Let $C$ be the extended complex plane; $G\subset C$ a finite Jordan with $0\in G$; $w=\varphi \left(z\right)$ the conformal mapping of $G$ onto the disk $B\left(0;{\rho }_0\right):=\left\}w\left|w\right|<{\rho }_0\right\{$ normalized by $\varphi \left(0\right)=0$ and ${\varphi }^{\text{'}}\left(0\right)=1$. Let us set ${\varphi }_p\left(z\right):={\int }_0^z{\left[{\varphi }^{\text{'}}\left(\zeta \right)\right]}^{2/p}\mathrm{d}\zeta$, and let ${\pi }_{n,p}\left(z\right)$ be the generalized Bieberbach polynomial of degree $n$ for the pair $(G,0)$, which minimizes the integral $\underset{G}{\int \int }{\left|{\varphi }_p^{\text{'}}\left(z\right)-P_n^{\text{'}}\left(z\right)\right|}^p\mathrm{d}{\sigma }_z$ in the class of all polynomials of degree not exceeding $\le n$ with $P_n\left(0\right)=0$, $P_n^{\text{'}}\left(0\right)=1$. In this paper we study the uniform convergence of the generalized Bieberbach polynomials ${\pi }_{n,p}\left(z\right)$ to ${\varphi }_p\left(z\right)$ on $\overline{G}$ with interior and exterior zero angles and determine its dependence on the...

Recently, A. W. Goodman introduced the geometrically defined class UCV of uniformly convex functions on the unit disk; he established some theorems and raised a number of interesting open problems for this class. We give a number of new results for this class. Our main theorem is a new characterization for the class UCV which enables us to obtain subordination results for the family. These subordination results immediately yield sharp growth, distortion, rotation and covering theorems plus sharp...

Recently, A. W. Goodman introduced the class UCV of normalized uniformly convex functions. We present some sharp coefficient bounds for functions f(z) = z + a₂z² + a₃z³ + ... ∈ UCV and their inverses $f^{-1}\left(w\right)=w+d₂w²+d₃w³+...$. The series expansion for $f^{-1}\left(w\right)$ converges when $\left|w\right|<{\varrho }_f$, where $0<{\varrho }_f$ depends on f. The sharp bounds on $|a_n|$ and all extremal functions were known for n = 2 and 3; the extremal functions consist of a certain function k ∈ UCV and its rotations. We obtain the sharp bounds on $|a_n|$ and all extremal functions for n = 4, 5,...

The aim of this paper is to find the necessary and sufficient conditions and inclusion relations for Pascal distribution series to be in the classes ${\mathrm{𝒮𝒫}}_p(\alpha ,\beta )$ and ${\mathrm{𝒰𝒞𝒱}}_p(\alpha ,\beta )$ of uniformly spirallike functions. Further, we consider an integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.

In this article, we aim to find sufficient conditions for a convolution of analytic univalent functions and the Pascal distribution series to belong to the families of uniformly starlike functions and uniformly convex functions in the open unit disk $𝕌$. We also state corollaries of our main results.