An extremal problem for univalent functions

Theorem 3.1. For a fixed z ∈D with |z| = r and a real number b, Φ+ z (b, S∗) = log(1 + b2) −2 log( p △−r) + 2b arctan br √△ and Φ− z (b, S∗) = log(1 + b2) −2 log( p △+ r) −2b arctan br √△, where △= 1 + b2(1 −r2).

Corollary 3.2. For a real number b, Φ+(b, S∗) = +∞ and Φ−(b, S∗) = log(1 + b2) −log 4 −2b arctan b.

Theorem 3.1 and Corollary 3.2 assure the following result.

Corollary 3.3. For a fixed z ∈D with |z| = r and a real number b, Φ+ z (b, K) = 1 2Φ+ z (b, S∗), Φ− z (b, K) = 1 2Φ− z (b, S∗) and Φ+(b, K) = +∞, Φ−(b, K) = 1 2 log(1 + b2) −log 2 −b arctan b. 4. Close-to-convex functions Biernacki [2] determined the variability region Wz(C) of log[f(z)/z] for the class C of

Lemma 4.1 (Theorem 1.4 in [5]). The full variability region W(C) for close-to-convex functions is the unbounded Jordan domain whose boundary is the Jordan arc −γ((−2π, 2π)). Here, γ(t) =    log(1 + 3eit) if |t| < π log(1 −eit) + t |t|πi if π ≤|t| < 2π. Note that the region W(C) is contained in the parallel strip {w : | Im w| < 3π/2} as was already shown by Biernacki [2]. By making use of the above lemma, we now describe Φ±(b, C).

Theorem 4.2. Let b be a real number. Then, Φ+(b, C) = +∞and Φ−(b, C) =            −1 2 log 2(5 −4b2 + 3

Corollary 4.3. For a real constant t with |t| < π/2, Ψ(t, C) =            

Theorem 5.1. Let c = a + bi be a complex number. If a > 0, inf f∈C z∈D log