Abstract:
In the present paper, we introduce a class of analytic functions in the open unit disc by using the analytic function q_{α}(z)=3/(3+(α-3)z-αz^{2}), which was investigated by Sokó? [1]. We find some properties including the growth theorem or the coefficient problem of this class and we find some relation with this new class and the class of convex functions.

Abstract:
The set S consists of complex functions f, univalent in the open unit disk, with f(0)=f ￠ € 2(0) ￠ ’1=0. We use the asymptotic behavior of the positive semidefinite FitzGerald matrix to show that there is an absolute constant N0 such that, for any f(z)=z+ ￠ ‘n=2 ￠ anzn ￠ S with |a3| ￠ ‰ ¤2.58, we have |an|N0.

Motivated and
stimulated especially by the work of Xu et
al. [1], in this paper, we introduce and discuss an
interesting subclass of
analytic and bi-univalent functions defined in the open unit disc U. Further, we find estimates on the
coefficients and for
functions in this subclass. Many relevant connections with known or new results
are pointed out.

Abstract:
The standard definition of a close-to-convex function involves a complex numerical factor ei 2 which is on occasion erroneously replaced by 1. While it is known to experts in the field that this replacement cannot be made without essentially changing the class, explicit reasons for this fact seem to be lacking in the literature. Our purpose is to fill this gap, and in so doing we are lead to a new coefficient problem which is solved for n=2, but is open for n>2.

Abstract:
In this paper, we define two new integral operators $L^k$ and $L_k$ which are iterative in nature. We show that for $f(z)=z+a_2z^2+ cdots +a_nz^n +cdots$ with radius of convergence larger than one, $L^kf(z)$ and $L_kf(z)$ when restricted on $E={z:|z|<1}$ will eventually be univalent for large enough $k$. We then show that these are the best possible results by demonstrating that there exists a holomorphic function $T(z)$ in normalized form and with radius of convergence equal to one such that $L^kT(z)$ and $L_kT(z)$ fail to be univalent when restricted to $E$ for every $kin mathbb{N}$.

Abstract:
We consider the class of univalent functions defined by the conditions f(z)/z ￠ ‰ 0 and |(z/f(z)) ￠ € 2 ￠ € ￠ € 2| ￠ ‰ ¤ ±,|z|<1, where f(z)=z+ ￠ ˉ is analytic in |z|<1 and 0< ± ￠ ‰ ¤2.