in the optimization of linear extremal problems. The functions studied are determined by a two-parameter collection of kernel functions integrated against measures on the torus. For specific choices of the parameters many families from classical geometric function theory are included. These families include the closed convex hull of the derivatives of normalized close-to-convex functions, the ratio of starlike functions of different orders, as well as many others. The main result introduces a surprising new class of extreme points.

We study some classes of planar harmonic mappings produced with the shear construction devised by Clunie and Sheil-Small in 1984. The first section reviews the basic concepts and describes the shear construction. The main body of the paper deals with the geometry of the classes constructed.

Let $\varOmega$ be a bounded, simply connected domain in $\mathbb{C}$ with $0\in\varOmega$ and $\partial\varOmega$ analytic. Let $S(\varOmega)$ denote the class of functions $F(z)$ which are analytic and univalent in $\varOmega$ with $F(0)=0$ and $F'(0)=1$. Let $\{\varPhi_{n}(z)\}_{n=0}^{\infty}$ be the Faber polynomials associated with $\varOmega$. If $F(z)\in S(\varOmega)$, then $F(z)$ can be expanded in a series of the form

$$ F(z)=\sum_{n=0}^{\infty}A_{n}\varPhi_{n}(z),\quad z\in\varOmega, $$

in terms of the Faber polynomials. Let

$$ E_{r}=\bigg\{(x,y)\in\mathbb{R}^{2}:\frac{x^{2}}{(1+(1/r^{2}))^{2}}+\frac{y^{2}}{(1-(1/r^{2}))^{2}}\lt1\bigg\}, $$

where $r\gt1$.

In this paper, we obtain sharp bounds for certain linear combinations of the Faber coefficients of functions $F(z)$ in $S(E_{r})$ and in certain related classes.

A sharp norm estimate will be given to the pre-Schwarzian derivatives of close-to-convex functions of specified type. In order to show the sharpness, we introduce a kind of maximal operator which may be of independent interest. We also discuss a relation between the subclasses of close-to-convex functions and the Hardy spaces.

We develop sharp conditions for various types of starlikeness for functions analytic in the unit disk with bounded derivatives. We also describe the precise range {zf′(z)/f(z): z ∈ D, f ∈ }, where f ∈ means f (0) = 0, f′(0) = 1, and |f′(z) - 1 |< ≦ λ in the unit disc D, and draw some cnoslusions from that.