Abstract
Let ${\mathcal S}$ be the class of all functions $f$ that are analytic and univalent in the unit disk $\ID$ with the normalization $f(0)=f'(0)-1=0$. Let $\mathcal{U} (λ)$ denote the set of all $f\in {\mathcal S}$ satisfying the condition $$|f'(z)(\frac{z}{f(z)})^{2}-1| <λ~for $z\in \ID$, $$ for some $λ\in (0,1]$. In this paper, among other things, we study a "harmonic mean" of two univalent analytic functions. More precisely, we discuss the properties of the class of functions $F$ of the form $$
Results & Lemmas (9)
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Theorem 1.
Theorem 1. Let f, g ∈S. Suppose that f(z)+g(z) z ̸= 0 for z ∈D and consider the function F defined by (3) F(z) = 2f(z)g(z) f(z) + g(z). Then…
Theorem 1. Let f, g ∈S. Suppose that f(z)+g(z) z ̸= 0 for z ∈D and consider the function F defined by (3) F(z) = 2f(z)g(z) f(z) + g(z). Then G, defined by G(z) = r−1F(rz), belongs to U(λ) for 0 < r ≤ p λ/(1 + λ). In particular, F belongs to U in the disk |z| < 1/ √ 2 ≈0.707107 (and hence, F is univalent in D1/ √
Theorem 2.
Theorem 2. Let f ∈U(λ1), g ∈U(λ2) (0 < λ1, λ2 ≤1) and f(z)+g(z) z ̸= 0 for z ∈D. Define F by (3) and G by G(z) = r−1F(rz). Then G belongs to…
Theorem 2. Let f ∈U(λ1), g ∈U(λ2) (0 < λ1, λ2 ≤1) and f(z)+g(z) z ̸= 0 for z ∈D. Define F by (3) and G by G(z) = r−1F(rz). Then G belongs to U(λ) whenever (4) 0 < r ≤ s −K2 + K √ K2 + 4 2 with K = p 2λ2/(λ1 + λ2). In particular, if f, g ∈U, then G ∈U for 0 < r ≤
Theorem 1
Theorem 1 may be generalized in the following form.
Theorem 1 may be generalized in the following form.
Theorem 3.
Theorem 3. Let fk ∈S for k = 1,..., m and Pm k=1 z fk(z) ̸= 0 for z ∈D. Define F by (5) z F(z) = 1 m m X k=1 z fk(z). Then we have
Theorem 3. Let fk ∈S for k = 1, . . . , m and Pm k=1 z fk(z) ̸= 0 for z ∈D. Define F by (5) z F(z) = 1 m m X k=1 z fk(z). Then we have
Theorem 4.
Theorem 4. Let fk ∈U(λk) (0 < λk ≤1) for k = 1,..., m, Pm k=1 z fk(z) ̸= 0 for z ∈D and F be defined by (5). Then G defined by G(z) =…
Theorem 4. Let fk ∈U(λk) (0 < λk ≤1) for k = 1, . . . , m, Pm k=1 z fk(z) ̸= 0 for z ∈D and F be defined by (5). Then G defined by G(z) = r−1F(rz) belongs to U(λ) whenever (6) 0 < r ≤ s −K2 + K √ K2 + 4 2 with K = s
Lemma 1.
Lemma 1. Let φ(z) = 1 + P∞ n=1 bnzn be a non-vanishing analytic function on D and let f be of the form (1). Then, we have the following:…
Lemma 1. Let φ(z) = 1 + P∞ n=1 bnzn be a non-vanishing analytic function on D and let f be of the form (1). Then, we have the following: (a) If P∞ n=2(n −1)|bn| ≤λ, then f ∈U(λ). (b) If P∞ n=2(n −1)|bn| ≤1 −|b1|, then f ∈S∗. (c) If f ∈U(λ), then P∞ n=2(n −1)2|bn|2 ≤λ2. The conclusion (a) in Lemma 1 is from [6, 7] whereas the (b) is due to Reade et al. [11, Theorem 1]. Finally, as f ∈U(λ), we have f ′(z) z f(z) 2
Lemma 2.
Lemma 2. Let f ∈A have the form (7) z f(z) = 1 + b1z + b2z2 + · · · with bn ≥0 for all n ≥2 and for all z in a neighborhood of z = 0. Then…
Lemma 2. Let f ∈A have the form (7) z f(z) = 1 + b1z + b2z2 + · · · with bn ≥0 for all n ≥2 and for all z in a neighborhood of z = 0. Then the following conditions are equiva- lent.
Lemma 1
Lemma 1(c), we have (11) ∞ X n=2 (n −1)2|bn|2 ≤λ2 1 and ∞ X n=2 (n −1)2|cn|2 ≤λ2 2. As in the proof of Theorem 1, for G belonging to U(λ),…
Lemma 1(c), we have (11) ∞ X n=2 (n −1)2|bn|2 ≤λ2 1 and ∞ X n=2 (n −1)2|cn|2 ≤λ2 2. As in the proof of Theorem 1, for G belonging to U(λ), it suffices to show by Lemma 1(a) that T =
Theorem 5.
Theorem 5. Let f, g ∈S have the form (7) with bn ≥0 and cn ≥0 for all n ≥2. Then the function F defined by (3) belongs to S. In particular,…
Theorem 5. Let f, g ∈S have the form (7) with bn ≥0 and cn ≥0 for all n ≥2. Then the function F defined by (3) belongs to S. In particular, if b1 + c1 = 0, then F is also starlike in D.
Function classes studied:
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