On a subclass of tilted starlike functions with respect to conjugate points / Nur Hazwani Aqilah Abdul Wahid - Abdul Wahid, Nur Hazwani Aqilah (2015)
This thesis is concerned with the function / defined on an open unit disk E = {z : |z| < l} of the complex plane. Let As be the class o f analytic functions defined on E which is normalized and has the Taylor series representation of the form oe f ( z ) = z + a2z 2 + ciyZ* H----- b anz" 4— = z + ^ anz ". Let H be the class of functions 11=2 co which are analytic and univalent in E of the form co(z)=txz + t2z 2 h— + tnz" h— = ^ t nz". We define the class S ’(a ,S ,A ,B ) for «= 1 which functions in the class S ' (a, S, A,B) satisfy the condition f ia zf'(z) . . ^ 1 1 + Aco(z) /x /(z)+ f(z ) e .— 8 - 1 sin a -----= ---------- 7- - , cog H where g\z) - - — and V J L , 1 + Bco(z) 2 I I 71 tm = c o s a - S with |ar| < —, cos« > <*>, 0 < S < 1 and - \< B < A < \ . Some of the basic properties are obtained for the class S '(a ,S ,A , B) such as distortion theorem, z f ' ( z ) growth theorem, argument of -—7—^. and coefficient bounds. The upper and lower g{z ) zf'(z) z f ' i2) bounds of Re—— and Im ■: .r~ for functions in the class S (a,S ,A,B) are also g \z ) g\z) given. This thesis also discusses on the radius problems which are the radius of convexity and the radius of starlikeness for the defined class. Lastly, the coefficient inequalities problems which are the upper bounds for the Second Hankel determinant a2aA- a ^ | and Fekete-Szego functional a3 - /w , 2 are determined for functions in the class S ’(a,S,A, B). Also included the coefficient determinant with Fekete-Szego parameter which is la,a, - f i a \