RADIUS, COEFFICIENT CONSTANTS AND DIFFERENTIAL SUBORDINATIONS OF CERTAIN UNIVALENT FUNCTIONS

Abstract

This thesis contributes several new results to the basic theory of the univalent function theory. Although there are several elegant and beautiful articles available in the literature, but it was Ma and Minda’s pa per “A unified treatment of some special classes of univalent functions. Proceedings of the Conference on Complex Analysis, Tianjin, Conf Proc Lecture Notes Anal., I Int Press, Cambridge, MA. 157–169 (1992)”, which paved the way for new research in univalent function theory. It is worth here to mention that the Ma-Minda class covers the classical classes of the univalent starlike and convex functions. After this, various problems were studied for a specific subclass of univalent functions, particular focus has been on the subclasses of starlike functions. In this thesis, we study a variety of problems for the Ma-Minda classes. Hence, either we generalize the known results or establish some new results for this class. In brief, we generalize certain results, which trace their origin to the following defining articles: “T.H. MacGregor, Majorization by univalent functions. Duke Math. J. 34, 95–102 (1967).” Interest in special functions in view of radius problem can be seen from the papers “R.K. Brown, Univalence of Bessel functions. Proc. Amer. Math. Soc. 11, 278–283 (1960)”, “R.K. Brown, Univalent solutions of W′′ + pW = 0. Canadian J. Math. 14, 69–78 (1962)”, “H.S. Wilf, The radius of univalence of certain entire functions. Illinois J. Math. 6, 242–244 (1962)” and “E. Kreyszig and J. Todd, The radius of univalence of Bessel functions. I, Illinois J. Math. 4, 143–149 (1960).” About absolute power series sum and its connection to the univalent function theory follows from the papers “H. Bohr, A Theorem Concerning Power Series, Proc. London Math. Soc. (2) 13 (1914), 1–5” and “Y.A. Muhanna, Bohr’s phenomenon in subordination and bounded harmonic classes, Complex Var. Elliptic Equ. 55 (2010), no. 11, 1071–1078.” The work on convolution and relevant radius problem comes from the papers “G. Szegö, Zur Theorie der schlichten Abbildungen, Math. Ann. 100 (1928), no. 1, 188–211” and “H. Silverman, Radii problems for sections of convex functions, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1191–1196.” In the context of the above, we prove the classical results and establish topics of current interest for the general Ma-Minda classes. In certain investigations, particularly chapter 3 and chapter 5 pave the way for future research.