A note on a subclass of bazilevič functions

Abstract: In this artcle, we introduce and investigate a subclass of Bazilevi{č} functions, denoted by $\mathcal{B}<em>{\varphi</em>{A,B}}(α<sup>{(m)})$. We determine the Hardy space to which this subclass of Bazilevi{č} functions belong to. Additionally, we provide a necessary condition for a particular case of this subclass. Finally, we obtain a sharp coefficient estimate for the functions associated with $\mathcal{B}_1(α).$

• The paper proves that functions in the Bazilevič subclass exhibit a Hardy space embedding of order 1/(2m) + ε based on careful integration estimates.
• It establishes necessary geometric conditions linking the subclass to close-to-convex functions via refined subordination methods.
• Sharp coefficient bounds for the Taylor expansion of the transformed function are derived, confirming conjectured domination results.

Subclasses of Bazilevič Functions: Hardy Classes, Necessary Conditions, and Coefficient Estimates

Introduction and Problem Framework

This paper focuses on the study of a subclass of Bazilevič functions, denoted as $\mathcal{B}_{\varphi_{A,B}}(\alpha^{(m)})$. Bazilevič functions generalize classical classes of univalent analytic functions in the unit disk $\mathbb{D}$ through an integral representation parameterized by $\alpha$, $\beta$, starlike functions $g$, and Carathéodory functions $P$. The subclass $\mathcal{B}_{\varphi_{A,B}}(\alpha^{(m)})$ is constructed by imposing additional subordination conditions involving the Janowski function $\varphi_{A,B}(z) = \frac{1 + Az}{1 + Bz}$, which is a Möbius transformation, and combining multiple starlike functions with potentially distinct positive exponents. The article investigates the Hardy spaces associated with this subclass, necessary conditions in the scalar-parameter case, and establishes sharp coefficient estimates.

Structural Foundation and Subordination

The function classes are built from analytic functions on $\mathbb{D}$, with normalization $f(0) = 0$ and $\mathbb{D}$0. The Bazilevič class $\mathbb{D}$1 uses the integral formula

$\mathbb{D}$2

with $\mathbb{D}$3 starlike and $\mathbb{D}$4 with positive real part. The subclass in question, $\mathbb{D}$5, admits $\mathbb{D}$6 starlike functions $\mathbb{D}$7 and corresponding exponents $\mathbb{D}$8, under a generalization described in Kim and Sugawa [sugawa]. The class is thus governed by integral operators and subordination constraints, taking advantage of inclusions and preservation properties of starlikeness and convexity through Möbius-type transforms.

Hardy Class Membership

The Hardy space $\mathbb{D}$9 for analytic functions is defined by the $\alpha$0-integrability of the boundary values, measured via integral means $\alpha$1. Previous work by Miller [miller] and Eenigenburg & Keogh [keogh] clarified the Hardy class embedding for various subclasses, and this paper extends those results systematically to $\alpha$2.

Main Theorem: For $\alpha$3 with $\alpha$4 (where $\alpha$5), there exists $\alpha$6 depending on $\alpha$7 such that

$\alpha$8

This result, achieved via careful integration estimates and use of Hölder's inequality along with structural lemmas from analytic function theory, strengthens earlier work and characterizes the space for multi-parameter Bazilevič classes. The result is sharp in the sense that the extremal function $\alpha$9 demonstrates the boundary; when $\beta$0, the Hardy index bound fails.

For the special case $\beta$1 and $\beta$2, the corollary yields $\beta$3.

Necessary Condition for $\beta$4 and Connections to Close-to-Convexity

The article investigates a link between $\beta$5 and close-to-convex functions in the sense of the class $\beta$6. By constructing explicit integral transforms between members of these classes, a bridge is established; specifically, if $\beta$7 for $\beta$8, there exists $\beta$9 related by differential and integral operations involving the exponents.

Key Necessary Condition: For $g$0, $g$1, the following integral inequality holds for all arcs in the disk:

$g$2

This concisely formalizes the geometric angular restriction imposed by the Bazilevič subclass structure.

Coefficient Estimates and Domination

Sharp coefficient estimates for analytic function classes are central in geometric function theory. Building on previous work for initial coefficients ([ram], [marjono]), the paper proves a general sharp bound for the Taylor coefficients of $g$3 when $g$4:

$g$5

where $g$6 is the $g$7-th coefficient of $g$8. The proof exploits the Carathéodory class bound for $g$9 with positive real part, and the sharpness is witnessed by the function constructed from an explicit integral using Janowski-type kernels. Furthermore, this result confirms two conjectures regarding the coefficient growth: for $P$0, the set of coefficients is dominated by those of an extremal starlike function; for $P$1, the bound $P$2 holds.

The approach uses functional and differential subordinations, yielding a coefficient domination result:

$P$3

with the symbol $P$4 denoting the Hadamard (coefficient-wise) ordering.

Implications and Future Directions

The systematic analysis of the Hardy space membership for $P$5 significantly clarifies the mapping properties and regularity of generalized Bazilevič functions composed through subordination and exponentiation. The identification of precise (and sharp) coefficient bounds for $P$6, together with necessary angular conditions, not only advances the geometric function theory of univalent maps but has secondary implications for the spectral theory of associated operators and potential theory on planar domains.

The explicit link constructed between Bazilevič subclasses and close-to-convexity opens avenues for further investigation of extremal problems, distortion, differential inequalities, and coefficient conjectures for broader parameter ranges. Extending the established coefficient bounds to non-integer values of $P$7 remains an outstanding open problem with potential ramifications for the analytic theory of subordinate semigroups and univalent operator families.

Conclusion

This work advances the theory of Bazilevič-type subclasses by establishing sharp Hardy space embeddings, necessary geometric conditions, and explicit sharp coefficient bounds. The multi-parameter formalism and use of Janowski-type subordination deepen the understanding of univalent function theory’s broader landscape. The presented results lay a foundation for further exploration of analytic, geometric, and operator-theoretic properties in complex analysis and related areas.