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Abstract
In this article, we determine two point distortion theorem and sharp coefficient estimates for the families of close-to-convex harmonic mappings whose analytic part is a convex function of order $α$. By making use of these results, we determine the radius of univalence of sections of these families in terms of zeros of certain equation. Lower bound for the radius of univalence has been obtained explicitly for the case $α= 1/2$. Comparison of radius of univalence of the sections have been shown b
Results & Lemmas (5)
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Theorem 1.
Theorem 1. (Two point distortion Theorem) Suppose that f = h + g ∈F(α) for some α (0 ≤α < 1). For each λ ∈C such that |λ| = 1 define Fλ(z):=…
Theorem 1. (Two point distortion Theorem) Suppose that f = h + g ∈F(α) for some α (0 ≤α < 1). For each λ ∈C such that |λ| = 1 define Fλ(z) := h(z) + λg(z) Then, for any t, ψ ∈R such that t ̸= ψ, f satisfies the following inequality
Theorem 1
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Theorem 1]) |h′(z)| ≥ 1 (1 + r)2(1−α), r = |z| < 1. The desired conclusion follows if we use the inequality in (8) in (7). □ Next, we…
Theorem 1]) |h′(z)| ≥ 1 (1 + r)2(1−α) , r = |z| < 1. The desired conclusion follows if we use the inequality in (8) in (7). □ Next, we provide the sharp coefficient estimates for the functions in the family F(α).
Theorem 2.
Theorem 2. Suppose that f = h + g ∈F(α) for some α (0 ≤α < 1) with the series representation as in (2). Then, for all n ≥2, the coefficients…
Theorem 2. Suppose that f = h + g ∈F(α) for some α (0 ≤α < 1) with the series representation as in (2). Then, for all n ≥2, the coefficients of f satisfy the following inequality (9) |an| ≤An(α) and |bn| ≤n −1 n −2αAn(α), where (10) An(α) = 1 n! n Y j=2 (j −2α). All these bounds are sharp and the equality in each inequality is attained for the
Theorem 3.
Theorem 3. Let f = h + g ∈F(α) with the series representation as in (2). Then for θ ∈R, the harmonic function sn,m(f; θ)(z) = sn(h)(z) +…
Theorem 3. Let f = h + g ∈F(α) with the series representation as in (2). Then for θ ∈R, the harmonic function sn,m(f; θ)(z) = sn(h)(z) + eiθsm(g)(z) univalent in the disk |z| < rn,m, where rn,m is the unique positive root of the equation µ(n, m, r, α) = 0 in (0, 1). Here (12) ψ(n, m, r, α) = A(r, α) − ∞ X k=n+1 kAk(α)rk−1 − ∞ X k=m+1
Corollary 1.
Corollary 1. Suppose that f ∈F(α). Then the value of n for which sn,n(f; θ)(z) is univalent in the disk |z| < ρ is formulated in Table 1:…
Corollary 1. Suppose that f ∈F(α). Then the value of n for which sn,n(f; θ)(z) is univalent in the disk |z| < ρ is formulated in Table 1: Value of ρ α = −1/2 α = 0 α = 1/4 α = 1/2 α = 3/4 1/4 n ≥4 n ≥3 n ≥2 n ≥2 n ≥2 1/2
Function classes studied:
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