Abstract
Let $\mathcal{A}$ denote the set of all analytic functions $f$ in the unit disk $\ID=\{z:\,|z|<1\}$ of the form $f(z)=z+\sum_{n=2}^{\infty}a_nz^n.$ Let $\mathcal{U}$ denote the set of all $f\in \mathcal{A}$, $f(z)/z\neq 0$ and satisfying the condition $$ | f'(z) (\frac{z}{f(z)})^{2}-1 | < 1 {for $z\in \ID$}. $$ Functions in ${\mathcal U}$ are known to be univalent in $\ID$. For $α\in [0,1]$, let $$ \mathcal{N}(α)= \{f_α:\, f_α(z)=(1-α)f(z)+α\int_0^z\frac{f(t)}{t}\,dt, {$f\in\mathcal{A}$ with $|a
Results & Lemmas (12)
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Theorem 1.
Theorem 1. If P∞ n=2 n|an| ≤1, then f(z) = z + P∞ n=2 anzn belongs to U. The result is sharp. At this place, it is worth recalling that the…
Theorem 1. If P∞ n=2 n|an| ≤1, then f(z) = z + P∞ n=2 anzn belongs to U. The result is sharp. At this place, it is worth recalling that the class U is neither included in S∗nor includes the class S∗. Also, the class U is neither contained in R1 nor contains the class R1 (see for example [10]). For instance, the function f defined by f(z) = z 1 + 1 2z + 1 2z3 is in U\S∗(see also [4]). Indeed for this function zf ′(z) f(z) =
Theorem 2.
Theorem 2. A function f(z) = z −P∞ n=2 |an|zn is in the class U if and only if P∞ n=2 n|an| ≤1. From the result of Silverman [15], we may…
Theorem 2. A function f(z) = z −P∞ n=2 |an|zn is in the class U if and only if P∞ n=2 n|an| ≤1. From the result of Silverman [15], we may now formulate the above discussion as
Corollary 1.
Corollary 1. Suppose that f(z) = z −P∞ n=2 |an|zn belongs to A. Then we have the following equivalent statements: f ∈U ⇐⇒f ∈S∗ 1 ⇐⇒f ∈R1 ⇐⇒…
Corollary 1. Suppose that f(z) = z −P∞ n=2 |an|zn belongs to A. Then we have the following equivalent statements: f ∈U ⇐⇒f ∈S∗ 1 ⇐⇒f ∈R1 ⇐⇒ ∞ X n=2 n|an| ≤1. In connection with a problem due to [7], Ruscheweyh and Wirths [14] discussed the univalency of functions in the set of convex linear combinations of the form µf(z) + (1 −µ)g(z), µ ∈[0, 1], when f, g belonging to suitable subsets of S. We shall consider a similar problem which is indeed a generalization of Theorem B. For α ∈[0, 1], let N (α
Theorem 3.
Theorem 3. The number rS(α) is the root in (0, 1) of the equation (1) 2(1 −r)3 + (2α −1)r −1 = 0. The extremal function is g(z) = 2z − αz 1…
Theorem 3. The number rS(α) is the root in (0, 1) of the equation (1) 2(1 −r)3 + (2α −1)r −1 = 0. The extremal function is g(z) = 2z − αz 1 −z −(1 −α)z (1 −z)2 , z ∈D. For a ready reference, the values of rS(α) for certain values of α ∈[0, 1] are listed in Table 1.
Corollary 2.
Corollary 2. Then f ∈N and each of its n-th partial sum sn(f) is univalent for |z| < r, where r ≈1− √ 2 2 ≈0.292893 is the root of the…
Corollary 2. Then f ∈N and each of its n-th partial sum sn(f) is univalent for |z| < r, where r ≈1− √ 2 2 ≈0.292893 is the root of the equation 2(1−r)3 +r −1 = 0 in (0, 1). The extremal function is g(z) = 2z − z 1 −z, z ∈D. We remark that N ⊃{f ∈A : Re (f(z)/z) > 1/2, z ∈D} ⊃{f : f ∈C}. In the case of α = 1/2, we see that if N (1/2) = f1/2(z) = 1
Theorem 4.
Theorem 4. Let f, g ∈U with f(z)+g(z) z ̸= 0 in the unit disk D, and F be defined by (5) F(z) = f(z) + g(z) 2, z ∈D. Then F ∈U for |z| < r0,…
Theorem 4. Let f, g ∈U with f(z)+g(z) z ̸= 0 in the unit disk D, and F be defined by (5) F(z) = f(z) + g(z) 2 , z ∈D. Then F ∈U for |z| < r0, where r0 ≈0.262453 is the root of the equation (6) r2 + (1 + r2) (2r + r2)2 1 −2r − π 4√
Theorem 5.
Theorem 5. Let f, g ∈S with f(z)+g(z) z ̸= 0 in the unit disk D. Then the function F define by (5) belongs to U for |z| < 1 √ 2r0, where r0…
Theorem 5. Let f, g ∈S with f(z)+g(z) z ̸= 0 in the unit disk D. Then the function F define by (5) belongs to U for |z| < 1 √ 2r0, where r0 ≈0.262453 is the root of the equation (6). At this place it worth recalling that S ⊂N (0). 3. A Lemma and an Example For the proof of Theorem 4 and the discussion in Example 1 below, we need the following lemma.
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 f(z) = z φ(z). Then, we have the…
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 f(z) = z φ(z). Then, we have the following: (a) If P∞ n=2(n −1)|bn| ≤1, then f ∈U. (b) If P∞ n=2(n −1)|bn| ≤1 −|b1|, then f ∈S∗. (c) If f ∈U, then P∞ n=2(n −1)2|bn|2 ≤1. The conclusion (a) in Lemma 1 is from [8, 11] whereas the (b) is due to Reade et. el. [13, Theorem 1]. Finally, as f ∈U, we have
Theorem 1
Theorem 1 show that Theorem 1 may be stated in an improved form.
Theorem 1 show that Theorem 1 may be stated in an improved form.
Corollary 3.
Corollary 3. Let f, g ∈U with f(z)+g(z) z ̸= 0 in the unit disk D, f ′′(0) + g′′(0) = 0, and F be defined by (5). Then F ∈U for |z| < r0,…
Corollary 3. Let f, g ∈U with f(z)+g(z) z ̸= 0 in the unit disk D, f ′′(0) + g′′(0) = 0, and F be defined by (5). Then F ∈U for |z| < r0, where r0 ≈0.3512 is the root of the equation r2 + (1 + r2) (2r + r2)2 1 − π 4√ 90 r2 4√ 1−r4
Corollary 4.
Corollary 4. Let f, g ∈U with f(z)+g(z) z ̸= 0 in the unit disk D, either f ′′(0) = 0 or g′′(0) = 0, and F be defined by (5). Then F ∈U for…
Corollary 4. Let f, g ∈U with f(z)+g(z) z ̸= 0 in the unit disk D, either f ′′(0) = 0 or g′′(0) = 0, and F be defined by (5). Then F ∈U for |z| < r0, where r0 ≈0.400502 is the root of the equation r2 + (1 + r2) (r + r2)2 1 −r − π 4√ 90 r2 4√ 1−r4
Corollary 5.
Corollary 5. Let f, g ∈U with f(z)+g(z) z ̸= 0 in the unit disk D, f ′′(0) = g′′(0) = 0, and F be defined by (5). Then F ∈U for |z| < r0,…
Corollary 5. Let f, g ∈U with f(z)+g(z) z ̸= 0 in the unit disk D, f ′′(0) = g′′(0) = 0, and F be defined by (5). Then F ∈U for |z| < r0, where r0 ≈0.667827 is the root of the equation r2 + (1 + r2) r4 1 − π 4√ 90 r2 4√ 1−r4
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