It has been conjectured for some time that the coefficients a^ of functions z-1 + a,z + a2z2 + ... , which are univalent in the unit disk, are subject to the inequality (*) j an| < 2(n+l)~ (which was known to hold for n =» 1, 2). This conjecture was recently disproved by M. Schiffer and P. Garabedian, who showed that the largest value of an is 2/1+e-6. E. Netanyahu showed, moreover, that (*) does not hold for any odd index larger than 3. In the present paper it is shown that if the univalent function f (z) is, in addition, starlike with respect to the origin, the sharp inequality (*) holds for n =» 3,4,5,6
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