Abstract
Our present investigation is motivated essentially by the fact that, in Geometric Function Theory, one can find many interesting and fruitful usages of a wide variety of special functions and special polynomials. The main purpose of this article is to make use of the Horadam polynomials and the generating function, in order to introduce three new subclasses of the bi-univalent function class. For functions belonging to the defined classes, we then derive coefficient inequalities and the Fekete S
Results & Lemmas (4)
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 · coeff
Theorem 2.1. Let be in the class. Then and for
Theorem 2.1. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{S}_{\Sigma}^*(\alpha, x)$ . Then
$$|a_2| \le \frac{|bx|\sqrt{|bx|}}{\sqrt{|[(1+4\alpha)b-p(1+2\alpha)^2]bx^2-qa(1+2\alpha)^2|}}, \quad and \quad |a_3| \le \frac{|bx|}{2+6\alpha} + \frac{b^2x^2}{(1+2\alpha)^2}$$
and for $\nu \in \mathbb{R}$
$$\left|a_{3}-\nu a_{2}^{2}\right| \leq \begin{cases} \frac{|bx|}{2+6\alpha} \\ if \quad |\nu-1| \leq \frac{\left|\left[\left(1+4\alpha\right)b-p(1+2\alpha)^{2}\right]bx^{2}-qa(1+2\alpha)^{2}\right|}{2b^{2}x^{2}\left(1+3\alpha\right)} \\ \frac{\left|bx\right|^{3}\left|\nu-1\right|}{\left|\left[\left(1+4\alpha\right)b-p(1+2\alpha)^{2}\right]bx^{2}-qa(1+2\alpha)^{2}\right|} \\ if \quad |\nu-1| \geq \frac{\left|\left[\left(1+4\alpha\right)b-p(1+2\alpha)^{2}\right]bx^{2}-qa(1+2\alpha)^{2}\right|}{2b^{2}x^{2}\left(1+3\alpha\right)}. \end{cases}$$
Theorem 2.2 · coeff
Theorem 2.2. Let be in the class. Then and and for
Theorem 2.2. Let
$$f(z) = z + \sum_{n=2}^{\infty} a_n z^n$$
be in the class $\mathcal{M}_{\Sigma}(\alpha, x)$ . Then $|a_2| \leq \frac{|bx|\sqrt{|bx|}}{\sqrt{|[(1+\alpha)b-p(1+\alpha)^2]bx^2-qa(1+\alpha)^2|}},$ and $|a_3| \leq \frac{|bx|}{2+4\alpha} + \frac{b^2x^2}{(1+\alpha)^2}$
and for $\nu \in \mathbb{R}$
$$|a_3 - \nu a_2^2| \le \begin{cases} \frac{|bx|}{2 + 4\alpha} \\ if \quad |\nu - 1| \le \frac{|[(1 + \alpha)b - p(1 + \alpha)^2]bx^2 - qa(1 + \alpha)^2|}{b^2x^2(2 + 4\alpha)} \\ \frac{|bx|^3|\nu - 1|}{|[(1 + \alpha)b - p(1 + \alpha)^2]bx^2 - qa(1 + \alpha)^2|} \\ if \quad |\nu - 1| \ge \frac{|[(1 + \alpha)b - p(1 + \alpha)^2]bx^2 - qa(1 + \alpha)^2|}{b^2x^2(2 + 4\alpha)}. \end{cases}$$
Theorem 2.3 · coeff
Theorem 2.3. Let be in the class. Then and for
Theorem 2.3. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{L}_{\Sigma}(\alpha, x)$ . Then
$$|a_2| \le \frac{|bx|\sqrt{2|bx|}}{\sqrt{|[(\alpha^2 - 3\alpha + 4)b - 2p(2 - \alpha)^2]bx^2 - 2qa(2 - \alpha)^2|}} \quad and \quad |a_3| \le \frac{|bx|}{6 - 4\alpha} + \frac{b^2x^2}{(2 - \alpha)^2}$$
and for $\nu \in \mathbb{R}$
$$\left|a_{3}-\nu a_{2}^{2}\right| \leq \begin{cases} \frac{|bx|}{6-4\alpha} \\ if \quad |\nu-1| \leq \frac{\left|\left[\left(\alpha^{2}-3\alpha+4\right)b-2p(2-\alpha)^{2}\right]bx^{2}-2qa(2-\alpha)^{2}\right|}{4b^{2}x^{2}\left(3-2\alpha\right)} \\ \frac{2\left|bx\right|^{3}\left|\nu-1\right|}{\left|\left[\left(\alpha^{2}-3\alpha+4\right)b-2p(2-\alpha)^{2}\right]bx^{2}-2qa(2-\alpha)^{2}\right|} \\ if \quad |\nu-1| \geq \frac{\left|\left[\left(\alpha^{2}-3\alpha+4\right)b-2p(2-\alpha)^{2}\right]bx^{2}-2qa(2-\alpha)^{2}\right|}{4b^{2}x^{2}\left(3-2\alpha\right)}. \end{cases}$$
Corollary 2.6 · coeff
Corollary 2.6. Let be in the class. Then and and for
Corollary 2.6. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ be in the class $\mathcal{L}_{\Sigma}(\alpha, t)$ . Then
$$|a_2| \le \frac{2t\sqrt{2t}}{\sqrt{|(2-\alpha)^2 - (\alpha^2 - 5\alpha + 4)t^2|}}$$
and $|a_3| \le \frac{t}{3 - 2\alpha} + \frac{4t^2}{(2-\alpha)^2}$
and for $\nu \in \mathbb{R}$
$$\left|a_{3}-\nu a_{2}^{2}\right| \leq \begin{cases} \frac{t}{3-2\alpha} & \text{if } |\nu-1| \leq \frac{\left|(2-\alpha)^{2}-(\alpha^{2}-5\alpha+4)\,t^{2}\right|}{8t^{2}\left(3-2\alpha\right)} \\ \frac{8t^{3}\left|\nu-1\right|}{\left|(2-\alpha)^{2}-(\alpha^{2}-5\alpha+4)\,t^{2}\right|} & \text{if } |\nu-1| \geq \frac{\left|(2-\alpha)^{2}-(\alpha^{2}-5\alpha+4)\,t^{2}\right|}{8t^{2}\left(3-2\alpha\right)}. \end{cases}$$
Function classes studied:
Coefficient bounds & claims (8)
Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.
coefficient_bound
S*_Sigma(alpha, x): |a2| <= |bx|*sqrt(|bx|) / sqrt(|[(1+4alpha)b - p(1+2alpha)^2]bx^2 - qa(1+2alpha)^2|) [Theorem 2.1]
coefficient_bound
S*_Sigma(alpha, x): |a3| <= |bx|/(2+6alpha) + b^2*x^2/(1+2alpha)^2 [Theorem 2.1]
coefficient_bound
S*_Sigma(alpha, x): |a3 - nu*a2^2| <= |bx|/(2+6alpha) if |nu-1| <= threshold, else |bx|^3|nu-1|/|denominator| [Theorem 2.1]
coefficient_bound
M_Sigma(alpha, x): |a2| <= |bx|*sqrt(|bx|) / sqrt(|[(1+alpha)b - p(1+alpha)^2]bx^2 - qa(1+alpha)^2|) [Theorem 2.2]
coefficient_bound
L_Sigma(alpha, x): |a3 - nu*a2^2| <= |bx|/(6-4alpha) if |nu-1| <= threshold, else 2|bx|^3|nu-1|/|denominator| [Theorem 2.3]
function_family
Class S*_Sigma(alpha, x): bi-univalent f: zf'/f + alpha*z^2*f''/f subordinate to Pi(x,z)+1-a for both f and f^{-1}
function_family
Class M_Sigma(alpha, x): bi-univalent f: (1-alpha)*zf'/f + alpha*(1+zf''/f') subordinate to Pi(x,z)+1-a for both f and f^{-1}
function_family
Class L_Sigma(alpha, x): bi-univalent f: (zf'/f)^alpha * (1+zf''/f')^(1-alpha) subordinate to Pi(x,z)+1-a for both f and f^{-1}
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