No abstract

For 0 ≤ α < 1, let Q α be the class of functions f which are normalised analytic and univalent in D = {z : |z| < 1} satisfying the conditionwhere g is a normalised convex function. For f ∈ Q α , sharp bounds are obtained for the Feketo-Szegö functional |a 3 − μa 2 2 | when μ is real. Mathematics Subject Classification: Primary 30C45

Let $ \mathcal{H} $ denote the class of functions $ f $ which are harmonic and univalent in the open unit disc $ {D=\{z:|z|<1\}} $. This paper defines and investigates a family of complex-valued harmonic functions that are orientation preserving and univalent in $ \mathcal{D} $ and are related to the functions convex of order $ \beta(0\leq \beta <1) $, with respect to symmetric points. We obtain coefficient conditions, growth result, extreme points, convolution and convex combinations for the above harmonic functions.

Let S to be the class of functions which are analytic, normalized and univalent in the unit disk U = {z : |z| < 1}. The main subclasses of S are starlike functions, convex functions, close-to-convex functions, quasiconvex functions, starlike functions with respect to (w.r.t.) symmetric points and convex functions w.r.t. symmetric points which are denoted by S * , K, C, C * , S * S , and K S respectively. In recent past, a lot of mathematicians studied about Hankel determinant for numerous classes of functions contained in S. The qth Hankel determinant for q ≥ 1 and n ≥ 0 is defined by H q (n). H 2 (1) = a 3 − a 2 2 is greatly familiar so called Fekete-Szegö functional. It has been discussed since 1930's. Mathematicians still have lots of interest to this, especially in an altered version of a 3 − µa 2 2. Indeed, there are many papers explore the determinants H 2 (2) and H 3 (1). From the explicit form of the functional H 3 (1), it holds H 2 (k) provided k from 1-3. Exceptionally, one of the determinant that is H 2 (3) = a 3 a 5 − a 4 2 has not been discussed in many times yet. In this article, we deal with this Hankel determinant H 2 (3) = a 3 a 5 − a 4 2. From this determinant, it consists of coefficients of function f which belongs to the classes S * S and K S so we may find the bounds of |H 2 (3)| for these classes. Likewise, we got the sharp results for S * S and K S for which a 2 = 0 are obtained.

Let S be the class of functions which are analytic and univalent in the open unit disc D = {z : |z| < 1} given by f (z) = z + ∞ n=2 a n z n and a n a complex number. Let T denote the class consisting of functions f of the form f (z) = z − ∞ n=2 a n z n where a n is a non negative real number.In this paper, we develop new subclass of S by adopting the original idea of Ramesha et al. [5] and Sudharsan et al. [7]. We give coefficient estimates, growth and extreme points for f belonging to this class. Mathematics Subject Classification: Primary 30C45