arXiv subjects

Hesam Mahzoon

Publications and source records attributed to Hesam Mahzoon.

4 recordsLinked to original sources

arXiv2019

On certain subclasses of close-to-convex functions related with the second-order differential subordination

Let $\mathcal{A}$ be the family of analytic and normalized functions in the open unit disc $|z|<1$. In this article we consider the following classes \begin{equation*} \mathcal{R}(α,β):=\left\{ f\in \mathcal{A}: {\rm Re}\left\{f'(z)+\frac{1+e^{iα}}{2}zf''(z)\right\}>β,\, |z|<1\right\} \end{equation*} and \begin{equation*} \mathcal{L}_α(b):=\left\{f\in\mathcal{A}:\left|f'(z) +\frac{1+e^{iα}}{2}zf''(z)-b\right|< b,\, |z|<1 \right\}, \end{equation*} where $-π<α\leq π$, $0\leq β<1$ and $b>1/2$. We show that if $f\in \mathcal{R}(α,β)$, then ${\rm Re}\{f'(z)\}$ and ${\rm Re}\{f(z)/z\}$ are greater than $β$, and if $f\in\mathcal{L}_α(b)$, then $0<{\rm Re}\{f'(z)\}<2b$. Also, some another interesting properties of the class $\mathcal{L}_α(b)$ are investigated. Finally, the radius of univalence of 2-th section sum of $f\in \mathcal{R}(α,β)$ is obtained.

math.CV↗

arXiv2019

Notes on the starlike log--harmonic mappings of order alpha

Let $h$ and $g$ be two analytic functions in the unit disc $Δ$ that $g(0)=1$. Also let $β$ be a complex number with ${\rm Re}\{β\}>-1/2$. A function $f$ is said to be log--harmonic mapping if it has the following representation \begin{equation*} f(z)=z |z|^{2β} h(z)\overline{g(z)}\quad (z\in Δ). \end{equation*} A log--harmonic mapping $f$ is said to be starlike log--harmonic mapping of order $α$, where $0\leq α<1$, if \begin{equation*} {\rm Re}\left\{\frac{zf_z -\overline{z}f_{\overline{z}}}{f}\right\}>α\quad(z\in Δ). \end{equation*} In this paper, by use of the subordination principle, we study some geometric properties of the starlike log--harmonic mappings of order $α$. Also, we estimate the Jacobian of log--harmonic mappings.

math.CV↗

arXiv2019

Coefficient and Fekete-Szegö problem estimates for certain subclass of analytic and bi-univalent functions

In this paper, we obtain the Fekete-Szegö problem for the $k$-th $(k\geq1)$ root transform of the analytic and normalized functions $f$ satisfying the condition \begin{equation*} 1+\frac{α-π}{2 \sin α}< {\rm Re}\left\{\frac{zf'(z)}{f(z)}\right\} < 1+\fracα{2\sin α} \quad (|z|<1), \end{equation*} where $π/2\leq α<π$. Afterwards, by the above two-sided inequality we introduce and investigate a certain subclass of analytic and bi-univalent functions in the disk $|z|<1$ and obtain upper bounds for the first few coefficients and Fekete-Szegö problem for functions belonging to this analytic and bi-univalent function class.

math.CV↗

arXiv2019

Further results for a subclass of univalent functions related with differential equation

Let $Ω$ denote the class of functions $f$ analytic in the open unit disc $Δ$, normalized by the condition $f(0)=f'(0)-1=0$ and satisfying the inequality \begin{equation*} \left|zf'(z)-f(z)\right|<\frac{1}{2}\quad(z\inΔ). \end{equation*} The class $Ω$ was introduced recently by Peng and Zhong (Acta Math Sci {\bf37B(1)}:69--78, 2017). Also let $\mathcal{U}$ denote the class of functions $f$ analytic and normalized in $Δ$ and satisfying the condition \begin{equation*} \left|\left(\frac{z}{f(z)}\right)^2f'(z)-1\right|<1\quad(z\inΔ). \end{equation*} In this article, we obtain some further results for the class $Ω$ including, an extremal function and more examples of $Ω$, inclusion relation between $Ω$ and $\mathcal{U}$, the radius of starlikeness, convexity and close--to--convexity and sufficient condition for function $f$ to be in $Ω$. Furthermore, along with the settlement of the coefficient problem and the Fekete--Szegö problem for the elements of $Ω$, the Toeplitz matrices for $Ω$ are also discussed in this article.

math.CV↗