Omitted values for some subclasses of univalent mappings

Abstract: We study the range of $\operatorname{Re}{a_2 f(z)}$ for normalized analytic functions $f$ in the unit disk belonging to several classes of conformal mappings. As our main contribution, we introduce the class $CC_α$ of completely convex mappings of order $α$, defined by a uniform two-point starlikeness condition, and we estimate the range of $\operatorname{Re}{a_2f(z)}$ in terms of $α$, for all $f\in CC_α$ and $z\in \mathbb{D}$, generalizing the classical result of Fournier--Ma--Ruscheweyh, which is recovered for $α=0$. We also determine omitted value sets for convex functions of order $α$, spherically convex mappings, uniformly starlike functions, and Nehari classes $\mathcal{N}_t$. The proofs rely primarily on the Schwarz--Pick lemma applied to auxiliary functions constructed from the two-point kernel $zf'(z)/(f(z)-f(x))$.

• The paper introduces the CCα class of completely convex mappings, yielding sharp omitted value bounds via a two-point geometric starlikeness condition.
• The study provides explicit, parameter-dependent lower bounds for coefficients like a₂f(z) across convex, spherically convex, and uniformly starlike function classes.
• The results refine classical convexity results by bridging Koebe’s theorem with modern geometric function theory techniques for omitted value analysis.

Omitted Values and Geometric Properties in Subclasses of Univalent Mappings

Introduction

This paper investigates the sets of values that are omitted by normalized univalent mappings in the unit disk, focusing on the image of the term $a_2 f(z)$ as $f$ varies over various function classes. The work is situated within the tradition stemming from Koebe’s one-quarter theorem and includes new generalizations and sharp estimates for several subclasses of univalent functions, especially convex and starlike families parameterized by order $\alpha$. A central contribution is the definition and study of the class $CC_\alpha$ of completely convex mappings of order $\alpha$, characterized via a two-point geometric starlikeness condition that uniformly refines standard convexity.

The Class $CC_\alpha$ of Completely Convex Mappings

The authors introduce $CC_\alpha$ as the set of analytic functions $f(z) = z + a_2 z^2 + \ldots$ on $D$ such that

$$\operatorname{Re}\left\{\frac{2z f'(z)}{f(z) - f(x)} - \frac{z + x}{z - x}\right\} \geq \alpha, \quad \forall\, z, x \in D$$

This condition ensures that the image domain $f$0 is starlike of order $f$1 with respect to any $f$2, encoding a uniform and global convexity property that is stricter than the traditional infinitesimal one-point condition for $f$3. When $f$4, the classical convex univalent functions $f$5 are recovered.

Key theoretical results established for $f$6 include sharp estimates:

• The modulus of $f$7 is bounded above and below in terms of $f$8 and $f$9.
• For all $\alpha$0,

$\alpha$1

with $\alpha$2 and explicit $\alpha$3.

• The sharp lower bound for $\alpha$4 generalizes the Fournier--Ma--Ruscheweyh result:

$\alpha$5

Equality is attained in the $\alpha$6 case, where previous results for convex functions are recovered.

The analysis leverages geometric function theory, particularly via the Schwarz--Pick lemma on suitably constructed two-point kernel functions, and ties the structural properties of $\alpha$7 to norm bounds and omitted value regions.

Other Subclasses: Convex, Spherically Convex, Uniformly Starlike Functions, and Nehari Classes

Convex Mappings of Order $\alpha$8 ($\alpha$9)

For functions in $CC_\alpha$0 defined by the local condition $CC_\alpha$1, the omitted value bounds are necessarily weaker than for $CC_\alpha$2. The authors exploit inclusion of $CC_\alpha$3 in certain starlike classes and derive explicit formulas for lower bounds on $CC_\alpha$4 in terms of a secondary parameter $CC_\alpha$5 that depends on $CC_\alpha$6, as well as further bounds for $CC_\alpha$7 involving a function of both $CC_\alpha$8 and $CC_\alpha$9.

Spherically Convex Mappings

For spherically convex mappings, mappings $\alpha$0 for which $\alpha$1 is spherically convex in the Riemann sphere, it is shown that for $\alpha$2,

$\alpha$3

The constant $\alpha$4 is proved to be sharp. This result synthesizes spherical metric properties with omitted value analysis via explicit extremal mappings.

Uniformly Starlike Mappings

Goodman’s class $\alpha$5, defined by a two-point starlikeness condition, is also examined. The principal numerical result is the lower bound:

$\alpha$6

for all $\alpha$7. The analysis is based on two-point kernel functions and the maximum modulus principle.

Nehari Classes $\alpha$8

The Nehari class $\alpha$9, defined using bounds on the Schwarzian derivative, interpolates between M\"obius transformations ($CC_\alpha$0) and the full univalent class ($CC_\alpha$1). For normalized $CC_\alpha$2, $CC_\alpha$3,

$CC_\alpha$4

The deviation from $CC_\alpha$5 quantifies the distortion attributable to the Schwarzian bound, and the result is quantitatively tight in the limiting cases.

Methodological Implications and Theoretical Significance

A distinctive methodological aspect of this work is the systematic use of two-point conditions, which provide a stronger, globally uniform geometric control compared to traditional one-point (infinitesimal) analytic conditions. This approach yields sharper omitted value estimates and more precise function-theoretic bounds, particularly for the $CC_\alpha$6 class and related families.

The development of explicit, parameter-dependent lower bounds for omitted values (in terms of $CC_\alpha$7) has multiple theoretical implications:

• It allows direct geometric comparison between function classes.
• The results quantify the extent to which additional geometric constraints (like complete convexity) tighten omitted value regions.
• For the Nehari class, the approach connects classical univalence criteria with modern omitted value problems.

The work also opens new paths for exploring global geometric properties of analytic mappings via multi-point kernel techniques, potentially impacting applications in conformal mapping, distortion theory, and the spectral theory of univalent functions.

Future Directions

Potential future developments include:

• Extension of these techniques to higher-order coefficient regions and to subclasses in higher-dimensional complex analysis.
• Systematic study of omitted value sets and sharp norm inequalities for classes defined by multi-point or nonlocal geometric constraints.
• Applications to geometric function theory problems arising in mathematical physics and complex dynamics.

Conclusion

The paper provides a comprehensive and sharp analysis of omitted values for several natural subclasses of univalent mappings, anchored by the introduction and study of the class $CC_\alpha$8 of completely convex mappings of order $CC_\alpha$9. The uniform two-point starlikeness condition enables strictly stronger bounds than classical convexity, and exact lower bounds for $CC_\alpha$0 are obtained for all considered subclasses. The techniques and results significantly enrich the toolkit of geometric function theory and have implications for the deeper understanding of extremal properties and value distributions in analytic mappings.

Reference: "Omitted values for some subclasses of univalent mappings" (2607.09925)