Related papers: On the initial coefficients for certain class of f…

Let $z_0$ and $w_0$ be given points in the open unit disk $\mathbb{D}$ with $|w_0| < |z_0|$. Let $\mathcal{H}_0$ be the class of all analytic self-maps $f$ of $\mathbb{D}$ normalized by $f(0)=0$, and $\mathcal{H}_0 (z_0,w_0) = \{ f \in…

In the present investigation, we consider two new subclasses N_{{\Sigma}}^{{\mu}}({\alpha},{\lambda}) and N_{{\Sigma}}^{{\mu}}({\beta},{\lambda}) of bi-univalent functions defined in the open unit disk U={z:|z|<1}. Besides, we find upper…

We consider a bounded radial function f. B. Korenblum and K.E.Zhu give a case where we have equality between the limit near the boundary of the unit disc of the Berezin transform and the limit of the normalized Mellin coefficient when one…

Let $z_0$ and $w_0$ be given points in the open unit disk $\mathbb{D}$ with $|w_0| < |z_0|$, and $\mathcal{H}_0$ be the class of all analytic self-maps $f$ of $\mathbb{D}$ normalized by $f(0)=0$. In this paper, we establish the fourth-order…

There are many results for sufficient conditions of functions f(z) which are analytic in the open unit disc U to be starlike and convex in U. The object of the present paper is to derive some interesting sufficient conditions for f(z) to be…

Function $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$, normalized, analytic and univalent in the unit disk $\mathbb D=\{z:|z|<1\}$, belongs to the class $\mathcal{U}$. if, and only if, \[ \left| \left(\frac{z}{f(z)}\right)^2 -1\right|<1 \quad\quad…

We consider three classes of functions defined using the class $\mathcal{P}$ of all analytic functions $p(z)=1+cz+\dotsb$ on the open unit disk having positive real part and study several radius problems for these classes. The first class…

Let $\mathcal{S}$ denote the class of analytic and univalent ({\it i.e.}, one-to-one) functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. For $f\in \mathcal{S}$, Ma proposed the…

Let $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ be the class of normalized analytic functions $f$ defined in the domain $|z|<1$ satisfying \begin{align*} {\rm Re\,}…

Local boundary smoothness of an analytic function f on the unit ball of C^n is compared to the smoothness of its modulus. We prove that different conditions imposed on the zeros of f imply different drops of the smoothness. We also show…

Inspired by the recent works of Srivastava et al. (2010), Frasin and Aouf (2011), and Caglar et al. (2013), we introduce and investigate in the present paper two new general subclasses of the class consisting of normalized analytic and…

For $\alpha\in\IC\setminus \{0\}$ let $\mathcal{E}(\alpha)$ denote the class of all univalent functions $f$ in the unit disk $\mathbb{D}$ and is given by $f(z)=z+a_2z^2+a_3z^3+\cdots$, satisfying $$ {\rm Re\,} \left (1+…

In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative $S(f)$ of a locally univalent analytic function $f$ in the unit disk satisfies that $\limsup_{|z|\to 1} |S(f)(z)| (1-|z|^2)^2 < 2$, then there exists a positive…

Let $\mathcal{H}(\mathbb{D})$ denote the space of analytic functions in the unit disc $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$. For $0<p<\infty$ and $f\in\mathcal{H}(\mathbb{D})$, let $M_p^p(r,f)=\int_0^{2\pi}|f(re^{i\theta})|^p…

Let $ \mathcal{H}(\mathbb{D}) $ be the class of analytic functions in the unit disk $ \mathbb{D} : =\{z\in\mathbb{C} : |z|<1\} $. The classical Bohr's inequality states that if a power series $ f(z)=\sum_{n=0}^{\infty}a_nz^n $ converges in…

The primary objective of this paper is to establish several sharp versions of Bohr inequalities for bounded analytic functions in the unit disk $\mathbb{D} := \{z\in\mathbb{C} : |z| < 1\}$ involving multiple Schwarz functions. Moreover, we…

Analytic functions in the Hardy class $H^2$ over the upper half-plane $\mathbb{H}_+$ are uniquely determined by their values on any curve $\Gamma$ lying in the interior or on the boundary of $\mathbb{H}_+$. The goal of this paper is to…

Let $f$ be analytic in the unit disk $\mathbb{D}= \{z \in \mathbb{C}~:~ |z| < 1\}$, and $\mathcal{S}$ be the subclass of normalized univalent functions given by $f(z)=\sum_{n=1}^{\infty}a_{n}z^{n},~a_{1}:=1$ for $z \in\mathbb{D}$. We…

For Banach spaces of analytic functions on the disc for which the polynomials are dense and their pointt evaluations continuous, we prove the following: If they contain a function such that the limit superior of its modulus is infinite…

Let $f:\mathbb{D}\to\mathbb{C}$ be a bounded analytic function. A set $K\subset\mathbb{D}$ which contains the point $1$ in its boundary is called a convergence set for $f$ at $1$ if $f(z)$ converges to some value $\zeta$ as $z\to1$ with…