Rosihan M. Ali’s research while affiliated with University of Science Malaysia and other places

The family $\mathcal {S}$ consists of functions that are analytic and univalent in the unit disc $\mathbb {D}$ and of the form $f(z)= z+\sum _{n=2}^\infty a_n z^n$. By the logarithmic coefficients of f in $\mathcal {S}$, one means the coefficients of the expansion $\log (f(z)/z)=2\sum _{n=1}^\infty \gamma _nz^n,\, z\in {\mathbb D}.$ I. M. Milin proposed a system of inequalities for the logarithmic coefficients of the family $\mathcal {S}$. Among those inequalities, one that is well-known as the Milin conjecture, was the key result in proving the Bieberbach conjecture by L. de Branges in 1984. In this study, we establish the logarithmic coefficient inequalities for a general family of starlike functions which are described by a subordination relation. Then, several special cases are deduced, which include one that corrects an earlier published result.

A normalized analytic function f is parabolic starlike if w(z) := zf′ (z)/f(z) maps the unit disk into the parabolic region {w : Re w > |w − 1|}. Sharp estimates on the third Hermitian-Toeplitz determinant are obtained for parabolic starlike functions. In addition, upper bounds on the third Hankel determinants are also determined.

Let ${\mathcal {A}}$ denote the class of all functions f analytic in the unit disc ${\mathbb {D}}:= \{z \in {\mathbb {C}}: |z| < 1\}$ with the normalization $f(0) = 0 = f'(0) - 1.$ For $\lambda \in (0,1]$ and complex $\mu ,$ a subclass of functions f in ${\mathcal {A}}$ satisfying $|f'(z) (z/f(z))^2 - \mu | < \lambda$ on ${\mathbb {D}}$ is considered. The conditions on $\lambda$ and $\mu$ such that the functions in this subclass to be univalent are given. It is shown that the subclass is preserved under rotation, dilation and conjugation. A necessary and sufficient condition for analytic functions to be in this subclass is given. With this, the sharp second coefficient bound and the growth theorem are determined.

The function Gα(z)=1+z/(1-αz2), 0≤α<1, maps the open unit disk D onto the interior of a domain known as the Booth lemniscate. Associated with this function Gα is the recently introduced class BS(α) consisting of normalized analytic functions f on D satisfying the subordination zf′(z)/f(z)≺Gα(z). Of interest is its connection with known classes M of functions in the sense g(z)=(1/r)f(rz) belongs to BS(α) for some r in (0, 1) and all f∈M. We find the largest radius r for different classes M, particularly when M is the class of starlike functions of order β, or the Janowski class of starlike functions. As a primary tool for this purpose, we find the radius of the largest disk contained in Gα(D) and centered at a certain point a∈R.

The function $G_\alpha(z)=1+ z/(1-\alpha z^2)$, \, $0\leq \alpha <1$, maps the open unit disc $\mathbb{D}$ onto the interior of a domain known as the Booth lemniscate. Associated with this function $G_\alpha$ is the recently introduced class $\mathcal{BS}(\alpha)$ consisting of normalized analytic functions f on $\mathbb{D}$ satisfying the subordination $zf'(z)/f(z) \prec G_\alpha(z)$. Of interest is its connection with known classes $\mathcal{M}$ of functions in the sense g(z)=(1/r)f(rz) belongs to $\mathcal{BS}(\alpha)$ for some r in (0,1) and all $f \in \mathcal{M}$. We find the largest radius r for different classes $\mathcal{M}$, particularly when $\mathcal{M}$ is the class of starlike functions of order $\beta$, or the Janowski class of starlike functions. As a primary tool for this purpose, we find the radius of the largest disc contained in $G_\alpha(\mathbb{D})$ and centered at a certain point $a \in \mathbb{R}$.

Let h be a nonvanishing analytic function in the open unit disc with h0=1. Consider the class consisting of normalized analytic functions f whose ratios fz/gz, gz/zpz, and pz are each subordinate to h for some analytic functions g and p. The radius of starlikeness of order α is obtained for this class when h is chosen to be either hz=1+z or hz=ez. Further, starlikeness radii are also obtained for each of these two classes, which include the radius of Janowski starlikeness, and the radius of parabolic starlikeness.

The Bohr operator Mr for a given analytic function f(z)=∑n=0∞anzn and a fixed z in the unit disk, |z|=r, is given byMr(f)=∑n=0∞|an||zn|=∑n=0∞|an|rn. Applying earlier results of Bohr and Rogosinski, the Bohr operator is used to readily establish the following inequalities: if f(z)=∑n=0∞anzn is subordinate (or quasi-subordinate) to h(z)=∑n=0∞bnzn in the unit disk, thenMr(f)≤Mr(h),0≤r≤1/3. Further, each k-th section sk(f)=a0+a1z+⋯+akzk satisfies|sk(f)|≤Mr(sk(h)),0≤r≤1/2, andMr(sk(f))≤Mr(sk(h)),0≤r≤1/3. Both constants 1/2 and 1/3 cannot be improved. From these inequalities, a refinement of Bohr's theorem is obtained in the subdisk |z|≤1/3. Also established are growth estimates in the subdisk of radius 1/2 for the k-th section sk(f) of analytic functions f subordinate to a concave wedge-mapping. A von Neumann-type inequality is established for the class consisting of Schwarz functions in the unit disk.

Let h be a non-vanishing analytic function in the open unit disc with h(0)=1. Consider the class consisting of normalized analytic functions f whose ratios f(z)/g(z), $g(z)/z p(z)$, and p(z) are each subordinate to h for some analytic functions g and p. The radius of starlikeness is obtained for this class when h is chosen to be either $h(z)=\sqrt{1+z}$ or $h(z)=e^z$. Further $\mathcal{G}$-radius is also obtained for each of these two classes when $\mathcal{G}$ is a particular widely studied subclass of starlike functions. These include $\mathcal{G}$ consisting of the Janowski starlike functions, and functions which are parabolic starlike.

For $f(z) = \sum_{n=0}^{\infty} a_n z^n$ and a fixed z in the unit disk, $|z| = r,$ the Bohr operator $\mathcal{M}_r$ is given by $\mathcal{M}_r (f) = \sum_{n=0}^{\infty} |a_n| |z^n| = \sum_{n=0}^{\infty} |a_n| r^n.$ This papers develops normed theoretic approaches on $\mathcal{M}_r$. Using earlier results of Bohr and Rogosinski, the following results are readily established: if $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$ is subordinate (or quasi-subordinate) to $h(z)=\sum_{n=0}^{\infty} b_{n}z^{n}$ in the unit disk, then $\mathcal{M}_{r}(f) \leq \mathcal{M}_{r}(h), \quad 0 \leq r \leq 1/3,$ that is, $\sum_{n=0}^{\infty} \ | a_{n}\ | |z|^{n} \leq \sum_{n=0}^{\infty} \ | b_{n}\ |t |z|^{n}, \quad 0 \leq |z| \leq 1/3.$ Further, each k-th section $s_k(f) = a_0 + a_1 z + \cdots + a_kz^k$ satisfies $\ | s_k(f)\ | \leq \mathcal{M}_r \ ( s_k(h)\ ), \quad 0 \leq r \leq 1/2,$ and $\mathcal{M}_{r}\ ( s_{k}(f) \ ) \leq \mathcal{M}_{r}(s_{k}(h)), \quad 0 \leq r \leq 1/3.$ A von Neumann-type inequality is also obtained for the class consisting of Schwarz functions in the unit disk.