The sharp bound for the third Hankel determinant for the coefficients of the inverse function of convex functions is obtained, thus answering a recent conjecture concerning invariance of coefficient functionals for convex functions.

Let $ \mathcal {B} $ be the class of analytic functions $ f $ in the unit disk $ \mathbb {D}=\{z\in \mathbb {C} : |z|<1\} $ such that $ |f(z)|<1 $ for all $ z\in \mathbb {D} $. If $ f\in \mathcal {B} $ of the form $ f(z)=\sum _{n=0}^{\infty }a_nz^n $, then $ \sum _{n=0}^{\infty }|a_nz^n|\leq 1 $ for $ |z|=r\leq 1/3 $ and $ 1/3 $ cannot be improved. This inequality is called Bohr inequality and the quantity $ 1/3 $ is called Bohr radius. If $ f\in \mathcal {B} $ of the form $ f(z)=\sum _{n=0}^{\infty }a_nz^n $, then $ |\sum _{n=0}^{N}a_nz^n|<1\;\; \mbox {for}\;\; |z|<{1}/{2} $ and the radius $ 1/2 $ is the best possible for the class $ \mathcal {B} $. This inequality is called Bohr–Rogosinski inequality and the corresponding radius is called Bohr–Rogosinski radius. Let $ \mathcal {H} $ be the class of all complex-valued harmonic functions $ f=h+\bar {g} $ defined on the unit disk $ \mathbb {D} $, where $ h $ and $ g $ are analytic in $ \mathbb {D} $ with the normalization $ h(0)=h^{\prime }(0)-1=0 $ and $ g(0)=0 $. Let $ \mathcal {H}_0=\{f=h+\bar {g}\in \mathcal {H} : g^{\prime }(0)=0\}. $ For $ \alpha \geq 0 $ and $ 0\leq \beta <1 $, let

$$ \begin{align*} \mathcal{W}^{0}_{\mathcal{H}}(\alpha, \beta)=\{f=h+\overline{g}\in\mathcal{H}_{0} : \mathrm{Re}\left(h^{\prime}(z)+\alpha zh^{\prime\prime}(z)-\beta\right)>|g^{\prime}(z)+\alpha zg^{\prime\prime}(z)|,\;\; z\in\mathbb{D}\} \end{align*} $$

be a class of close-to-convex harmonic mappings in $ \mathbb {D} $. In this paper, we prove the sharp Bohr–Rogosinski radius for the class $ \mathcal {W}^{0}_{\mathcal {H}}(\alpha , \beta ) $.

A subset E of a metric space X is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $\mathbb{R}^n$ for some n, sending E to a starlike set. A subset $E\subset X$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets $\{E_i\}_{i=0}^{N+1}$ such that $E_{i}/E_{i+1}\subset X/E_{i+1}$ is starlike-equivalent for each i and $E_{N+1}$ is a point. A decomposition $\mathcal{D}$ of a metric space X is said to be recursively starlike-equivalent, if there exists $N\geq 0$ such that each element of $\mathcal{D}$ is recursively starlike-equivalent of filtration length N. We prove that any null, recursively starlike-equivalent decomposition $\mathcal{D}$ of a compact metric space X shrinks, that is, the quotient map $X\to X/\mathcal{D}$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman–Starbird and Freedman and is applicable to the proof of Freedman’s celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological 4-manifolds, including the four-dimensional Poincaré conjecture.

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 ${\mathcal{B}}_{{\mathcal{H}}}^{0}(M)$.

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

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

$f(z)$

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

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

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

$f$

$0<r\leq 1$

$L_{1}(r,f)$

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

$r$

$f$

$m$

Let ${\mathcal{S}}$ denote the set of all univalent analytic functions $f$ of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ on the unit disk $|z|<1$. In 1946, Friedman [‘Two theorems on Schlicht functions’, Duke Math. J.13 (1946), 171–177] found that the set ${\mathcal{S}}_{\mathbb{Z}}$ of those functions in ${\mathcal{S}}$ which have integer coefficients consists of only nine functions. In a recent paper, Hiranuma and Sugawa [‘Univalent functions with half-integer coefficients’, Comput. Methods Funct. Theory13(1) (2013), 133–151] proved that the similar set obtained for functions with half-integer coefficients consists of only 21 functions; that is, 12 more functions in addition to these nine functions of Friedman from the set ${\mathcal{S}}_{\mathbb{Z}}$. In this paper, we determine the class of all normalized sense-preserving univalent harmonic mappings $f$ on the unit disk with half-integer coefficients for the analytic and co-analytic parts of $f$. It is surprising to see that there are only 27 functions out of which only six functions in this class are not conformal. This settles the recent conjecture of the authors. We also prove a general result, which leads to a new conjecture.

Let S(m, M) be the set of functions regular and satisfying │zf′(z)/f(z) – m│< M in │z│ <1, where│m –│ <M;≦ m; and let S*(p) be the set of starlike functions of order p, 0≦ p <1. In this paper we obtain integral operators which map S(m, M) into S(mM) and S* (p) × S(mM) into S*(p). Our results improve and generalize many recent results.