Estimates for the Koebe Constant and the Second Coefficient for Some Classes of Univalent Functions

LEMMA 1. ([1]). For f Ç S(d) and \z\ < 1, we have -k(-\z\,d) ^ \f(z)\ ^ k(\z\,d).

LEMMA 2. ([6]). For f(z) = z + a2z2 + ... in S(d), we have M ^|(i-v^)(3v^-i). Equality occurs if and only if f(z) — e~iak(eiaz, d) for some real a. In [5,6] Netanyahu actually considered the class S(d)\Ud>dS(d) in terms of our notation. However, since the bound in Lemma 2 is a decreasing function of d, it is valid for the full class S(d). Example 2. The functions f(z) = — (l/d)k( — k~l(dz, d),d') belong to c/?d'id^ (nere fc-i is with respect to the first argument.) They will be useful in the exp

LEMMA 3. Let f belong to Sf9. Then for 1 ^ p ^ 4 (a) min — - k( — k~l(\z\x, x), px) g |/(s)| S max -k( — k~(—\z\x,x),px) l/^px,x^l x and for p > 4 (b) min - -k(-k~\\z\x, I), px) ^ |/(z)| l / 4 ^ P . r ^ l

THEOREM 1. The Koebe constants for the families 5^p are 2(1 + V^)2 - ( 2 v 7 - i)VQ + v a ( 4 + V a P P2(l+V^) 2+ ( 2 V P - 1 ) V ( 1 + A / P ) ( 4 + V P ) <w i ^ P ^ 4 * V 4 + P - V P , . A V4 + P + V P

THEOREM 2. For fixed p, 0 < p < oo, define x0 = y0 = 0 and yn = max {yni | } , * n +i = pk~l\~ ,yn) (4) x ; p ; xn = max {xn, J}, 3Vfi = - k (p, x„) https://doi.org/10.4153/CJM-1980-101-9 Published online by Cambridge University Press

THEOREM 3. For f(z) — z + a2z2 -f ... in the class Sf9 we have f- (AV~p - 1 - P) for i £ p ^ 4 Sp 2/p for p > 4 and the estimates are sharp.

THEOREM 4. Iff(z) = z + a2s2 + ... belongs to Sp, then \a2\ S min)-,g(p), A(p)f . vp / In the following table we have computed this estimate of \a2\ in the class Sp for various values of p. p 1/4 2.000 1/2 1.973 3/4 1.782 1