Hyperbolically convex functions II

Lemma 2.1. Let p(z) ∈P. Then we have the sharp estimates: for 0 < |z| < 1, (2.1) |2zp′(z) + 1 −p(z)2| + p(z) −1 + |z|2 1 −|z|2

Theorem 3.1 (Mejia–Pommerenke). Let f(z) be holomorphic and locally univalent in D with f(D) ⊂D. Then f(z) is hyperbolically convex if and only if (3.1) Re  2zf ′(z) f(z) −f(a) −z + a z −a + 2f(a)zf ′(z) 1 −f(a)f(z)  > 0 (z ∈D, a ∈D).

Theorem 3.1 can be used to prove that the nonholomorphic function p(z) = 1 + zf ′′(z) f ′(z) + 2f(z)zf ′(z) 1 −|f(z)|2 still satisfies p(z) −1 + |z|2 1 −|z|2 ≤ 2|z| 1 −|z|2 , which is a well known result for P. We state the result for hyperbolically convex functions; a similar characterization with strict inequality was given in [3], and its proof was quite different from what we have here.

Corollary 3.2. Suppose f(z) is holomorphic and locally univalent in D with f(D) ⊂D. Then f(z) is hyperbolically convex if and only if

Theorem 4.1. If f(z) is hyperbolically convex with f(0) = 0, then for every a ∈D, Fa(z) = za f(a) f(z) −f(a) (z −a)(1 −f(a)f(z)) is starlike of order 1/2. P r o o f. Fa(z) is starlike of order 1/2 if and only if Re 2zF ′ a(z) Fa(z) −1 

Theorem 3.1 then implies the desired inequality. Mejia and Pommerenke [6] showed that for hyperbolically convex func- tions f(z) with f(0) = 0, Re  a f(a) f(z) −f(a) z −a  > 1 2. As a corollary of Theorem 4.1, we now state a similar result. We use the fact that Re{F(z)/z} > 1/2 if F(z) is starlike of order 1/2 (see [7, p. 49]).

Corollary 4.2. If f(z) is hyperbolically convex with f(0) = 0, then for every a ∈D, Re  a f(a) f(z) −f(a) (z −a)(1 −f(a)f(z))  > 1 2 (z ∈D).

Corollary 4.3. If f(z) is hyperbolically convex with f(0) = 0, then for every a ∈D, Fa(z)2 z = za2 f(a)2 (f(z) −f(a))2 (z −a)2(1 −f(a)f(z))2 is starlike in D. When a = 0, this result is due to Mejia and Pommerenke [6]. As another application of Theorem 3.1, we provide a lower bound on Re{a2f(z)/α2} for hyperbolically convex functions.

Theorem 4.4. Let f(z) = αz +a2z2 +. . . be hyperbolically convex. Then Re a2f(z) α2  > −1 (z ∈D). P r o o f. For any a ∈D, we define p(z) by p(z) = 2zf ′(z) f(z) −f(a) −z + a z −a + 2f(a)zf ′(z) 1 −f(a)f(z) , which belongs to P by Theorem 3.1. Direct differentiation then yields

Theorem 5.1. Suppose f(z) (not necessarily normalized) is holomorphic and locally univalent in D with f(D) ⊂D. Then f(z) is hyperbolically convex if and only if (1 −|z|2)2|Sf(z)| + 3 4

Theorem 3.1 tells us that p(z) ∈P. Long but straightforward calculations result in p′(z) = 2(f ′(z) + zf ′′(z))(f(z) −f(a)) −2zf ′(z)2 (f(z) −f(a))2 + 2a (z −a)2 + 2f(a) (1 −f(a)f(z))2 [(f ′(z) + zf ′′(z))(1 −f(a)f(z)) + f(a)zf ′(z)2]

Corollary 5.2. Let f(z) = αz + a2z2 + a3z3 + . . . be hyperbolically convex in D. Then

Theorem 6.1. Let f(z) = αz +a2z2 +. . . be hyperbolically convex. Then for all z ∈D, |f ′(z)| ≥k′ α(−|z|), and for |z| ≤ √ 2 −1, |f ′(z)| ≤k′ α(|z|). P r o o f. In [3], we obtained the sharp growth theorem −kα(−|z|) ≤ |f(z)| ≤kα(|z|) for hyperbolically convex f(z) = αz + a2z2 + . . . That is, if f(z) is hyperbolically convex with f(0) = 0, then 2|f ′(0)||z| 1 + |z| +