Abstract:
In 1984, Clunie and Sheil-Small proved that a sense-preserving harmonic function whose analytic part is convex, is univalent and close-to-convex. In this paper, certain cases are discussed under which the conclusion of this result can be strengthened and extended to fully starlike and fully convex harmonic mappings. In addition, we investgate the properties of functions in the class $\mathcal{M}(\alpha)$ $(|\alpha|\leq 1)$ consisting of harmonic functions $f=h+\overline{g}$ with $g'(z)=\alpha zh'(z)$, $\RE (1+{zh''(z)}/{h'(z)})>-{1}/{2} $ $ \mbox{for} |z|<1 $. The coefficient estimates, growth results, area theorem and bounds for the radius of starlikeness and convexity of the class $\mathcal{M}(\alpha)$ are determined. In particular, the bound for the radius of convexity is sharp for the class $\mathcal{M}(1)$.

Abstract:
For an analytic function f(z)=z+\sum_{n=2}^\infty a_n z^n satisfying the inequality \sum_{n=2}^\infty n(n-1)|a_n|\leq \beta, sharp bound on $\beta$ is determined so that $f$ is either starlike or convex of order $\alpha$. Several other coefficient inequalities related to certain subclasses are also investigated.

Abstract:
For $0\leq \alpha <1$, the sharp radii of starlikeness and convexity of order $\alpha$ for functions of the form $f(z)=z+a_2z^2+a_3z^3+...$ whose Taylor coefficients $a_n$ satisfy the conditions $|a_2|=2b$, $0\leq b\leq 1$, and $|a_n|\leq n $, $M$ or $M/n$ ($M>0$) for $n\geq 3$ are obtained. Also a class of functions related to Carath\'eodory functions is considered.

Abstract:
In this paper, we first find an estimate for the range of polyharmonic mappings in the class $HC_{p}^{0}$. Then, we obtain two characterizations in terms of the convolution for polyharmonic mappings to be starlike of order $\alpha$, and convex of order $\beta$, respectively. Finally, we study the radii of starlikeness and convexity for polyharmonic mappings, under certain coefficient conditions.

Abstract:
The object of the present paper is to prove some interesting results for the bounds of starlikeness and convexity of certain multivalent functions.

Abstract:
We give coefficient estimates for a class of close-to-convex harmonic mappings, and discuss the Fekete-Szeg\H{o} problem of it. We also introduce two classes of polyharmonic mappings $\mathcal{HS}_{p}$ and $\mathcal{HC}_{p}$, consider the starlikeness and convexity of them, and obtain coefficient estimates on them. Finally, we give a necessary condition for a mapping $F$ to be in the class $\mathcal{HC}_{p}$.

Abstract:
Sufficient conditions on a sequence of nonnegative numbers are obtained that ensures is starlike of nonnegative order in the unit disk. A result of Vietoris on trigonometric sums is extended in this pursuit. Conditions for close to convexity and convexity in the direction of the imaginary axis are also established. These results are applied to investigate the starlikeness of functions involving the Gaussian hypergeometric functions. 1. Introduction Let denote the class of analytic functions defined in the unit disk normalized by the conditions . Denote by the subclass of consisting of functions univalent in . A function is starlike if is starlike with respect to the origin and convex if is a convex domain. These classes denoted by and , respectively, are subsets of . The generalized classes and of starlike and convex functions of order are defined, respectively, by the analytic characterizations with and . An extension of starlike functions is the class of close-to-convex functions of order defined analytically by for some real . The family of close-to-convex functions of order with respect to is denoted by , with . Exposition on the geometric properties of functions in these classes can be found in [1, 2]. A function satisfying in is said to be typically real, and is convex in the direction of the imaginary axis if every line parallel to the imaginary axis either intersects in an interval or has an empty intersection. For with real coefficients, Robertson [3] proved that being convex in the direction of the imaginary axis is equivalent to being typically real, which in turn is equivalent to . For satisfying is typically real and in , Ruscheweyh [4] proved that it is necessarily starlike. The latter result is extended in [5] to include starlike functions of a nonnegative order. Lemma 1.1 (see [5]). For , let satisfy and be typically real in . If in , then . Trigonometric series, in particular the cosine and sine series along with their partial sums, have found widely important applications in many works, for example, those of [4–9]. Vietoris [10] (also see [11]) showed that if , , then for any positive integer . Here the Pochhammer symbol is defined by , ？and？？ . Using Abel’s partial summation formula equation (1.3) yields the following classical result on the positivity of cosine and sine sums. Theorem 1.2 (see [10]). Let be a decreasing sequence of nonnegative real numbers satisfying and . Then for any positive integer . Using Theorem 1.2, Ruscheweyh [4] obtained sufficient coefficient conditions for functions to be starlike which can readily be

Abstract:
Our purpose is to derive some sufficient conditions for starlikeness and close-to-convexity of order α of certain analytic functions in the open unit disk.

Abstract:
The object of the present paper is to derive a property of certain meromorphic functions in the punctured unit disk. Our main theorem contains certain sufficient conditions for starlikeness and close-to-convexity of order α of meromorphic functions.