Uniformly convex functions

Theorem 1. Suppose p is a normalized (p(0) = 1) holomorphic function in D. Then p ∈PAR if and only if Re{p(z)} ≥|p(z) −1|/|z|. P r o o f. If p satisfies Re{p(z)} ≥|p(z) −1|/|z|, then clearly Re{p(z)} ≥ |p(z) −1|, so p ∈PAR. Next we want to prove that each p ∈PAR satisfies the inequality. It suffices to verify the inequality at any fixed point a in D. Consider any r with |a| < r < 1. Then for |z| = r we have Re{p(z)} ≥|p(z) −1| = r|p(z) −1|/|z| . Because Re{p(z)} is harmonic and r|p(z) −1|/|z| is subh

Theorem 2. Assume that f(z) is holomorphic and locally univalent in D with f(0) = f ′(0) −1 = 0. Then the following are equivalent: (i) f ∈UCV; (ii) 1 + zf ′′(z)/f ′(z) ∈PAR; (iii) 1 + Re{zf ′′(z)/f ′(z)} ≥|f ′′(z)/f ′(z)| (z ∈D).

Theorem 3. Assume that f ∈UCV. Then 1 + zf ′′(z)/f ′(z) ≺1 + zk′′(z)/k′(z) and f ′(z) ≺k′(z). P r o o f. Let p(z) = 1 + zf ′′(z)/f ′(z). Then 1 + zf ′′(z)/f ′(z) ≺1 + zk′′(z)/k′(z) is the same as p(z) ≺q(z), which follows from Theorem 2(ii). Note that q(z) −1 is a convex univalent function in D. By using a result of Goluzin [G] (see also [P, p. 50]) we may conclude that log f ′(z) = zR 0 p(ζ) −1 ζ dζ ≺ zR 0

Corollary 1 (Distortion Theorem). Assume f ∈UCV and |z| = r < 1. Then k′(−r) ≤|f ′(z)| ≤k′(r) . Equality holds for some z ̸= 0 if and only if f is a rotation of k. P r o o f. Since q(z)−1 is convex univalent in D, it follows that log k′(z) is also convex univalent in D. In fact, the power series for log k′(z) has positive

Corollary 2 (Growth Theorem). Let f ∈UCV and |z| = r < 1. Then −k(−r) ≤|f(z)| ≤k(r) . Equality holds for some z ̸= 0 if and only if f is a rotation of k.

Corollary 3 (Covering Theorem). Suppose f ∈UCV. Then either f is a rotation of k or f(D) ⊇{w : |w| ≤−k(−1)}. Recall that k is continuous on D so k(−1) makes sense. Also, −k(−1) < k(1) since the power series for k has positive coefficients. From Corollary 2 it follows that the functions in UCV are uniformly bounded above by the sharp constant k(1). The following rotation theorem follows from the sub- ordination f ′ ≺k′ given in Theorem 3.

Corollary 4 (Rotation Theorem). Let f ∈UCV and |z0| = r < 1. Then |Arg {f ′(z0)}| ≤max |z|=r Arg{k′(z)} . Equality holds for some z0 ̸= 0 if and only if f is a rotation of k.

Theorem 4. Let f(z) = z + a2z2 + a3z3 + . . . ∈UCV. Then we have the sharp bounds |a2| ≤A2 = 4/π2 , |a3| ≤A3 = 8/(9π2) + 32/(3π4) . Equality holds in either inequality if and only if f is a rotation of k. P r o o f. Since p ≺q, we have |b1| ≤B1 with equality if and only if p(z) is q(eiθz) for some θ ∈R. This implies that |a2| ≤A2 with equality if and only if f is a rotation of k. Rønning [Rø, Thm. 5] also obtained this bound. Moreover, a result of Rogosinski for subordinate functions ([R], see a

Theorem 5. Let f(z) = z + a2z2 + a3z3 + . . . ∈UCV. Then we have the sharp order of growth |an| = O(1/n2). P r o o f. From Example 3, we see that this order is best possible. We now show that there exists a constant M such that n2(n −1)2|an|2 ≤M 2. Because p ≺q and f ′ ≺k′, we have n−1 X k=1 |bk|2 ≤ n−1 X k=1 B2 k and