Some characteristics and theorems associated with analytic and univalent functions




Srivastava, H.M.

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Operational techniques based largely upon various families of linear operators are becoming increasingly useful in Geometric Function Theory which is the study of the relationship between the analytic properties of a given function and the geometric properties of its image domain. On the other hand, in immensely useful class of special functions (namely, the generalized hypergeometric function) played a rather crucial role in Louis de Branges' proof of the celebrated Bieberbach, Robertson, and Milin conjectures in the theory of analytic and univalent functions. These latter developments in an area other than the so-called traditional areas of applications of generalized hypergeometric functions have naturally provided a new impetus for the study of such an important class of special functions. With these points in view, we first illustrate the usefulness (in the study of univalent, starlike, and convex generalized hypergeometric functions) of certain families of linear operators which are defined in terms of (for example) fractional derivatives and fractional integrals, Hadamard product or convolution, and so on. We also present a systematic discussion of some characteristics and theorems involving starlike functions and various families of integral operators considered here. Finally, we consider several inclusion theorems associated with the Hardy space of analytic functions, which hold true for various classes of generalized hypergeometric functions whose derivative has a positive real part.