Abstract
M. Biernacki gave concrete forms of the variability regions of $z/f(z)$ and $zf'(z)/f(z)$ of close-to-convex functions $f$ for a fixed $z$ with $|z|<1$ in 1936. The forms are, however, not necessarily convenient to determine the shape of the full variability region of $zf'(z)/f(z)$ over all close-to-convex functions $f$ and all points $z$ with $|z|<1.$ We will propose a couple of other forms of the variability regions and see that the full variability region of $zf'(z)/f(z)$ is indeed the comple
Results & Lemmas (18)
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Lemma 1.1.
Lemma 1.1. (1) Xz is a compact subset of C for each z ∈D and X = U, V, W, LU, LV, LW. (2) Xz = exp(LXz) for each z ∈D and X = U, V, W. (3)…
Lemma 1.1. (1) Xz is a compact subset of C for each z ∈D and X = U, V, W, LU, LV, LW. (2) Xz = exp(LXz) for each z ∈D and X = U, V, W. (3) Xz = Xr for |z| = r < 1 and X = U, V, W, LU, LV, LW. (4) Xr ⊂Xs for 0 ≤r < s < 1 and X = U, V, W, LU, LV, LW.
Lemma 1.2
Lemma 1.2 (Biernacki (1936)). For 0 < r < 1, the following hold: (1) Ur = (1+s)2/(1+ s+t 2 ): |s| ≤r, |t| ≤r = 2u2/(u+v): |u−1| ≤r, |v −1|…
Lemma 1.2 (Biernacki (1936)). For 0 < r < 1, the following hold: (1) Ur = {(1+s)2/(1+ s+t 2 ) : |s| ≤r, |t| ≤r} = {2u2/(u+v) : |u−1| ≤r, |v −1| ≤r}. (2) Wr = (1 −r2)−2Ur. (3) U1−= D(1, 3) \ {0} and LU1−⊂{w ∈C : |Im w| < 3π/2}. (4) LW1−⊂{w ∈C : |Im w| < 3π/2}. The above expressions of Ur and Wr are simple but somewhat implicit. For instance, the parametrization of the boundary curve cannot be obtained immediately and the shape of the limit W1−is not clear (as we will see below, this set is equal
Theorem 1.3.
Theorem 1.3. For 0 < r < 1, Ur = F(D(0, r)), where F(z) = (3 + ¯z)(1 + z)3 3 + 3z + ¯z + z2, z ∈D.
Theorem 1.3. For 0 < r < 1, Ur = F(D(0, r)), where F(z) = (3 + ¯z)(1 + z)3 3 + 3z + ¯z + z2, z ∈D.
Theorem 1.4.
Theorem 1.4. The variability region LU1−is an unbounded Jordan domain with the boundary curve γ(t), −2π < t < 2π, given by γ(t) = Log…
Theorem 1.4. The variability region LU1−is an unbounded Jordan domain with the boundary curve γ(t), −2π < t < 2π, given by γ(t) = Log (1 + 3eit) if |t| < π Log (1 −eit) + t |t|πi if π ≤|t| < 2π. Here and hereafter, Log w = log |w|+iArg w denotes the principal branch of log w with −π < Im Log w = Arg w ≤π. As we will see in the next section, the function log F is univalent in D. Therefore, the last theorem tells us that F : D →U1−covers the disk D(−1, 1) bivalently whereas it
Theorem 1.5.
Theorem 1.5. For 0 < r < 1, Wr = 2u v(u + v): |u −1| ≤r, |v −1| ≤r . Indeed, as an application of the last theorem, we can show the…
Theorem 1.5. For 0 < r < 1, Wr = 2u v(u + v) : |u −1| ≤r, |v −1| ≤r . Indeed, as an application of the last theorem, we can show the following result.
Theorem 1.6.
Theorem 1.6. LW1−= w ∈C: |Im w| < 3π/2. Since W1−= exp(LW1−), we obtain the following corollary, which was used in [7].
Theorem 1.6. LW1−= {w ∈C : |Im w| < 3π/2}. Since W1−= exp(LW1−), we obtain the following corollary, which was used in [7].
Corollary 1.7.
Corollary 1.7. The full variability region zf ′(z)/f(z): z ∈D, f ∈C is equal to C 0. The corollary means that W1−= C 0. We note here that…
Corollary 1.7. The full variability region {zf ′(z)/f(z) : z ∈D, f ∈C} is equal to C\{0}. The corollary means that W1−= C \ {0}. We note here that this does not seem to follow immediately from Lemma 1.2. Krzy˙z [9] showed that LVr is convex and determined its shape for 0 < r < 1.
Proposition 1.8
Proposition 1.8 (Krzy˙z). For 0 < r < 1, the variability region LVr is convex and its boundary is described by the curve σr(t) = log(1…
Proposition 1.8 (Krzy˙z). For 0 < r < 1, the variability region LVr is convex and its boundary is described by the curve σr(t) = log(1 −reiθ2(t))/(1 −reiθ1(t))3, −π ≤t ≤π. Here, θ1(t) = t −arcsin(r sin t), θ2(t) = π + t + arcsin(r sin t). He also proved that LVr is contained in the domain |Im w| < 4 arcsin r for each 0 < r < 1 and that this bound is sharp. (See also [4, Chap. 11].) In particular, LV1−⊂{w : |Im w| < 2π}. Since Re σr(t) →+∞as r →1−for |t| < π/2 and Re σr(t) →−∞as r →1−for π/2 < |t
Theorem 1.9.
Theorem 1.9. Vr = (1 + s)/(1 + t)3: |s| ≤r, |t| ≤r = u/v3: |u −1| ≤r, |v −1| ≤r for 0 < r < 1. Moreover, LV1−= w: |Im w| < 2π and V1−= C 0.…
Theorem 1.9. Vr = {(1 + s)/(1 + t)3 : |s| ≤r, |t| ≤r} = {u/v3 : |u −1| ≤r, |v −1| ≤r} for 0 < r < 1. Moreover, LV1−= {w : |Im w| < 2π} and V1−= C \ {0}. One might expect that LUr and LWr would also be convex for each 0 < r < 1. This is, however, not true unlike LVr.
Theorem 1.10.
Theorem 1.10. The variability regions LUr and LWr are closed Jordan domains for each 0 < r < 1. Moreover, there exists a number 0 < r0 < 1…
Theorem 1.10. The variability regions LUr and LWr are closed Jordan domains for each 0 < r < 1. Moreover, there exists a number 0 < r0 < 1 such that both LUr and LWr are not convex for every r with r0 < r < 1. We prove the above results in Section 3. Section 2 will be devoted to the study of mapping properties of the function G = log F that are necessary to show our results. 2. Univalence of the function G = log F In order to analyze the shape of LUr or LWr, we need to investigate mapping proper
Theorem 2.1.
Theorem 2.1. The function G = log F is a homeomorphism of the unit disk D onto the domain LU1−. For r ∈(0, 1) and x ∈(0, π), we set Φr(x) =…
Theorem 2.1. The function G = log F is a homeomorphism of the unit disk D onto the domain LU1−. For r ∈(0, 1) and x ∈(0, π), we set Φr(x) = Arg (1 + reix). We will use the following elementary properties of the function Φr.
Lemma 2.2.
Lemma 2.2. Let r ∈(0, 1). Then Φ′ r(x) = r(r + cos x) 1 + 2r cos x + r2, x ∈(0, π). In particular, Φr(x) is increasing in 0 < x < xr and…
Lemma 2.2. Let r ∈(0, 1). Then Φ′ r(x) = r(r + cos x) 1 + 2r cos x + r2, x ∈(0, π). In particular, Φr(x) is increasing in 0 < x < xr and decreasing in xr < x < π, where xr = π −arccos r. Furthermore, Φ′ r(x) is decreasing in 0 < x < π and therefore Φr is concave in (0, π). We also need the following information.
Lemma 2.3.
Lemma 2.3. Let 0 < r < 1. Then the inequalities 0 < θ + 2φ < π hold for 0 < θ < π and φ = Arg (3 + reiθ).
Lemma 2.3. Let 0 < r < 1. Then the inequalities 0 < θ + 2φ < π hold for 0 < θ < π and φ = Arg (3 + reiθ).
Lemma 2.4.
Lemma 2.4. The function G is orientation-preserving and locally univalent in D.
Lemma 2.4. The function G is orientation-preserving and locally univalent in D.
Lemma 2.5.
Lemma 2.5. For a fixed 0 < r < 1, the real part of G(reiθ) is a decreasing function in 0 ≤θ ≤π.
Lemma 2.5. For a fixed 0 < r < 1, the real part of G(reiθ) is a decreasing function in 0 ≤θ ≤π.
Lemma 2.6.
Lemma 2.6. Im G(z) > 0 for z ∈D with Im z > 0.
Lemma 2.6. Im G(z) > 0 for z ∈D with Im z > 0.
Lemma 2.7.
Lemma 2.7. The function gr defined in (2.2) satisfies g′ r(θ) > 0 for 0 < θ < xr = π −arccos r.
Lemma 2.7. The function gr defined in (2.2) satisfies g′ r(θ) > 0 for 0 < θ < xr = π −arccos r.
Lemma 3.1
Lemma 3.1 ([5, Lemma 5.1]). Let λ1 and λ2 be continuous linear functionals on A such that λ2 does not vanish on C. Then for every f ∈C…
Lemma 3.1 ([5, Lemma 5.1]). Let λ1 and λ2 be continuous linear functionals on A such that λ2 does not vanish on C. Then for every f ∈C there exist complex numbers a, b with |a| ≤1, |b| ≤1 such that λ1(f) λ2(f) = λ1(fa,b) λ2(fa,b).
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