On the generalization of some classes of close-to-convex and typically real functions

The paper introduces a class C(lambda, a, gamma) of functions f(z), analytic in the unit disk E = {z :vertical bar z vertical bar 1}, having a power series expansion of the form f (z) = z + a(2)z(2) + a(3)z(3) +...,, z is an element of E, and satisfying the condition vertical bar[(1- lambda z(2)) f'(z)](1/gamma) - a vertical bar <= a, where 0 <= lambda <= 1, 0 < gamma <= 1, and a > 1/2. It is proved that all functions of class C(lambda, a, gamma) are close-to-convex of the order of gamma. Class C(lambda, a, gamma) generalizes classes of functions with bounded turning Re f'(z)>= 0 (as, a -> +infinity, lambda = 0) and functions f(z) convex in the direction of the imaginary axis Re[(1 - lambda z(2))f'(z)] >= 0 (a -> +infinity, lambda =1 ) and creates a simple parametric passage from one class to another. Based on the subordination method in class C(lambda, a, gamma) and its subclasses, exact estimates are obtained for vertical bar f'(z)vertical bar,vertical bar f(z)vertical bar, vertical bar zf(n)(z) / f'(z) and the exact radii of the convexity. In particular cases, they yield previously known results for functions with bounded turning and functions convex in the direction of the imaginary axis. Using the relationship f (z) is an element of C(lambda, a, gamma) double left right arrow F(z) = zf'(z) is an element of T(lambda, a, gamma), the article introduces the class T(lambda, a, gamma) = {F(z): vertical bar[(1 - lambda z(2))F(z)/z](1/gamma) - a vertical bar <= a}functions generalizing the class of typically real functions and the class of functions satisfying the condition Re[F(z) / z] >= 0. In the class T(lambda, a, gamma) and its subclasses, exact estimates of vertical bar F(z)vertical bar, vertical bar zF'(z) / F(z)vertical bar are found and the exact radii of starlikeness are determined, which generalizes the classical results for typically real functions.