arXiv subjects

Rosihan M. Ali

Publications and source records attributed to Rosihan M. Ali.

15 recordsLinked to original sources

arXiv2021

Starlikeness of Analytic Functions with Subordinate Ratios

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.

math.CV↗

arXiv2019

Bohr operator on analytic functions

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.

math.CV↗

arXiv2019

Differential Inequalities and Univalent Functions

Let ${\mathcal M}$ be the class of analytic functions in the unit disk $\ID$ with the normalization $f(0)=f'(0)-1=0$, and satisfying the condition $$\left |z^2\left (\frac{z}{f(z)}\right )''+ f'(z)\left(\frac{z}{f(z)} \right)^{2}-1\right |\leq 1, \quad z\in \ID. $$ Functions in $\mathcal{M}$ are known to be univalent in $\ID$. In this paper, it is shown that the harmonic mean of two functions in ${\mathcal M}$ are closed, that is, it belongs again to ${\mathcal M}$. This result also holds for other related classes of normalized univalent functions. A number of new examples of functions in $\mathcal{M}$ are shown to be starlike in $\ID$. However we conjecture that functions in $\mathcal{M}$ are not necessarily starlike, as apparently supported by other examples.

math.CV↗

arXiv2018

The spherical metric and univalent harmonic mappings

Let $f=h+\overline{g}$ be a harmonic univalent map in the unit disk $\mathbb{D}$, where $h $ and $g$ are analytic. We obtain an improved estimate for the second coefficient of $h$. This indeed is the first qualitative improvement after the appearance of the papers by Clunie and Sheil-Small in 1984, and by Sheil-Small in 1990. Also, when the sup-norm of the dilatation is less than $1$, it is shown that the spherical area of the covering surface of $h$ is dominated by the spherical area of the covering surface of $f.$

math.CV↗

arXiv2017

Characterizing Rotationally Typically Real Logharmonic Mappings

This paper treats the class of normalized logharmonic mappings f(z) = zh(z)bar{g(z)} in the unit disk satisfying ϕ(z) = zh(z)g(z) is analytically typically real. Every such mapping f is shown to be a product of two particular logharmonic mappings, each of which admits an integral representation. Also obtained is the radius of starlikeness and an upper estimate for arclength. Additionally, it is shown that f maps the unit disk into a domain symmetric with respect to the real axis when it is univalent and its second dilatation has real coefficients.

math.CV↗

arXiv2017

Starlikeness of the generalized Bessel function

For a fixed $a \in \{1, 2, 3, \ldots\},$ the radius of starlikeness of positive order is obtained for each of the normalized analytic functions \begin{align*} \mathtt{f}_{a, ν}(z)&:= \bigg(2^{a ν-a+1} a^{-\frac{a(aν-a+1)}{2}} Γ(a ν+1) {}_a\mathtt{B}_{2a-1, a ν-a+1, 1}(a^{a/2} z)\bigg)^{\tfrac{1}{a ν-a+1}},\\ \mathtt{g}_{a, ν}(z)&:= 2^{a ν-a+1} a^{-\frac{a}{2}(aν-a+1)} Γ(a ν+1) z^{a-aν} {}_a\mathtt{B}_{2a-1, a ν-a+1, 1}(a^{a/2} z),\\ \mathtt{h}_{a, ν}(z)&:= 2^{a ν-a+1} a^{-\frac{a}{2}(aν-a+1)} Γ(a ν+1) z^{\frac{1}{2}(1+a-aν)} {}_a\mathtt{B}_{2a-1, a ν-a+1, 1}(a^{a/2} \sqrt{z}) \end{align*} in the unit disk, where ${}_a\mathtt{B}_{b, p, c}$ is the generalized Bessel function \begin{align*} {}_a\mathtt{B}_{b, p, c}(z):= \sum_{k=0}^\infty \frac{(-c)^k}{k! \; \mathrmΓ{\left( a k +p+\frac{b+1}{2}\right)} } \left(\frac{z}{2}\right)^{2k+p}. \end{align*} The best range on $ν$ is also obtained for a fixed $a$ to ensure the functions $\mathtt{f}_{a, ν}$ and $\mathtt{g}_{a, ν}$ are starlike of positive order in the unit disk. When $a=1,$ the results obtained reduced to earlier known results.

math.CV↗

arXiv2016

On the Bohr inequality

The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\sum_{n=0}^\infty |a_n|r^n \leq 1$ holds whenever $|\sum_{n=0}^\infty a_nz^n|\leq 1$ in the unit disk $\mathbb{D}$ of the complex plane. The exact value of this largest radius, known as the \emph{Bohr radius}, has been established to be $1/3.$ This paper surveys recent advances and generalizations on the Bohr inequality. It discusses the Bohr radius for certain power series in $\mathbb{D},$ as well as for analytic functions from $\mathbb{D}$ into particular domains. These domains include the punctured unit disk, the exterior of the closed unit disk, and concave wedge-domains. The analogous Bohr radius is also studied for harmonic and starlike logharmonic mappings in $\mathbb{D}.$ The Bohr phenomenon which is described in terms of the Euclidean distance is further investigated using the spherical chordal metric and the hyperbolic metric. The exposition concludes with a discussion on the $n$-dimensional Bohr radius.

math.CV↗

arXiv2012

Second-order differential subordinations for analytic functions with fixed initial coefficient

Functions with fixed initial coefficient have been widely studied. A new methodology is proposed in this paper by making appropriate modifications and improvements to the theory of second-order differential subordination. Several interesting examples are given. The results obtained are applied to the classes of convex and starlike functions with fixed second coefficient.

math.CV↗

arXiv2012

On the radius constants for classes of analytic functions

Radius constants for several classes of analytic functions on the unit disk are obtained. These include the radius of starlikeness of a positive order, radius of parabolic starlikeness, radius of Bernoulli lemniscate starlikeness, and radius of uniform convexity. In the main, the radius constants obtained are sharp. Conjectures on the non-sharp constants are given.

math.CV↗

arXiv2012

Radius Constants for Analytic Functions with Fixed Second Coefficient

Let $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ be analytic in the unit disk with second coefficient $a_2$ satisfying $|a_2|=2b$, $0\leq b\leq1$. Sharp radius of Janowski starlikeness and other radius constants are obtained when $|a_n|\leq cn+d$ ($c,d\geq0$) or $|a_n|\leq c/n$ ($c>0$) for $n\geq3$.

math.CV↗

arXiv2011

Coefficient Inequalities for Starlikeness and Convexity

For an analytic function f(z)=z+\sum_{n=2}^\infty a_n z^n satisfying the inequality \sum_{n=2}^\infty n(n-1)|a_n|\leq β, sharp bound on $β$ is determined so that $f$ is either starlike or convex of order $α$. Several other coefficient inequalities related to certain subclasses are also investigated.

math.CV↗

arXiv2011

Inclusion Criteria for Subclasses of Functions and Gronwall's Inequality

A normalized analytic function f is shown to be univalent in the open unit disk D if its second coefficient is sufficiently small and relates to its Schwarzian derivative through a certain inequality. New criteria for analytic functions to be in certain subclasses of functions are established in terms of the Schwarzian derivatives and the second coefficients. These include obtaining a sufficient condition for functions to be strongly $α$-Bazileviĉ of order $β$.

math.CV↗

arXiv2011

Radii of Starlikeness Associated with the Lemniscate of Bernoulli and the Left-Half Plane

A normalized analytic function f defined on the open unit disk in the complex plane is in the class SL if zf'(z)/f(z) lies in the region bounded by the right-half of the lemniscate of Bernoulli given by |w^2 - 1| < 1. In the present investigation, the SL-radii for certain well-known classes of functions are obtained. Radius problems associated with the left-half plane are also investigated for these classes

math.CV↗

arXiv2011

A first-order differential double subordination with applications

Let $q_1$ and $q_2$ belong to a certain class of normalized analytic univalent functions in the open unit disk of the complex plane. Sufficient conditions are obtained for normalized analytic functions $p$ to satisfy the double subordination chain $q_1(z)\prec p(z)\prec q_2(z)$. The differential sandwich-type result obtained is applied to normalized univalent functions and to $Φ$-like functions.

math.CV↗

arXiv2011

Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions

Estimates on the initial coefficients are obtained for normalized analytic functions $f$ in the open unit disk with $f$ and its inverse $g=f^{-1}$ satisfying the conditions that $zf'(z)/f(z)$ and $zg'(z)/g(z)$ are both subordinate to a starlike univalent function whose range is symmetric with respect to the real axis. Several related classes of functions are also considered, and connections to earlier known results are made.

math.CV↗