Related papers: Schwarzian Norm Estimate for Functions in Generali…

This paper aims to pursue some classes of normalized analytic functions $f$ with fixed second coefficient defined on open unit disk, such that ${(1+z)^2f(z)}/{z}$ and ${(1+z)f(z)}/{z}$ are functions having positive real part. The radius of…

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…

This article contains several results for \lambda-Robertson functions, i.e., analytic functions $f$ defined on the unit disk $D$ satisfying $f(0) = f'(0)-1=0$ and $Re e^{-i\lambda} {1+zf"(z)/f'(z)} > 0$ in $D$, where $\lambda \epsilon…

In this article, we study preserving properties of certain nonlinear integral transforms in some classical families of normalized analytic univalent functions defined in the unit disk. Also, we find sharp pre-Schwarzian norm estimates of…

For an analytic function $f$ defined on the unit disk $|z|<1$, let $\Delta(r,f)$ denote the area of the image of the subdisk $|z|<r$ under $f$, where $0<r\le 1$. In 1990, Yamashita conjectured that $\Delta(r,z/f)\le \pi r^2$ for convex…

In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp estimates for $f(A)-f(B)$ were obtained for self-adjoint operators $A$ and $B$ and for various classes of functions $f$ on the real line $\R$. In this paper we extend…

We prove that if D is a domain in C, alpha>1 and c>0, then the family F of functions meromorphic in D such that |f'(z)|/(1+|f(z)|^alpha)>c for every z in D is normalin D. For alpha=1, the same assumptions imply quasi-normality but not…

Estimates are obtained for the initial coefficients of a normalized analytic function $f$ in the unit disk $\mathbb{D}$ such that $f$ and the analytic extension of $f^{-1}$ to $\mathbb{D}$ belong to certain subclasses of univalent…

In this paper we study the class $\mathcal{U}$ of functions that are analytic in the open unit disk ${\mathbb D}=\{z:|z|<1\}$, normalized such that $f(0)=f'(0)-1=0$ and satisfy \[\left|\left [\frac{z}{f(z)} \right]^{2}f'(z)-1…

Let $f$ be analutic in the unit disk $\mathbb D$ and normalized so that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we give sharp bound of Hankel determinant of the second order for the class of analytic unctions satisfying \[ \left|\arg…

\noindent In the present investigation, we find the sharp bound of fifth coefficient of analytic normalized function $f$ satisfying $z f'(z)/f(z) \prec \varphi(z)$ when coefficients of $\varphi$ satisfy certain conditions. For an…

Let $\mathcal{S}$ denote the class of functions analytic and univalent (i.e. one-to-one) in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:\, |z|<1\}$ normalized by $f(0)=0=f'(0)-1$. The logarithmic coefficients $\gamma_n$ of $f\in\mathcal{S}$…

In this article, we consider the family $\mathcal{F}(\alpha)$ defined for $\alpha \in (0, 3]$ by \begin{align*} {\rm Re}\left(1+\frac{zf''(z)}{f'(z)}\right) > 1 - \frac{\alpha}{2} \quad \text{for } z \in \mathbb{D}. \end{align*} Our primary…

For every countable ordinal number $\alpha$ we construct an entire function $f=f_\alpha$ such that the family $\left\{f(nz):n\in\mathbb{N}\right\}$ is exactly $Q_\alpha$-normal in the unit disk.

Suppose $\alpha>-1$ and $1\leq p \leq \infty$. Let $f=P_{\alpha}[F]$ be an $\alpha$-harmonic mapping on $\mathbb{D}$ with the boundary $F$ being absolute continuous and $\dot{F}\in L^p(0,2\pi)$, where…

In 2011, Sok\'{o}{\l} (Comput. Math. Appl. 62, 611--619) introduced and studied the class $\mathcal{SK}(\alpha)$ as a certain subclass of starlike functions, consists of all functions $f$ ($f(0)=0=f'(0)-1$) which satisfy in the following…

One of the classical problems concerns the class of analytic functions $f$ on the open unit disk $|z|<1$ which have finite Dirichlet integral $\Delta(1,f)$, where $$\Delta(r,f)=\iint_{|z|<r}|f'(z)|^2 \, dxdy \quad (0<r\leq 1). $$ The class…

The logarithmic coefficients $\gamma_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ is defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty}…

For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=0, S. S. Miller and P. T. Mocanu (Integral Transform. Spec. Funct. 19(2008)) have considered some sufficient problems for starlikeness. The object of the present paper is…

Let $p(z)=zf'(z)/f(z)$ for a function $f(z)$ analytic on the unit disk $|z|<1$ in the complex plane and normalized by $f(0)=0, f'(0)=1.$ We will provide lower and upper bounds for the best constants $\delta_0$ and $\delta_1$ such that the…