We introduce and study certain subclasses of analytic functions which are defined by differential subordination. Coefficient inequalities, some properties of neighborhoods, distortion and covering theorems, radius of starlikeness, and convexity for these subclasses are given.

Let [Formula: see text] be the family of analytic and normalized functions [Formula: see text] in the open unit disc [Formula: see text]. In this paper, we consider the following classes: [Formula: see text] and [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text]. We show that if [Formula: see text], then [Formula: see text] and [Formula: see text] are greater than [Formula: see text], and if [Formula: see text], then [Formula: see text]. Also, some another interesting properties of the class [Formula: see text] are investigated. Finally, the radius of univalence of 2nd section sum of [Formula: see text] is obtained.

In this paper, we obtain the Fekete-Szeg? problem for the k-th (k ? 1) root transform of the analytic and normalized functions f satisfying the condition 1+ ? ??/2sin? < Re{z f'(z)/f(z)) < 1+ ?/2sin? (|z| < 1), where ? ? [?/2,?). Afterwards, by the above two-sided inequality we introduce a certain subclass of analytic and bi-univalent functions in the disk |z| < 1 and obtain upper bounds for the first few coefficients and Fekete-Szeg? problem for functions f belonging to this class.

We consider the so-called covariance set of Moore-Penrose inverses in rings with an involution. We deduce some new results concerning covariance set. We will show that ifais a regular element in aC∗-algebra, then the covariance set ofais closed in the set of invertible elements (with relative topology) ofC∗-algebra and is a cone in theC∗-algebra.

Let h and g be two analytic functions in the unit disc ∆ that g(0) = 1. Also let β be a complex number with Re{β} > −1/2. A function f is said to be log-harmonic mapping if it has the following representationA log-harmonic mapping f is said to be starlike log-harmonic mapping ofIn this paper, by use of the subordination principle, we study some geometric properties of the starlike log-harmonic mappings of order α. Also, we estimate the Jacobian of log-harmonic mappings.2010 Mathematics Subject Classification. 30C35;30C45;35Q30.