GEOMETRIC PROPERTIES OF STARLIKE FUNCTIONS ASSOCIATED WITH A NEPHROID DOMAIN

Abstract

Let A denote the collection of all analytic functions f(z) defined on the open unit disk D satisfying f(0) = f0(0) − 1 = 0, and let the symbol denote subordination. The central idea of this thesis is to introduce a novel class of analytic functions S Ne given by S Ne := ( f 2 A : zf0(z) f(z) 1 + z − z3 3 ) , where the function 'Ne(z) := 1+z−z3/3 maps D univalently onto a 2-cusped, heartshaped curve called nephroid. The 2-cusped nature of the nephroid domain 'Ne(D) makes the geometries of the functions in S N e more fascinating and appealing, and this thesis is an exploration (rather visual) of these geometries. In this thesis, we consider the function class S Ne and discuss several interesting geometrical and analytical properties of its members. For instance, it is proved that for each f 2 S N e, the region f(D) is starlike with respect to origin containing the disk n w 2 C : |w| < e−8/9 o and contained in n w 2 C : |w| < e8/9 o . Further, it is verified that the quantity zf0(z)/f(z) assumes values from the sector {w 2 C : | arg w| < 0 /2}, where 0 0.929. Inclusion type relations of S N e with other geometrically defined function classes are discussed in detail. Sharp estimates on the Fekete-Szeg¨o functional |a3−μa22 | and the first few Taylor coefficients for the function f(z) = z + P1 n=2 anzn 2 S N e are obtained. Employing the first-order differential subordination techniques, a number of sufficient conditions are established in order that the function f 2 A happens to be a member of S N e. Generally, these techniques involve the determination of 2 R so i that the first-order differential subordination 1 + zp0(z) pj(z) P(z), z 2 D, j 2 {0, 1, 2} implies the subordination p(z) 'Ne(z), where P(z) is some Carath´eodory function with certain specific geometries (convex, non-convex, cusped, dimpled, etc.), and p(z) is analytic satisfying p(0) = 1. The famous radius problems related to the function class, S N e, have also been discussed in this thesis. Redefining, in geometrical terms, the radius problem for S Ne, we compute sharp estimates on r 2 (0, 1) so that in the disk |z| < r, each member of some well-known function family G A continues to have the properties possessed by the members of S Ne. Calling such a radius as the nephroid-radius for the function family G, we investigate nephroid-radius for many classical and newly introduced function families in the theory of univalent functions. In particular, the nephroidradii for the starlike class S , the convex class C, and the Sok´o l-Stankiewicz class S L associated with the lemniscate of Bernoulli are obtained to be 1/4, 2/5 and 8/9, respectively. More importantly, the sharpness of the radius estimates is illustrated graphically besides the analytic verification. Finally, we consider another function class 0 whose members f(z) satisfy the differential inequality |zf0(z)−f(z)| < 1/2 in D rather than being characterized by zf0(z)/f(z) P(z) for some Carath´eodory function P(z). Apart from determining sufficient conditions for 0, we study inclusion properties of 0 concerning functions that map D onto certain parabolic regions. The techniques used are different from the ones employed to study the class S N e. Moreover, some radii problems associated with 0 are solved, and the sharp nephroid-radius for 0 is found to be 4/5.