RESEARCH

PAST AND ONGOING RESEARCH IN GEOMETRIC FUNCTION THEORY

• Geometric Properties of Harmonic Mappings Harmonic mappings in the plane are univalent (one-to-one) complex-valued functions of a complex variable that satisfies the Laplace's equation. We construct examples of harmonic mappings by using harmonic shears and also investigate closure properties under Hadamard product (or convolution) of various classes of harmonic univalent function such as the classes of starlike, convex and close-to-convex harmonic functions.

• Radii Problems in Geometric Function Theory: A convex function is necessarily starlike but not conversely. However, the starlike functions maps a smaller disk into convex domains. The largest radius with this property is known as the radius of convexity of starlike functions. For any two geometric properties, onecan talk about radius problems. We have investigated several radii problems associated with starlikeness, convexity and uniform convexity.

• Uniformly Convex Functions and Related Topics: A convex (or starlike) functions need not maps a disk inside the unit disk to a convex (or a starlike) do-main. Such a strong requirement leads to the investigation of uniformly convex (or starlike) functions. Among other things, We have introduced and investigated uniformly spiral functions. Convolutions and Neighborhood problems: An old conjecture of Polya-Schoenberg says that convex, starlike and close-to-convex functions are known to be closed under convolutions with convex functions. This was proved Ruscheweyh and Sheil-Small. We have investigated this convolution problem for several more general classes of functions.

• Starlike/Convex wrt Symmetric and Conjugate Points: Motivated by the investigation of the class of starlike functions with respect to symmetric points by Sakaguchi, we have investigated several related classes of functions. In particular, we have obtained radii results, distortion and growth estimates, Koebe domain, estimates for the coefficients and integral representations for functions starlike and convex with respect to symmetric, conjugate and symmetric conjugate points.

• Differential Subordinations and Superordinations: Concept subordination is the analogue of inequalities in real line to the complex plane. Subordination is defined by containment of regions. A differential subordination is extension of differential inequality while superordination is a dual concept. We have obtained several conditions that ensure analytic functions to have certain geometric properties, like starlikeness, convexity.

• Radii Problems by Fixing Second Coefficient: The second coefficient of univalent functions plays important role in GFT; it leads to growth and distortion estimates, and Koebe domains. We have investigated several radii problems by considering functions where the second coefficient is fixed.

• Subordination Theory for Functions with Preassigned Initial Coefficient: My aim here is to develop the theory of differential subordination in the same line as that of Miller and Mocanu for functions with preassigned initial coefficient. This theory is gives us better results than that of Miller and Mocanu.