Abstract
We study the class ${\mathcal C}(Ω)$ of univalent analytic functions $f$ in the unit disk $\mathbb{D} = \{z \in \mathbb{C} :\,|z|<1 \}$ of the form $f(z)=z+\sum_{n=2}^{\infty}a_n z^n$ satisfying \[ 1+\frac{zf"(z)}{f'(z)} \in Ω, \quad z\in \mathbb{D}, \] where $Ω$ will be a proper subdomain of ${\mathbb C}$ which is starlike with respect to $1 (\in Ω)$. Let $φ_Ω$ be the unique conformal mapping of ${\mathbb D}$ onto $Ω$ with $φ_Ω(0)=1$ and $φ_Ω'(0) > 0$ and $ k_Ω(z) = \int_0^z \exp \left(\int_0^t
Results & Lemmas (10)
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Theorem 2
Theorem 2] and [6, p. 39], and moreover, m(r) ≥ √ 6 2 πr (1 −r)2. The extremal problem (1.1) stimulated much research in the theory of…
Theorem 2] and [6, p. 39], and moreover, m(r) ≥ √ 6 2 πr (1 −r)2. The extremal problem (1.1) stimulated much research in the theory of univa- lent functions, and the problem of determining of the maximum value and the extremal functions in S remains open. However, the extremal problem (1.2) max f∈F Lr(f) has been solved for a number of subclasses F of S. In order to motivate these known results and also for our further discussion on this topic, we need to intro-
Theorem 1.1.
Theorem 1.1. If Ωis starlike with respect to 1, then, for f ∈C(Ω), we have (1.4) Lr(f) ≤Lr(kΩ) with equality if and only if f(z) = εkΩ(εz)…
Theorem 1.1. If Ωis starlike with respect to 1, then, for f ∈C(Ω), we have (1.4) Lr(f) ≤Lr(kΩ) with equality if and only if f(z) = εkΩ(εz) for some ε ∈∂D. Let f and F be analytic functions in D. Then f is said to be subordinate to F (f ≺F, or f(z) ≺F(z) in D, in short) if there exists an analytic function ω in D with |ω(z)| ≤|z| and f(z) = F(ω(z)) in D. In particular f(D) ⊂F(D) holds, if f ≺F. Notice that when F is univalent in D, f ≺F if and only if f(D) ⊂F(D) and f(0) = F(0). Furthermore, by m
Theorem 1.2.
Theorem 1.2. If Ωis starlike with respect to 1, then log k′ Ω(z) is convex uni- valent in D and log f ′(z) ≺log k′ Ω(z) holds for f ∈C(Ω).…
Theorem 1.2. If Ωis starlike with respect to 1, then log k′ Ω(z) is convex uni- valent in D and log f ′(z) ≺log k′ Ω(z) holds for f ∈C(Ω). Furthermore for any subharmonic function u in the domain log k′ Ω(D) and r ∈(0, 1) Z π −π u(log f ′(reiθ)) dθ ≤ Z π −π u(log k′ Ω(reiθ)) dθ holds with equality for some u and r ∈(0, 1) if and only if u is harmonic in log k′
Corollary 1.3.
Corollary 1.3. If Ωis starlike with respect to 1, then for any f ∈C(Ω) and r ∈(0, 1) the following inequalities hold. Z π −π log | log f…
Corollary 1.3. If Ωis starlike with respect to 1, then for any f ∈C(Ω) and r ∈(0, 1) the following inequalities hold. Z π −π log | log f ′(reiθ)| dθ ≤ Z π −π log | log k′ Ω(reiθ)| dθ, (1.5) Z π −π | log f ′(reiθ)|p dθ ≤ Z π −π
Theorem 1.4.
Theorem 1.4. If Ωis starlike with respect to 1 and symmetric with respect to R, then log |k′ Ω(reiθ)| is a symmetric function of θ and…
Theorem 1.4. If Ωis starlike with respect to 1 and symmetric with respect to R, then log |k′ Ω(reiθ)| is a symmetric function of θ and nonincreasing on [0, π], and for any convex and nondecreasing function Φ in R and any Lebesgue measurable set E ⊂[−π, π] of Lebesgue measure 2θ, we have Z E Φ(log |f ′(reis)|) ds ≤ Z θ −θ Φ(log |k′ Ω(reis)|) ds. In particular Z E
Lemma 2.1.
Lemma 2.1. Let f, F ∈A with f ≺F. Then for any subharmonic function u in F(D) and r ∈(0, 1) (2.1) Z π −π u(f(reiθ) dθ ≤ Z π −π u(F(reiθ))…
Lemma 2.1. Let f, F ∈A with f ≺F. Then for any subharmonic function u in F(D) and r ∈(0, 1) (2.1) Z π −π u(f(reiθ) dθ ≤ Z π −π u(F(reiθ)) dθ with equality if and only if f(z) = F(εz) for some ε ∈∂D or u is harmonic in F(D(0, r)).
Theorem 1.2.
Theorem 1.2. Furthermore the subharmonic function reRe w is not harmonic in any domain in C. Thus if equality holds in (1.4), then log f…
Theorem 1.2. Furthermore the subharmonic function reRe w is not harmonic in any domain in C. Thus if equality holds in (1.4), then log f ′(z) = log k′ Ω(εz) for some ε ∈∂D and hence we obtain that f(z) = εQΩ(εz). □
Lemma 3.1.
Lemma 3.1. ([2, p. 150]) For h, H ∈L1[−π, π], the following statements are equivalent: (a) For every convex nondecreasing function Φ on R Z…
Lemma 3.1. ([2, p. 150]) For h, H ∈L1[−π, π], the following statements are equivalent: (a) For every convex nondecreasing function Φ on R Z π −π Φ(h(θ)) dθ ≤ Z π −π Φ(H(θ)) dθ, (b) For every t ∈R Z π −π (h(θ) −t)+ dθ ≤ Z π −π
Lemma 3.2.
Lemma 3.2. Let f, F ∈A with f ≺F. Then for any subharmonic function u in F(D) and r ∈(0, 1) (u ◦f)∗(reiθ) ≤(u ◦F)∗(reiθ), 0 ≤θ ≤π.
Lemma 3.2. Let f, F ∈A with f ≺F. Then for any subharmonic function u in F(D) and r ∈(0, 1) (u ◦f)∗(reiθ) ≤(u ◦F)∗(reiθ), 0 ≤θ ≤π.
Lemma 2.1
Lemma 2.1 thatZ π −π (u ◦f(reiθ) −t)+ dθ ≤ Z π −π (u ◦F(reiθ) −t)+ dθ. Therefore it follows from Lemma 3.1 that (u ◦f)∗(reiθ) ≤(u…
Lemma 2.1 thatZ π −π (u ◦f(reiθ) −t)+ dθ ≤ Z π −π (u ◦F(reiθ) −t)+ dθ. Therefore it follows from Lemma 3.1 that (u ◦f)∗(reiθ) ≤(u ◦F)∗(reiθ) for 0 ≤θ ≤π. □
Function classes studied:
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