Continue from this paper
Indexing is not verification. Extracted statements should be checked against the source.
Abstract
In the present investigation, certain subclasses of close-to-convex functions are investigated. In particular, we obtain an estimate for the Fekete-Szegö functional for functions belonging to the class, distortion, growth estimates and covering theorems.
Results & Lemmas (6)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Theorem 2.1
Theorem 2.1 (Fekete-Szegö Inequality). For a function f(z) = z + a2z2 + a3z3 + · · · belonging to the class Ks(ϕ), the following sharp…
Theorem 2.1 (Fekete-Szegö Inequality). For a function f(z) = z + a2z2 + a3z3 + · · · belonging to the class Ks(ϕ), the following sharp estimate holds: |a3 −µa2 2| ≤1/3 + max(B1/3, |B2/3 −µB2 1/4|) (µ ∈C).
Corollary 2.1.
Corollary 2.1. Let f ∈Ks(ϕ). Then the coefficients d2 and d3 of the inverse function f −1(w) = w + d2w2 + d3w3 + · · · satisfy the inequality…
Corollary 2.1. Let f ∈Ks(ϕ). Then the coefficients d2 and d3 of the inverse function f −1(w) = w + d2w2 + d3w3 + · · · satisfy the inequality |d3 −µd2 2| ≤1/3 + max(B1/3, |B2/3 −(2 −µ)B2 1/4|) (µ ∈C).
Theorem 3.1.
Theorem 3.1. Let ϕ be an analytic univalent functions with positive real part and φ(−r) = min |z|=r<1|φ(z)|, φ(r) = max |z|=r<1|φ(z)|. If f…
Theorem 3.1. Let ϕ be an analytic univalent functions with positive real part and φ(−r) = min |z|=r<1|φ(z)|, φ(r) = max |z|=r<1|φ(z)|. If f ∈Ks(ϕ), the following sharp inequalities holds: ϕ(−r) 1 + r2 ≤|f ′(z)| ≤ϕ(r) 1 −r2 (|z| = r < 1), Z r 0 ϕ(−t) 1 + t2 dt ≤|f(z)| ≤ Z r
Theorem 7
Theorem 7, p. 67]. To prove the sharpness of our results, we consider the functions (3.2) f0(z) = Z z 0 ϕ(w) 1 −w2dw, f1(z) = Z z 0 ϕ(w) 1…
Theorem 7, p. 67]. To prove the sharpness of our results, we consider the functions (3.2) f0(z) = Z z 0 ϕ(w) 1 −w2dw, f1(z) = Z z 0 ϕ(w) 1 + w2dw. Define the function g0 and g1 by g0(z) = z/(1−z) and g1(z) = z/ √
Lemma 4.1.
Lemma 4.1. Let F(z, t) be analytic in D for each t ∈(0, δ), F(z, 0) = f(z), f ∈S and F(0, t) = 0 for each t ∈(0, δ). Suppose that F(z, t)…
Lemma 4.1. Let F(z, t) be analytic in D for each t ∈(0, δ), F(z, 0) = f(z), f ∈S and F(0, t) = 0 for each t ∈(0, δ). Suppose that F(z, t) ≺f(z) and that lim t→0+ F(z, t) −f(z) ztρ = F(z) exists for some ρ > 0. If F is analytic and Re(F(z)) ̸= 0, then Re F(z) f ′(z) < 0.
Theorem 4.1.
Theorem 4.1. Let f ∈S and g ∈S∗(1/2). Let δ > 0 and f(z) + tg(z)g(−z)/z ≺f(z), t ∈(0, δ). Then f ∈Ks.
Theorem 4.1. Let f ∈S and g ∈S∗(1/2). Let δ > 0 and f(z) + tg(z)g(−z)/z ≺f(z), t ∈(0, δ). Then f ∈Ks.
Function classes studied:
Related Papers