NEW CRITERIA FOR UNIVALENT FUNCTIONS

NEW CRITERIA FOR UNIVALENT FUNCTIONS

Volume 49, Number 1, May 1975 | STEPHAN RUSCHEWEYH
The paper introduces and explores the classes \( K_n \) of univalent functions \( f(z) \) regular in the unit disc \( \mathcal{U} \) with specific conditions involving higher derivatives. The main results include: 1. **Definition and Properties of \( K_n \)**: The class \( K_n \) is defined by the condition that the real part of a specific ratio of derivatives must exceed \( (n+1)/2 \). It is shown that \( K_{n+1} \subset K_n \) for all \( n \in \mathbb{N}_0 \). Since \( K_0 \) is the class of starlike functions of order \( 1/2 \), all functions in \( K_n \) are univalent. 2. **Coefficient Estimates**: Sharp estimates for the coefficients \( a_k \) of functions in \( K_n \) are derived, particularly focusing on the estimate \( |a_3 - a_2^2| \leq (1 - |a_2|^2) / (n+2) \). This implies that \( a_3 = a_2^2 \) for functions in \( \bigcap_{n \in \mathbb{N}_0} K_n \). 3. **Special Elements of \( K_n \)**: Examples of functions in \( K_n \) are provided, such as \( b_\gamma(z) \), which have interesting convolution properties. The paper extends a result by G. Pólya and I. J. Schoenberg to \( K_n \). 4. **Conjectures and Open Problems**: The paper concludes with several conjectures and open problems, including the behavior of \( K_\alpha \) for real \( \alpha \geq -1 \), the closure of \( K_\alpha \) under Hadamard products, and the determination of the smallest values \( \delta_n \) that guarantee univalence. The paper contributes to the understanding of univalent functions and their properties, particularly in the context of higher derivatives and Hadamard products.The paper introduces and explores the classes \( K_n \) of univalent functions \( f(z) \) regular in the unit disc \( \mathcal{U} \) with specific conditions involving higher derivatives. The main results include: 1. **Definition and Properties of \( K_n \)**: The class \( K_n \) is defined by the condition that the real part of a specific ratio of derivatives must exceed \( (n+1)/2 \). It is shown that \( K_{n+1} \subset K_n \) for all \( n \in \mathbb{N}_0 \). Since \( K_0 \) is the class of starlike functions of order \( 1/2 \), all functions in \( K_n \) are univalent. 2. **Coefficient Estimates**: Sharp estimates for the coefficients \( a_k \) of functions in \( K_n \) are derived, particularly focusing on the estimate \( |a_3 - a_2^2| \leq (1 - |a_2|^2) / (n+2) \). This implies that \( a_3 = a_2^2 \) for functions in \( \bigcap_{n \in \mathbb{N}_0} K_n \). 3. **Special Elements of \( K_n \)**: Examples of functions in \( K_n \) are provided, such as \( b_\gamma(z) \), which have interesting convolution properties. The paper extends a result by G. Pólya and I. J. Schoenberg to \( K_n \). 4. **Conjectures and Open Problems**: The paper concludes with several conjectures and open problems, including the behavior of \( K_\alpha \) for real \( \alpha \geq -1 \), the closure of \( K_\alpha \) under Hadamard products, and the determination of the smallest values \( \delta_n \) that guarantee univalence. The paper contributes to the understanding of univalent functions and their properties, particularly in the context of higher derivatives and Hadamard products.
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