3 CONJECTURE FOR STARLIKE FUNCTIONS1

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which are regular and univalent for |z|
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 34, Number 2, August 1972

A NOTE ON THE 2/3 CONJECTURE FOR STARLIKE FUNCTIONS1 CARL P. MCCARTY AND DAVID E. TEPPER2

Abstract. Let w=f(z)=z+ 2«=2 a«z" be regu'ar and univalent for |z|0 if /?'> —(135/1792). Hence, if (17)

R' = R(l + R2)(4R3 - 3aR2 - 12R - a)-1 > -(1/14),

then we have W(R)>0 and the proof is complete. However, using the

fact that 4R3-3aR2-l2R-a 0.

Recalling l^a^2,

we have:

-18.R3 + 3R2 - 2R + 1 > ((1/3) - R)(ISR2 + 3R + R) > 0

because l/4^Ä

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