SHARP BOUNDS FOR INITIAL COEFFICIENTS AND THE SECOND HANKEL DETERMINANT

For functions f (z) = z + a(2)z(2) + a(3)z(3) + ... belonging to particular classes, this paper finds sharp bounds for the initial coefficients a(2), a(3), a(4), as well as the sharp estimate for the second order Hankel determinant H-2 (2) = a(2)a(4) - a(3)(2). Two classes are treated: first is the class consisting of f (z) = z + a(2)z(2) + a(3)z(3) + ... in the unit disk D satisfying vertical bar(z/f(z)(1+alpha) f'(z) - 1 vertical bar < lambda, 0 < alpha < 1, 0 < lambda <= 1. The second class consists of Bazilevic functions 1(z) = z+a(2)z(2)+a(3)z(3)+... in D satisfying Re {(f(z)/z(alpha-1 )f'(z)} > 0, alpha > 0.