COEFFICIENT INEQUALITIES AND YAMASHITA'S CONJECTURE FOR SOME CLASSES OF ANALYTIC FUNCTIONS

For any real number beta with beta > 1, let M (beta) (N (beta) respectively) denote the class of analytic functions f in the unit disk D : = { z is an element of C : vertical bar z vertical bar < 1} of the form f (z) = z + Sigma(infinity)(n=2) a(n)z(n) and satisfying Re P-f < beta (Re Q(f) < beta respectively) in D, where P-f = zf'(z) = f (z) and Q(f) = 1 + zf ''(z) = f'(z). Also, for beta > 1, let M Sigma(beta) (N Sigma(beta) respectively) denote the class of analytic functions g of the form g(z) = z(1 + Sigma(infinity)(n=1) b(n)z(-n)) and satisfying Re P-g < beta (Re Q(g) < beta respectively) for z is an element of Delta = {z is an element of C : 1 < vertical bar z vertical bar < infinity}. In this paper, we shall determine the coefficient bounds, inverse coefficient bounds, the growth and distortion theorem and the upper bounds for the Fekete-Szego functional Lambda(lambda) (f) = a(3) - lambda a(2)(2) for functions f in the classes M (beta) and N (beta). Further, we shall solve the maximal area problem for functions of the type z = f (z) when f is an element of M (beta), which is Yamashita's conjecture for the class M(beta). We shall obtain the radius of convexity for the class N Sigma(beta). We shall also determine the coefficient bounds for functions g in the classes M Sigma(beta) and N Sigma(beta) and the inverse coefficient bounds for functions g in the class M Sigma(beta). All the results are sharp.