Research Areas

The differential subordination is the complex analogue of the differential inequality in the real line and differential superordination is its dual concept. We have developed the theory of differential subordination for functions with preassigned initial coefficient by making appropriate modifications and improvements to the existing Miller and Mocanu’s subordination theory. This new theory has several interesting applications in univalent function theory.

In 1992, Ma and Minda unified various subclasses of starlike functions in terms of subordination. Using this notion, we have introduced and defined various classes of starlike function which are associated with exponential function, right half of the shifted lemniscate of Bernoulli and cardioid. Various inclusion relations, coefficient bounds and radius problems are derived for these newly defined classes of univalent functions.

A planar harmonic univalent mapping is a complex-valued function that does not take the same value twice and whose real and imaginary parts have continuous second partial derivatives satisfying the Laplace equation. The study of planar harmonic univalent mappings initiated by Clunie and Sheil-Small in 1984, is a fairly active area of research. We investigate the properties of various subclasses of harmonic univalent functions defined by natural geometric conditions such as the classes of starlike, convex and close-to-convex harmonic functions.  Apart from that, we have recently introduced a new product for harmonic mappings which is expected to play a more important role than the existing concept of harmonic convolution.