On a New Class of Concave Bi-Univalent Functions Associated with Bounded Boundary Rotation
Abstract
:1. Introduction
1.1. Concave Function
- and satisfies the condition
- g maps conformally onto a concave set, i.e., g maps conformally, a set whose complement with is convex.
- The opening angle of the image of f (i.e., ) at ∞ is equal to or less than
1.2. Function with Bounded Variation
2. Examples for the Class Concave Functions with Bounded Boundary Rotation
2.1. Concave Functions with Bounded Boundary Rotation
2.2. Integral Representation of
2.3. Relation Between Class and
2.4. Examples
3. Concave Bi-Univalent Functions with Bounded Boundary Rotation
4. Concluding Remarks and Observations
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Sharma, P.; Sivasubramanian, S.; Murugusundaramoorthy, G.; Cho, N.E. On a New Class of Concave Bi-Univalent Functions Associated with Bounded Boundary Rotation. Mathematics 2025, 13, 370. https://doi.org/10.3390/math13030370
Sharma P, Sivasubramanian S, Murugusundaramoorthy G, Cho NE. On a New Class of Concave Bi-Univalent Functions Associated with Bounded Boundary Rotation. Mathematics. 2025; 13(3):370. https://doi.org/10.3390/math13030370
Chicago/Turabian StyleSharma, Prathviraj, Srikandan Sivasubramanian, Gangadharan Murugusundaramoorthy, and Nak Eun Cho. 2025. "On a New Class of Concave Bi-Univalent Functions Associated with Bounded Boundary Rotation" Mathematics 13, no. 3: 370. https://doi.org/10.3390/math13030370
APA StyleSharma, P., Sivasubramanian, S., Murugusundaramoorthy, G., & Cho, N. E. (2025). On a New Class of Concave Bi-Univalent Functions Associated with Bounded Boundary Rotation. Mathematics, 13(3), 370. https://doi.org/10.3390/math13030370