GEOMETRIC PROPERTIES OF HYPERGEOMETRIC TYPE FUNCTIONS USING POSITIVITY OF TRIGONOMETRIC SUMS

Abstract

The study of the class of analytic univalent functions and its subclasses is an interesting part of Geometric Function Theory (GFT). The present work deals various conditions and criteria under which the class of analytic functions in unit disk is in various sub-classes of the class of univalent function. This work also deals with the application of these results to hypergeometric type functions, e.g, Cesar° means, Polylogarithm (also the generalization), Bessel functions (also the generalization). For this study, among the various methods available in the literature such as positivity of trigonometric cosine and sine sums, subordination and differential subordination, convolution, duality technique, variational technique, integral operator technique and negative coefficient techniques, the positivity technique is considered because this method gives the coefficient condi-tions that are better than the other methods. At the beginning a brief introduction about univalent function theory is provided. By the influence of the famous Riemann mapping theorem, unit disk is considered as the domain of definition and the class of analytic functions with particular normalization is considered. Definitions for the subclasses of the class of univalent functions, generalized hypergeometric functions and their properties that are prerequisite for the present work are given. A brief account of the related work made by various authors in this direction are presented. Standard notations are used for all the definitions, classes and results that exist in the literature. Explanations for new notations were given wherever they first appear. At first, the n-th Cesaro means of order (3 of the normalized analytic functions are defined using the well known convolution or Hadamard product. Using certain known results on the positivity of certain cosine and sine sums conditions on the parameters such that the Cesexo means is starlike univalent are obtained. By means of examples it has been shown that these results cover some more range than the results available in the literature. These results are applied to find the _conditions such that the Hohlov operator is starlike univalent. Whenever the Taylor coefficients of the analytic function are unity, from above results the conditions for the starlike univalency of normalized Gaussian hypergeometric functions are obtained. In particular results on the incomplete beta functions are deduced from the existing results or obtained directly. The generalized polylogarithm function is studied in the sequel. A relation between the generalized polylogarithm and the generalized hypergeometric functions is given. Using known results, simplified conditions on the parameters such that the generalized polylogarithm is starlike, close-to-convex with respect to particular starlike functions, whose Taylor's series expansion have integer coefficients, are given. Also the conditions for the convexity of the generalized polylogarithm are deduced. Positivity of trigonometric cosine sum and sine sum has its influence in geometric func-tion theory. Hence the generalization of certain well-known results in this direction are considered. The condition on the Taylor coefficients of the analytic function such that the partial sum of certain trigonometric sums involving either sine or cosine terms are simultaneously positive has been obtained and hence generalizing an existing result. An application of this result not only gives a simple proof of an existing result related to the monotonicity of cosine sum available in the literature, but also generalizes the existing result in certain range. The celebrated Vietoris' conditions for cosine sum are extended and some inequality for sine sum are also deduced. Further, a relation between the typically real functions and functions of starlike of order ,u, is proved. Using this result, together with the results obtain earlier, coefficient conditions for the starlikeness of orderμ are deduced. As a first application, conditions on the parameters such that the generalized polylogarithm function is starlike of order ,u, are obtained. For further applications, the Cesar° means of normalized analytic functions is revisited and the coefficient conditions for which these means are starlike and close-to-convex are obtained. By means of examples, it has been established that the Cesar° means of a non-univalent function may be univalent and also starlike. ii To find further applications, the generalized Bessel functions are considered. Here the solution of the generalized Bessel differential equation and its normalized form are con-sidered. Using the results on positivity of trigonometric sums obtained earlier, the con-dition on the parameters such that these Bessel functions are starlike univalent and also convex are obtained. In particular, taking particular values, the geometric properties of the modified Bessel functions are derived. Finally, the n-th Cesar() means of non-normalized analytic functions are considered. A subordination result for these means is given which is the generalization of a known sub-ordination result for the partial sum of the analytic function available in the literature. The concept of stable functions is extended by defining a new concept called Cesar° stable functions. Concluding remarks are given at the end of each chapter. Open problems and further scope for research are underlined wherever possible.