On the second Hankel determinant of some analytic functions

Abstract

Let the function f be analytic in zD  z : z 1 and be given by   2 n . n n f z z az      For 0   1, denote by V   and U  , the sets of functions analytic in D, satisfying         '' Re 1 ' 1 0 ' zf z f z f z                   and         ' Re 1 0 f z zf z z fz            respectively, so that f V   zf 'U  . We give sharp bounds for the Hankel determinant 2 2 2 4 3 H  a a  a for f V   and f U  .