Ma-Minda φ-classes studied in this paper:
Abstract
For an analytic function $f(z)$ on the unit disk $|z|<1$ with $f(0)=f'(0)-1=0$ and $f(z)\ne0, 0<|z|<1,$ we consider the power deformation $f_c(z)=z(f(z)/z)^c$ for a complex number $c.$ We determine those values $c$ for which the operator $f\mapsto f_c$ maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity $zf'(z)/f(z),~|z|<1,$ for the class in most cases which we
Results & Lemmas (8)
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Theorem 1.1.
Theorem 1.1. (1) [S∗, S]K = [S∗, SP]K = D( 1 2, 1 2). (2) [S∗(α), S]K = [S∗(α), SP]K = D 1 2(1−α), 1 2(1−α) for 0 ≤α < 1. (3) [K, S]K =…
Theorem 1.1. (1) [S∗, S]K = [S∗, SP]K = D( 1 2, 1 2). (2) [S∗(α), S]K = [S∗(α), SP]K = D 1 2(1−α), 1 2(1−α) for 0 ≤α < 1. (3) [K, S]K = [K, SP]K = D(1, 1). (4) [SP(λ), S]K = [SP(λ), SP]K = D 1−i tan λ 2
Theorem 1.2.
Theorem 1.2. Let f be a strongly spirallike function. Then Log f(z)/z is bounded on D. In particular, f(z) is bounded on D. We note that…
Theorem 1.2. Let f be a strongly spirallike function. Then Log f(z)/z is bounded on D. In particular, f(z) is bounded on D. We note that boundedness of strongly starlike functions is due to Brannan and Kirwan [3]. 2. Fundamental facts In this section, we collect fundamental properties of the operators Ic, Jc, Kc and the sets [M, N ]X of exponents for X = I, J, K. We first observe that the Alexander transformation J1 maps the class ZF of zero-free functions onto LU, the class of locally univalent
Lemma 2.1.
Lemma 2.1. Let X represent one of I, J, K and let M, M′, Mλ ⊂DX (λ ∈Λ), N, N ′ ⊂ RX. Then the following hold: (1) [M, N ]X ⊃[M′, N ]X if M…
Lemma 2.1. Let X represent one of I, J, K and let M, M′, Mλ ⊂DX (λ ∈Λ), N , N ′ ⊂ RX. Then the following hold: (1) [M, N ]X ⊃[M′, N ]X if M ⊂M′. (2) [M, N ]X ⊂[M, N ′]X if N ⊂N ′. (3) [S λ∈Λ Mλ, N ]X = T λ∈Λ[Mλ, N ]X. (4) [T λ∈Λ Mλ, N ]X ⊃S λ∈Λ[Mλ, N ]X. (5) [Xc(M), N ]X = 1 c [M, N ]X for c ∈C \ {0} and for X = I, K.
Lemma 2.2.
Lemma 2.2. For a function f ∈ZF and c ∈C, let fc = Kc[f]. (1) If Log f(z)/z is bounded in D, then so is Log fc(z)/z for every c ∈C. (2) If…
Lemma 2.2. For a function f ∈ZF and c ∈C, let fc = Kc[f]. (1) If Log f(z)/z is bounded in D, then so is Log fc(z)/z for every c ∈C. (2) If log |f(z)/z| is unbounded in D, then so is log |fc(z)/z| for every c > 0. (3) Suppose that f is unbounded and univalent in D and that Arg f(z)/z is bounded in D. Then fc is never univalent when Re c < 0 while fc is unbounded when Re c > 0.
Lemma 2.3.
Lemma 2.3. For a subclass M of ZF, the set [M, LU]K and the variability region V (M) of zf ′(z)/f(z) are related by [M, LU]K = C T(V (M)).
Lemma 2.3. For a subclass M of ZF, the set [M, LU]K and the variability region V (M) of zf ′(z)/f(z) are related by [M, LU]K = C \ T(V (M)).
Corollary 2.4.
Corollary 2.4. Let M be a subclass of ZF which contains a function f ̸= id. Then [M, LU]K is a compact subset of C.
Corollary 2.4. Let M be a subclass of ZF which contains a function f ̸= id. Then [M, LU]K is a compact subset of C.
Lemma 2.5.
Lemma 2.5. One has the following relations: (1) V (S∗) = w: Re w > 0. (2) V (S∗(α)) = w: Re w > α. (3) V (K) = w: Re w > 1/2. (4) V (SP(λ))…
Lemma 2.5. One has the following relations: (1) V (S∗) = {w : Re w > 0}. (2) V (S∗(α)) = {w : Re w > α}. (3) V (K) = {w : Re w > 1/2}. (4) V (SP(λ)) = {w : Re e−iλw > 0}. (5) V (SP) = C \ (−∞, 0]. (6) V (SS(α)) = {w : | arg w| < πα/2}. (7) V (SP(λ, α)) = {w : | arg w −λ| < πα/2}. (8) V (S) = V (C) = C \ {0}.
Lemma 3.1
Lemma 3.1 (Goodman [6]). |Arg f(z)/z| ≤2 arcsin |z| < π, |z| < 1, for f ∈S∗. We are now ready to prove Theorem 1.2.
Lemma 3.1 (Goodman [6]). |Arg f(z)/z| ≤2 arcsin |z| < π, |z| < 1, for f ∈S∗. We are now ready to prove Theorem 1.2.
Function classes studied:
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