Research Areas

Harmonic Functions

Harmonic univalent mappings are an important class of functions in complex analysis that generalize conformal (analytic) mappings by allowing both analytic and co-analytic components. They arise naturally in geometric function theory and have applications in minimal surface theory, fluid dynamics, and elasticity. The study of these mappings focuses on geometric properties, including univalence, distortion, growth, and coefficient bounds. Their rich structure makes them a central topic in modern harmonic mapping theory.

Differential Subordination

Differential subordination serves as the complex-analytic counterpart of differential inequalities on the real line. We have developed a theory of differential subordination for functions whose images are confined to regions bounded by the exponential function. This framework yields several new and meaningful applications within univalent function theory.

Special Functions

Special functions are distinguished mathematical functions with well-established names and standard notations, owing to their fundamental role in mathematical analysis, functional analysis, geometry, physics, and related applications. Examples include the confluent hypergeometric function, Bessel function, and Wright function. The geometric properties of these functions have been extensively investigated, particularly in connection with regions associated with the exponential function.

Radius Problems

Every convex function is necessarily starlike, though the converse does not hold. Nevertheless, starlike functions map a sufficiently small disk onto convex domains. The largest radius for which this property is preserved is known as the radius of convexity of starlike functions. More generally, radius problems can be formulated for any pair of geometric regions, and we have investigated several such radius problems associated with the exponential function.

Hankel Determinants

Hankel determinants, constructed from the Taylor coefficients of analytic or univalent functions, play an important role in geometric function theory. They generalize several classical coefficient problems, including the Fekete–Szegő problem and are closely related to coefficient inequalities arising from the Zalcman conjecture. Estimates of Hankel determinants provide deeper insight into the growth, distortion, and geometric behavior of univalent and related function classes.

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