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Abstract
An analytic function $f$ defined on the open unit disk $\mathbb{D}=\{z:|z|<1\}$ is bi-univalent if the function $f$ and its inverse $f^{-1}$ are univalent in $\mathbb{D}$. Estimates for the initial coefficients of bi-univalent functions $f$ are investigated when $f$ and $f^{-1}$ respectively belong to some subclasses of univalent functions. Some earlier results are shown to be special cases of our results.
Results & Lemmas (7)
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 1.1 · coeff
Theorem 1.1. [10] Let with Taylor series and. Then We need the following classes investigated in [1–3].
Theorem 1.1. [10] Let $f \in \sigma$ with Taylor series $f(z) = z + a_2 z^2 + \cdots$ and $g = f^{-1}$ . Then
$$|a_2| \le \begin{cases} 1.5894 & \text{if } f \in \mathcal{S}, \ g \in \mathcal{S}, \\ 2 & \text{if } f \in \mathcal{S}^, \ g \in \mathcal{S}^, \\ 1.507 & \text{if } f \in \mathcal{S}^*, \ g \in \mathcal{S}, \\ 1.224 & \text{if } f \in \mathcal{C}, \ g \in \mathcal{S}. \end{cases}$$
We need the following classes investigated in [1–3].
Theorem 2.1 · coeff
Theorem 2.1. Let and. If, and f of the form (2.3) then (2.4) and where.
Theorem 2.1. Let $f \in \sigma$ and $g = f^{-1}$ . If $f \in \mathcal{P}(\alpha, \varphi)$ , $g \in \mathcal{P}(\beta, \psi)$ and f of the form
(2.3)
$$f(z) = z + \sum_{n=2}^{\infty} a_n z^n,$$
then
(2.4)
$$|a_2| \le \frac{B_1 D_1 \sqrt{B_1 (1 + 3\beta) + D_1 (1 + 3\alpha)}}{\sqrt{|\sigma B_1^2 D_1^2 - (1 + 2\alpha)^2 (1 + 3\beta)(B_2 - B_1) D_1^2 - (1 + 2\beta)^2 (1 + 3\alpha)(D_2 - D_1) B_1^2|}}$$
and
$$(2.5) \ 2\sigma|a_3| \le B_1(3+10\beta) + D_1(1+2\alpha) + (3+10\beta)|B_2 - B_1| + \frac{(1+2\beta)^2 B_1^2 |D_2 - D_1|}{D_1^2(1+2\alpha)}$$
where $\sigma := 2 + 7\alpha + 7\beta + 24\alpha\beta$ .
Theorem 2.2 · coeff
Theorem 2.2. Let and. If and, then (2.17) and where.
Theorem 2.2. Let $f \in \sigma$ and $g = f^{-1}$ . If $f \in \mathcal{P}(\alpha, \varphi)$ and $g \in \mathcal{M}(\beta, \psi)$ , then (2.17)
$$|a_2| \le \frac{B_1 D_1 \sqrt{B_1 (1 + 2\beta) + D_1 (1 + 3\alpha)}}{\sqrt{|\sigma B_1^2 D_1^2 - (1 + 2\alpha)^2 (1 + 2\beta)(B_2 - B_1) D_1^2 - (1 + \beta)^2 (1 + 3\alpha)(D_2 - D_1) B_1^2|}}$$
and
$$(2.18) \quad 2\sigma|a_3| \le B_1(3+5\beta) + D_1(1+2\alpha) + (3+5\beta)|B_2 - B_1| + \frac{(1+\beta)^2 B_1^2 |D_2 - D_1|}{D_1^2 (1+2\alpha)}$$
where $\sigma := 2 + 7\alpha + 3\beta + 11\alpha\beta$ .
Theorem 2.3 · coeff
Theorem 2.3. Let and. If and, then (2.25) and (2.26) where.
Theorem 2.3. Let $f \in \sigma$ and $g = f^{-1}$ . If $f \in \mathcal{P}(\alpha, \varphi)$ and $g \in \mathcal{L}(\beta, \psi)$ , then (2.25)
$$|a_2| \le \frac{B_1 D_1 \sqrt{2[B_1(3-2\beta)+D_1(1+3\alpha)]}}{\sqrt{|\sigma B_1^2 D_1^2 - 2(1+2\alpha)^2(3-2\beta)(B_2-B_1)D_1^2 - 2(2-\beta)^2(1+3\alpha)(D_2-D_1)B_1^2|}}$$
and
$$|\sigma a_3| \le \frac{1}{2} B_1(\beta^2 - 11\beta + 16) + D_1(1 + 2\alpha) + \frac{1}{2} (\beta^2 - 11\beta + 16) |B_2 - B_1|$$
$$+ \frac{(2 - \beta)^2 B_1^2 |D_2 - D_1|}{D_1^2 (1 + 2\alpha)}$$
(2.26)
where $\sigma := 10 + 36\alpha - 7\beta - 25\alpha\beta + \beta^2 + 3\alpha\beta^2$ .
Theorem 2.4 · coeff
Theorem 2.4. Let and. If,, then (2.33) and where.
Theorem 2.4. Let $f \in \sigma$ and $g = f^{-1}$ . If $f \in \mathcal{M}(\alpha, \varphi)$ , $g \in \mathcal{M}(\beta, \psi)$ , then (2.33)
$$|a_2| \le \frac{B_1 D_1 \sqrt{B_1 (1 + 2\beta) + D_1 (1 + 2\alpha)}}{\sqrt{|\sigma B_1^2 D_1^2 - (1 + \alpha)^2 (1 + 2\beta)(B_2 - B_1) D_1^2 - (1 + \beta)^2 (1 + 2\alpha)(D_2 - D_1) B_1^2|}}$$
and
$$(2.34) \ 2\sigma|a_3| \le B_1(3+5\beta) + D_1(1+3\alpha) + (3+5\beta)|B_2 - B_1| + \frac{(1+\beta)^2(1+3\alpha)B_1^2|D_2 - D_1|}{D_1^2(1+\alpha)^2}$$
where $\sigma := 2 + 3\alpha + 3\beta + 4\alpha\beta$ .
Theorem 2.5 · coeff
Theorem 2.5. Let and. If,, then (2.41) and where.
Theorem 2.5. Let $f \in \sigma$ and $g = f^{-1}$ . If $f \in \mathcal{M}(\alpha, \varphi)$ , $g \in \mathcal{L}(\beta, \psi)$ , then (2.41)
$$|a_2| \le \frac{B_1 D_1 \sqrt{2[B_1(3-2\beta) + D_1(1+2\alpha)]}}{\sqrt{|\sigma B_1^2 D_1^2 - 2(1+\alpha)^2 (3-2\beta)(B_2-B_1)D_1^2 - 2(2-\beta)^2 (1+2\alpha)(D_2-D_1)B_1^2|}}$$
and
$$|\sigma a_3| \le \frac{B_1}{2} (\beta^2 - 11\beta + 16) + D_1 (1 + 3\alpha) + \frac{1}{2} (\beta^2 - 11\beta + 16) |B_2 - B_1|$$
$$+ \frac{(2 - \beta)^2 (1 + 3\alpha) B_1^2 |D_2 - D_1|}{D_1^2 (1 + \alpha)^2}$$
where $\sigma := 10 + 14\alpha - 7\beta + \beta^2 + 2\alpha\beta^2 - 10\alpha\beta$ .
Theorem 2.6 · coeff
Theorem 2.6. Let and. If,, then (2.49) and where.
Theorem 2.6. Let $f \in \sigma$ and $g = f^{-1}$ . If $f \in \mathcal{L}(\alpha, \varphi)$ , $g \in \mathcal{L}(\beta, \psi)$ , then (2.49)
$$|a_2| \le \frac{B_1 D_1 \sqrt{2[B_1(3-2\beta) + D_1(3-2\alpha)]}}{\sqrt{|\sigma B_1^2 D_1^2 - 2(2-\alpha)^2 (3-2\beta)(B_2-B_1)D_1^2 - 2(2-\beta)^2 (3-2\alpha)(D_2-D_1)B_1^2|}}$$
and
$$2|\sigma a_3| \le B_1(\beta^2 - 11\beta + 16) + D_1(8 - 5\alpha - \alpha^2) + (\beta^2 - 11\beta + 16)|B_2 - B_1|$$
$$+ \frac{(2 - \beta)^2(\alpha^2 + 5\alpha - 8)B_1^2|D_2 - D_1|}{D_1^2(2 - \alpha)^2}$$
where $\sigma := 24 + 3\alpha^2 + 3\beta^2 - 17\alpha - 17\beta - 2\beta\alpha^2 - 2\alpha\beta^2 - 12\alpha\beta$ .
Definitions (1)
Def 1.1
Definition 1.1. Let be analytic and with and. For, let In this paper, we obtain the estimates for the second and third coefficients of…
Definition 1.1. Let $\varphi : \mathbb{D} \to \mathbb{C}$ be analytic and $\varphi(z) = 1 + B_1 z + B_2 z^2 + \cdots$ with $B_1 > 0$ and $B_2 \in \mathbb{R}$ . For $\alpha \geq 0$ , let
$$\mathcal{M}(\alpha,\varphi) := \left\{ f \in \mathcal{S} : (1-\alpha) \frac{zf'(z)}{f(z)} + \alpha \left( 1 + \frac{zf''(z)}{f'(z)} \right) \prec \varphi(z) \right\},$$
$$\mathcal{L}(\alpha,\varphi) := \left\{ f \in \mathcal{S} : \left( \frac{zf'(z)}{f(z)} \right)^{\alpha} \left( 1 + \frac{zf''(z)}{f'(z)} \right)^{1-\alpha} \prec \varphi(z) \right\},$$
$$\mathcal{P}(\alpha,\varphi) := \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f(z)} + \alpha \frac{z^2f''(z)}{f(z)} \prec \varphi(z) \right\}.$$
In this paper, we obtain the estimates for the second and third coefficients of functions f when
- (1) $f \in \mathcal{P}(\alpha, \varphi)$ and $g := f^{-1} \in \mathcal{P}(\beta, \psi)$ , or $g \in \mathcal{M}(\beta, \psi)$ , or $g \in \mathcal{L}(\beta, \psi)$ ,
- (2) $f \in \mathcal{M}(\alpha, \varphi)$ and $g \in \mathcal{M}(\beta, \psi)$ , or $g \in \mathcal{L}(\beta, \psi)$ ,
- (3) $f \in \mathcal{L}(\alpha, \varphi)$ and $g \in \mathcal{L}(\beta, \psi)$ .
Function classes studied:
Coefficient bounds & claims (6)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
bi-univalent P(alpha,phi) / P(beta,psi): |a2| <= B1*D1*sqrt(B1*(1+3beta)+D1*(1+3alpha)) / sqrt(|sigma*B1^2*D1^2 - (1+2alpha)^2*(1+3beta)*(B2-B1)*D1^2 - (1+2beta)^2*(1+3alpha)*(D2-D1)*B1^2|) where sigma = 2+7alpha+7beta+24*alpha*beta. [Theorem 2.1]
coefficient_bound
bi-univalent M(alpha,phi) / M(beta,psi): |a2| <= B1*D1*sqrt(B1*(1+2beta)+D1*(1+2alpha)) / sqrt(|sigma*B1^2*D1^2 - (1+alpha)^2*(1+2beta)*(B2-B1)*D1^2 - (1+beta)^2*(1+2alpha)*(D2-D1)*B1^2|) where sigma = 2+3alpha+3beta+4*alpha*beta. [Theorem 2.4]
coefficient_bound
bi-univalent L(alpha,phi) / L(beta,psi): |a2| <= B1*D1*sqrt(2*(B1*(3-2beta)+D1*(3-2alpha))) / sqrt(|sigma*B1^2*D1^2 - 2*(2-alpha)^2*(3-2beta)*(B2-B1)*D1^2 - 2*(2-beta)^2*(3-2alpha)*(D2-D1)*B1^2|) where sigma = 24+3alpha^2+3beta^2-17alpha-17beta-2*beta*alpha^2-2*alpha*beta^2-12*alpha*beta. [Theorem 2.6]
function_family
Class P(alpha, phi): f in S satisfying zf'(z)/f(z) + alpha*z^2*f''(z)/f(z) subordinate to phi(z), alpha >= 0
function_family
Class M(alpha, phi): f in S satisfying (1-alpha)*zf'(z)/f(z) + alpha*(1 + zf''(z)/f'(z)) subordinate to phi(z)
function_family
Class L(alpha, phi): f in S satisfying (zf'(z)/f(z))^alpha * (1 + zf''(z)/f'(z))^(1-alpha) subordinate to phi(z)
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