Related papers: Coefficients of the inverse of functions for the s…

Let ${\mathcal A}$ be the class of functions that are analytic in the unit disc ${\mathbb D}$, normalized such that $f(z)=z+\sum_{n=2}^\infty a_nz^n$, and let class ${\mathcal U}(\lambda)$, $0<\lambda\le1$, consists of functions…

We consider functions of the type $f(z)=z+a_2z^2+a_3z^3+\cdots$ from a family of all analytic and univalent functions in the unit disk. Let $F$ be the inverse function of $f$, given by $F(z)=w+\sum_{n=2}^{\infty}A_nw^n$ defined on some…

Let $\mathcal{U(\alpha, \lambda)}$, $0<\alpha <1$, $0 < \lambda <1$ be the class of functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ satisfying $$\left|\left(\frac{z}{f(z)}\right)^{1+\alpha}f'(z)-1\right|<\lambda$$ in the unit disc ${\mathbb…

Let ${\mathcal A}$ be the class of functions analytic in the unit disk ${\mathbb D} := \{ z\in {\mathbb C}:\, |z| < 1 \}$ and normalized such that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we study the class $\mathcal{U}(\lambda)$,…

Let $f$ be function that is analytic in the unit disk ${\mathbb D}=\{z:|z|<1\}$, normalized such that $f(0)=f'(0)-1=0$, i.e., of type $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. If additionally, \[ \left| \left(\frac{z}{f(z)}\right)^2 f'(z)…

In this article, we consider the family of functions $f$ analytic in the unit disk $|z|<1$ with the normalization $f(0)=0=f'(0)-1$ and satisfying the condition $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<\lambda $ for some $0<\lambda \leq…

For $ -1 \leq B \leq 1$ and $A>B$, let $\mathcal{S}^*[A,B]$ denote the class of generalized Janowski starlike functions consisting of all normalized analytic functions $f$ defined by the subordination $z f'(z)/f(z) \prec (1+ A z)/(1+ B z)$…

Let ${\mathcal U}^+$ be the class of analytic functions $f$ such that $\frac{z}{f(z)}$ has real and positive coefficients and $f^{-1}$ be its inverse. In this paper we give sharp estimates of the initial coefficients and initial logarithmic…

Let function $f$ be analytic in the unit disk ${\mathbb D}$ and be normalized so that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we give sharp bounds of the modulus of its second, third and fourth coefficient, if $f$ satisfies \[…

For $-1\le B<A\le 1$, let $\mathcal{S}^*(A,B)$ denote the class of normalized analytic functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in $|z|<1$ which satisfy the subordination relation $zf'(z)/f(z)\prec (1+Az)/(1+Bz)$ and $\Sigma^*(A,B)$…

Let $\es$ be the class of analytic and univalent functions in the unit disk $|z|<1$, that have a series of the form $f(z)=z+ \sum_{n=2}^{\infty}a_nz^n$. Let $F$ be the inverse of the function $f\in\es$ with the series expansion %in a disk…

In this paper, we investigate the inverse logarithmic coefficients associated with the class $\mathcal{C}_e$ of analytic and univalent functions satisfying the subordination condition \[ 1+\frac{z f''(z)}{f'(z)} \prec e^z, \quad…

Let $\mathcal{A}$ denote the class of analytic functions in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ satisfying $f(0)=0$ and $f'(0)=1$. Let $\mathcal{U}$ be the class of functions $f\in\mathcal{A}$ satisfying…

For $\alpha\ge 0$, let $\mathcal{W}(\alpha)$ be the class of all analytic functions in the unit disk $\mathbb{D}$ with normalization $f(0) = 0 $ and $ f'(0) = 1 $ that satisfy the relation $Re\,\{f'(z) + \alpha z f''(z)\} > 0$. This article…

For $0<\lambda \leq 1$, let ${\mathcal U}(\lambda)$ denote the family of functions $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ analytic in the unit disk $\ID$ satisfying the condition $\left |\left (\frac{z}{f(z)}\right )^{2}f'(z)-1\right |<\lambda…

Ma-Minda class (of starlike functions) consists of all normalized analytic functions $f$ on the unit disk for which the image of $zf'(z)/f(z)$ is contained in the some starlike region in the right-half plane. We obtain the best possible…

An analytic function $f$ defined on the open unit disk $\mathbb{D}=\{z:|z|<1\}$ is bi-univalent if the function $f$ and its inverse $f^{-1}$ are univalent in $\mathbb{D}$. Estimates for the initial coefficients of bi-univalent functions $f$…

Let $f$ be analytic in the unit disk and $\mathcal{S}$ be the subclass of normalized univalent functions with $f(0) = 0$, and $f'(0) = 1$. Let $F$ be the inverse function of $f$, given by $F(w)=w+\sum_{n=2}^{\infty}A_nw^n$ defined on some…

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

Let $\mathcal{A}$ denote the set of all analytic functions $f$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z| < 1\}$ normalized by $f (0) = 0$ and $f'(0) = 1.$ The logarithmic coefficients $\gamma_n$ of $f \in \mathcal{A}$ are…