In this paper, we investigate the sharp bounds of second Hankel determinant Logarithmic coefficients for starlike and convex functions with respect to symmetric points in open unit disk.

An analytic functions f(z) defined on 4 = {z : |z| < 1} and normalized by f(0) = 0, f ′(0) = 1 is starlike with respect to conjugate points

Let Ss*(ez) denote the class of starlike functions with respect to symmetric points subordinate exponential function, i.e., which satisfy in unit disk U condition 2zf′(z)f(z)−f(−z)≺ez(z∈U). We obtained sharp estimate second-order Hankel determinants H2,3(f) and improved third-order H3,1(f) for this Ss*(ez).

We introduce new subclasses S σ,s k λ, δ, φ and K σ,s k λ, δ, φ of analytic functions with respect to k-symmetric points defined by differential operator. Some interesting properties for these classes are obtained.

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means subordination, analytic parts which are reciprocal starlike (or convex) functions. Further, combining graph function, discuss bound Bloch constant and norm pre-Schwarzian derivative for classes.

Making use of Srivastava-Wright operator we introduced a new class of complexvalued harmonic functions with respect to symmetric points which are orientation preserving, univalent and starlike. We obtain coefficient conditions, extreme points, distortion bounds, convex combination. Mathematics subject classification (2010): 30C45.