Properties of tilted univalent analytic functions of order δ / Sidik Rathi - Rathi, Sidik (2015)
This thesis deals with the functions f9 analytic and univalent in the open unit disk denoted as U = {z e C: Izl < lj. Let A be the class of analytic functions/defined in U and S be the subclass of A normalized by /(o) = 0, / ' ( o ) = 1 and has the Taylor series expansion of the form 00 f(z)= z + a2z2 +a3z3 +••• = z+Janzn. n = 2 Also, let P be the subclass of A consisting functions h9 such that Re{/z(z)} > 0 , h(o) = 1 and has the form of 00 h(z) = \ + hxz + h2z2 +••- = !+2_\KZ"< n = \ In this thesis, we investigate on the class P{X9S) of A- tilted Caratheodory functions of order 8 and the subclasses of S denoted by C{A9S)of X-close-toconvex functions of order 8 . Such functions in P {X98) and Cg (A,S) satisfies Re{euh(z)}> 8 and Rejea ^ - ^ 1 > 8 (zeU) respectively with \A\ < — , cos(/l) >8, 0 < 8 < 1 and ga (z) = —f or 0 < a < 1. 2 (l - az) Some basic properties such as representation function, coefficient bounds, distortion theorem and growth theorem for the class P{A,9S) and Cg {X9 8) = C (X9 8) are obtained. The bounds for real and imaginary part of he P(X98) and / ' e Cg(X98) are also determined. We also discuss on the coefficient inequalities which consist of the upper bounds for the second Hankel determinant, a2a4 -a3 and the Feketeszego functional, jua2 . We determined the upper bound for a2a4 •a. for function in C (A98) and the upper bound for # 3 - / ^ 2 for function in C (Z98).Lastly, we discuss on the radius problems which focuses on finding the radius of convexity and the radius of starlikeness for the class C (A98).