Alpha-Starlike Functions of Order Beta

• Alpha-starlike functions of order beta are analytic functions defined on the unit disk that generalize classical starlike functions through subordination with parameters α and β.
• Sharp coefficient estimates and extremal function models reveal precise bounds and mapping properties that extend traditional Bieberbach-type inequalities.
• Subordination techniques and operator theory provide sufficient conditions for univalence and drive applications in geometric function theory and complex analysis.

An alpha-starlike function of order beta is an analytic function defined on the unit disk whose geometric and coefficient-theoretic properties generalize classical starlike functions through two parameters, $\alpha$ and $\beta$. This two-parameter class unifies a variety of function families appearing in geometric function theory, encompassing and interpolating between the classical starlike, strongly starlike, and convex function classes, as well as the Janowski and Padmanabhan subclasses. The study of these functions involves sharp representation and subordination, coefficient estimates, extremal problems, and function-theoretic invariants, with applications to geometric shape, mapping properties, and extremal metrics in complex analysis.

1. Definitions and Fundamental Characterizations

A function $f$ analytic in the unit disk $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$, normalized by $f(0) = 0$ and $f'(0) = 1$, is called alpha-starlike of order beta (commonly denoted $f \in S^*(\alpha,\beta)$ with $0 < \alpha \leq 1$, $0 \leq \beta < 1$) if it satisfies the subordination: $$\frac{z f'(z)}{f(z)} \prec \Phi_{\alpha,\beta}(z) := \frac{1 + (1 - 2\beta)\alpha z}{1 - \alpha z}, \qquad z \in \mathbb{D}.$$ Equivalently,

$\beta$0

Special cases include:

• $\beta$1 is the Padmanabhan class $\beta$2,
• $\beta$3 is the classical starlike-of-order-$\beta$4 class $\beta$5,
• $\beta$6 recovers the classical starlike class $\beta$7.

The canonical extremal (model) function is

$\beta$8

which satisfies $\beta$9 (Sahoo et al., 2014).

2. Coefficient Estimates and Extremal Functions

For $f$0 in $f$1, the study of sharp bounds for Taylor coefficients reveals several regimes depending on $f$2:

• In the $f$3-valent case, the best constants for the coefficients $f$4 are given as follows:
  • $f$5,
  • For $f$6, two cases are distinguished:
  • If $f$7, then $f$8,
  • If $f$9, then $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$0,
  • with sharpness attained for the explicit extremal functions of the type

$\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$1

For the univalent ($\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$2) case, these reduce to classical Bieberbach-type bounds, and the extremal functions coincide with rotations of the generalized Koebe function (Sahoo et al., 2014). These results settle precisely the sharp bounds previously conjectured for these classes.

Coefficient functionals such as $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$3 further yield necessary and sufficient conditions for membership in $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$4 and $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$5 and relate tightly to starlikeness and convexity of corresponding order (Ali et al., 2011).

3. Subordination Principles and Structural Properties

The alpha-starlike classes are defined intrinsically via subordination. For $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$6, the mapping

$\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$7

ensures that the image of the unit disk under this function lies within an explicitly described region, which depends parametrically on $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$8 and $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$9. For $f(0) = 0$0, this region is a disk; for general $f(0) = 0$1, the boundary is a conic section or lemniscate (Malik et al., 2022, Kargar et al., 2018). The mapping properties induce growth, distortion, and covering theorems, as well as geometric criteria (such as the sector and conical containment) that are crucial to Sheil-Small, Janowski, and strongly starlike subclasses (Kargar et al., 2018).

The framework further generalizes via classes such as $f(0) = 0$2: under suitable parameterization, these recover the rotation-order-determined alpha-starlike of order beta classes, and the subordination admits generalizations toward convex conic domains in the complex plane. Explicit coefficient criteria for these broader classes are

$f(0) = 0$3

for $f(0) = 0$4 in the univalent case, using suitable normalization $f(0) = 0$5, $f(0) = 0$6 (Kargar et al., 2018).

4. Area-Maximization and Extremal Problems

The Yamashita area problem, seeking the maximal value of

$f(0) = 0$7

as $f(0) = 0$8 ranges over $f(0) = 0$9, is settled for these families. The sharp maximum is

$f'(0) = 1$0

where $f'(0) = 1$1 is the Gauss hypergeometric function. Equality is attained precisely for rotations of $f'(0) = 1$2. Subordination and Rogosinski’s lemma, together with Clunie’s method and partial-sum arguments, establish these results (Sahoo et al., 2014). Special cases reproduce all known previous bounds for convex, starlike, and Padmanabhan function classes.

5. Connections to Strongly Starlike and Janowski Classes

The class $f'(0) = 1$3 bridges classical starlike, strongly starlike, and Janowski-type classes. For $f'(0) = 1$4, $f'(0) = 1$5 consists of those $f'(0) = 1$6 satisfying

$f'(0) = 1$7

with equivalent subordination to a convex-univalent mapping. For $f'(0) = 1$8, $f'(0) = 1$9 coincides with the strongly starlike functions of order $f \in S^*(\alpha,\beta)$0, and

$f \in S^*(\alpha,\beta)$1

with sharp coefficient and logarithmic coefficient bounds established in this setting (Kargar et al., 2018). The Janowski class $f \in S^*(\alpha,\beta)$2, with the right choice of parameters, encompasses $f \in S^*(\alpha,\beta)$3.

6. Sufficient Conditions and Operator Theory

Operator-theoretic and differential-inequality methods provide sufficient conditions for membership in $f \in S^*(\alpha,\beta)$4. The usage of Jack’s Lemma (or Miller–Mocanu’s version), combined with suitable control of the modulus of linear combinations of $f \in S^*(\alpha,\beta)$5 and $f \in S^*(\alpha,\beta)$6, yields explicit domains of univalence and starlikeness (Shiraishi et al., 2013). Hypergeometric and generalized Bessel operators preserve membership in these classes under sharp coefficient-sum inequalities, generating further convolution-closed subclasses (Kargar et al., 2018).

7. Geometric Extremality and Radius Problems

For analytic $f \in S^*(\alpha,\beta)$7 the Booth-lemniscate starlikeness radius is determined as the largest $f \in S^*(\alpha,\beta)$8 such that the disk $f \in S^*(\alpha,\beta)$9 (parameterized via growth and covering properties of $0 < \alpha \leq 1$0) lies within the Booth-lemniscate domain $0 < \alpha \leq 1$1. Explicit two-branch formulas for $0 < \alpha \leq 1$2 are obtained, depending on critical tangency configurations, and extremality is again achieved for the generalized Koebe function (Malik et al., 2022).

---

In summary, the theory of alpha-starlike functions of order beta comprehensively refines and extends classical geometric function theory, providing precise characterizations, sharp inequalities, extremal function models, and constructive operators, with deep connections to subordination, coefficient problems, and extremal geometric function theory (Sahoo et al., 2014, Sahoo et al., 2014, Kargar et al., 2018, Kargar et al., 2018, Shiraishi et al., 2013, Malik et al., 2022, Ali et al., 2011).