Abstract
For $0<λ\le 1$, let $\mathcal{U}(λ)$ be the class analytic functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}$ satisfying $|f'(z)(z/f(z))^2-1|<λ$ and $\mathcal{U}:=\mathcal{U}(1)$. In the present article, we prove that the class $\mathcal{U}$ is contained in the closed convex hull of the class of starlike functions and using this fact, we solve some extremal problems such as integral mean problem and arc length problem for functions in $\mathcal{U}$. By means of the so-c
Results & Lemmas (16)
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Proposition 2.1.
Proposition 2.1. [1] Let F be an analytic and univalent function in the unit disk D. If F(z) ̸= 0 in D and C F(D) is convex domain, then…
Proposition 2.1. [1] Let F be an analytic and univalent function in the unit disk D. If F(z) ̸= 0 in D and C \ F(D) is convex domain, then any analytic function f in s(F n) := {g : g ≺F n},
Proposition 2.2.
Proposition 2.2. [8, Theorem 5.8] The set co S∗consists of all functions represented by f(z) = Z |x|=1 z (1 −xz)2 dµ(x), where µ varies…
Proposition 2.2. [8, Theorem 5.8] The set co S∗consists of all functions represented by f(z) = Z |x|=1 z (1 −xz)2 dµ(x), where µ varies over the set of all probability measures on the unit circle |x| = 1. Also, E(co S∗) = z (1 −xz)2 : |x| = 1 . Using Propositions 2.1 and 2.2, we prove the following integral representation for functions in U.
Theorem 2.1.
Theorem 2.1. Each function f ∈U has an integral representation of the form (2.1) f(z) = Z |x|=1 z (1 −xz)2 dµ(x) for some probability…
Theorem 2.1. Each function f ∈U has an integral representation of the form (2.1) f(z) = Z |x|=1 z (1 −xz)2 dµ(x) for some probability measure µ on the unit circle |x| = 1. Moreover, U ⊊co S∗.
Corollary 2.1.
Corollary 2.1. If f ∈U is of the form (1.1) then |an| ≤n. Equality holds if, and only if, f is a rotation of the Koebe function. If G is a…
Corollary 2.1. If f ∈U is of the form (1.1) then |an| ≤n. Equality holds if, and only if, f is a rotation of the Koebe function. If G is a convex subset of H and J : H →R then J is called convex on G if J(tf +(1−t)g) ≤ tJ(f) + (1 −t)J(g) whenever f, g ∈G and 0 ≤t ≤1. We note that co S∗is a compact subset of H and E(co S∗) ⊊U. Hence, for any real-valued, continuous and convex functional J on co S∗, by [8, Theorems 4.5 and 4.6], we have max f∈U J(f) ≤max f∈co S∗J(f) = max f∈E(co S∗) J(f) ≤max f∈U
Theorem 2.2.
Theorem 2.2. If f ∈U and k(z) = z/(1 −z)2 then (2.5) 1 2π Z 2π 0 |f (n)(reiθ)|p dθ ≤1 2π Z 2π 0 |k(n)(reiθ)|p dθ, whenever 0 < r < 1, p ≥1…
Theorem 2.2. If f ∈U and k(z) = z/(1 −z)2 then (2.5) 1 2π Z 2π 0 |f (n)(reiθ)|p dθ ≤1 2π Z 2π 0 |k(n)(reiθ)|p dθ, whenever 0 < r < 1, p ≥1 and n = 0, 1, 2, . . .. Moreover, for n = 0, the inequality (2.5) holds for any real number p.
Corollary 2.2.
Corollary 2.2. If f ∈U then L(r, f) ≤L(r, k), where k is the Koebe function. We now review some of the standard facts on the theory of…
Corollary 2.2. If f ∈U then L(r, f) ≤L(r, k), where k is the Koebe function. We now review some of the standard facts on the theory of *-functions developed by Baern- stein [4]. For more on *-functions we refer to Duren [6]. For g ∈L1[−π, π], the *-function of g is defined by g∗(θ) = sup |E|=2θ Z E g(x) dx, 0 ≤θ ≤π, where |E| denote the Lebesgue measure of the set E. Here the supremum is taken over all Lebesgue measurable subsets of [−π, π] with |E| = 2θ.
Lemma 2.1.
Lemma 2.1. For g, h ∈L1[−π, π], the following statements are equivalent. (a) For every convex nondecreasing function Φ on R, Z π −π Φ(g(θ))…
Lemma 2.1. For g, h ∈L1[−π, π], the following statements are equivalent. (a) For every convex nondecreasing function Φ on R, Z π −π Φ(g(θ)) dθ ≤ Z π −π Φ(h(θ)) dθ. (b) For every t ∈R, Z π −π [g(θ) −t]+ dθ ≤ Z π −π [h(θ) −t]+ dθ.
Lemma 2.2.
Lemma 2.2. [11, Lemma 2] Let u and v be two subharmonic functions in D and there exists an analytic function ω: D →D with ω(0) = 0 such…
Lemma 2.2. [11, Lemma 2] Let u and v be two subharmonic functions in D and there exists an analytic function ω : D →D with ω(0) = 0 such that u(z) = v(ω(z)) for z ∈D. Then for each 0 < r < 1, u∗(reiθ) ≤v∗(reiθ). By using Lemmas 2.1 and 2.2 we prove the following interesting result.
Theorem 2.3.
Theorem 2.3. Let Φ be a convex and nondecreasing function in R. Then for f ∈U(λ) with 0 < λ ≤1 and 0 < r < 1, we have Z π −π Φ(± log…
Theorem 2.3. Let Φ be a convex and nondecreasing function in R. Then for f ∈U(λ) with 0 < λ ≤1 and 0 < r < 1, we have Z π −π Φ(± log |f(reiθ)|) dθ ≤ Z π −π Φ(± log |kλ(reiθ)|) dθ where kλ is defined by (1.3).
Corollary 2.3.
Corollary 2.3. If f ∈U(λ) and kλ is defined by (1.3) then for any 0 < r < 1, p ∈R, we have 1 2π Z 2π 0 |f(reiθ)|p dθ ≤1 2π Z 2π 0…
Corollary 2.3. If f ∈U(λ) and kλ is defined by (1.3) then for any 0 < r < 1, p ∈R, we have 1 2π Z 2π 0 |f(reiθ)|p dθ ≤1 2π Z 2π 0 |kλ(reiθ)|p dθ.
Theorem 2.3
Theorem 2.3, we obtain the desired result. □ For a locally univalent function f in D, the pre-Schwarzian derivative Tf is defined by Tf = f…
Theorem 2.3, we obtain the desired result. □ For a locally univalent function f in D, the pre-Schwarzian derivative Tf is defined by Tf = f ′′/f ′ and the pre-Schwarzian norm of f is defined by ∥f∥= sup z∈D (1 −|z|2)|Tf(z)|. The pre-Schwarzian norm has significant meaning in the theory of Teichmüller spaces. It is interesting to note that the pre-Schwarzian norm of f is nothing but the Bloch seminorm of the function log f ′. In 1976, Yamashita [22] observed that ∥f∥is finite if, and only if, f is un
Theorem 2.4.
Theorem 2.4. For f ∈U(λ), 0 < λ ≤1, let Jα[f] be defined by (1.2) where α ∈C. Then (2.8) ∥Jα[f]∥≤∥Jα[kλ]∥= 2|α| for 0 < λ ≤1 3 3+λ−2√…
Theorem 2.4. For f ∈U(λ), 0 < λ ≤1, let Jα[f] be defined by (1.2) where α ∈C. Then (2.8) ∥Jα[f]∥≤∥Jα[kλ]∥= 2|α| for 0 < λ ≤1 3 3+λ−2√ 2(1−λ2) λ |α| for
Corollary 2.4.
Corollary 2.4. For f ∈U, let Jα[f] be defined by (1.2) where α ∈C. Then ∥Jα[f]∥≤∥Jα[k]∥= 4|α| where k is the Koebe function. Our next result…
Corollary 2.4. For f ∈U, let Jα[f] be defined by (1.2) where α ∈C. Then ∥Jα[f]∥≤∥Jα[k]∥= 4|α| where k is the Koebe function. Our next result deals with the Fekete-Szegö problem for functions in the class U(λ).
Theorem 2.5.
Theorem 2.5. Let f ∈U(λ), 0 < λ ≤1 be of the form (1.1) and µ be a complex number. Then (2.9) |a3 −µa2 2| ≤ |(1 + λ + λ2) −µ(1 +…
Theorem 2.5. Let f ∈U(λ), 0 < λ ≤1 be of the form (1.1) and µ be a complex number. Then (2.9) |a3 −µa2 2| ≤ |(1 + λ + λ2) −µ(1 + λ)2| for µ −1+λ+λ2 (1+λ)2 ≥
Corollary 2.5.
Corollary 2.5. Let f ∈U(λ) be of the form (1.1) and µ be a real number. Then the following sharp inequality |a3 −µa2 2| ≤|(1 + λ + λ2) −µ(1…
Corollary 2.5. Let f ∈U(λ) be of the form (1.1) and µ be a real number. Then the following sharp inequality |a3 −µa2 2| ≤|(1 + λ + λ2) −µ(1 + λ)2| for µ ∈ −∞, (λ/(1 + λ))2 ∪(1, ∞) . holds and equality attains for the function kλ. 3. Properties of meromorphic functions associated with U(λ) Let Σ be the class of meromorphic and univalent functions g on ∆:= {ζ ∈bC : |ζ| > 1} of the form (3.1) g(ζ) = ζ + ∞
Theorem 3.1.
Theorem 3.1. Let g ∈M0(λ) with area (E(g)) = π(1 −λ2). Then g is an extreme point of M0(λ).
Theorem 3.1. Let g ∈M0(λ) with area (E(g)) = π(1 −λ2). Then g is an extreme point of M0(λ).
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