This paper introduces new criteria for univalent functions. The classes $ K_n $ consist of functions $ f(z) $ regular in the unit disc $ \mathcal{U} $ with $ f(0) = 0 $, $ f'(0) = 1 $, satisfying $ \mathrm{Re}\left[(z^n f)^{(n+1)}/(z^{n-1} f)^{(n)}\right] > (n+1)/2 $ in $ \mathcal{U} $. It is shown that $ K_{n+1} \subset K_n $ for $ n \in \mathbb{N}_0 $, and since $ K_0 $ is the class of functions starlike of order $ 1/2 $, all functions in $ K_n $ are univalent. Coefficient estimates and special elements of $ K_n $ are also given. The paper discusses the classes $ K_n $, showing that they are natural extensions of known classes like $ S_{1/2}^* $ and $ K $. Theorem 1 proves $ K_{n+1} \subset K_n $, and Theorem 2 relates $ K_n $ to other classes. Coefficient estimates are derived, showing that for $ f \in K_n $, $ |a_k - \mu a_2^{k-1}| \leq 1 - \mu $ for $ \mu \leq \gamma(n, k) $, where $ \gamma(n, k) $ is a binomial coefficient. These estimates are sharp and provide insights into the structure of $ K_n $. Special elements of $ K_n $ include functions like $ b_\gamma(z) = \sum_{j=1}^\infty \frac{\gamma+1}{\gamma+j} z^j $ with $ \mathrm{Re} \gamma \geq (n-1)/2 $. These functions have interesting convolution properties and extend results by Pólya and Schoenberg. The paper also addresses open problems, such as whether $ K_\alpha \subset K_\beta $ for $ \alpha > \beta $, and whether $ K_\alpha $ is closed under Hadamard products. It conjectures that these properties hold for all $ \alpha \geq -1 $. Additionally, it explores the smallest values $ \delta_n $ such that $ \mathrm{Re}(D^{n+1}f/D^n f) > \delta_n $ guarantees univalence of $ f \in A $. It is known that $ \delta_0 = 0 $, $ \delta_1 = 1/4 $. The paper concludes with further conjectures and open questions about the geometric structure of $ K_n $.This paper introduces new criteria for univalent functions. The classes $ K_n $ consist of functions $ f(z) $ regular in the unit disc $ \mathcal{U} $ with $ f(0) = 0 $, $ f'(0) = 1 $, satisfying $ \mathrm{Re}\left[(z^n f)^{(n+1)}/(z^{n-1} f)^{(n)}\right] > (n+1)/2 $ in $ \mathcal{U} $. It is shown that $ K_{n+1} \subset K_n $ for $ n \in \mathbb{N}_0 $, and since $ K_0 $ is the class of functions starlike of order $ 1/2 $, all functions in $ K_n $ are univalent. Coefficient estimates and special elements of $ K_n $ are also given. The paper discusses the classes $ K_n $, showing that they are natural extensions of known classes like $ S_{1/2}^* $ and $ K $. Theorem 1 proves $ K_{n+1} \subset K_n $, and Theorem 2 relates $ K_n $ to other classes. Coefficient estimates are derived, showing that for $ f \in K_n $, $ |a_k - \mu a_2^{k-1}| \leq 1 - \mu $ for $ \mu \leq \gamma(n, k) $, where $ \gamma(n, k) $ is a binomial coefficient. These estimates are sharp and provide insights into the structure of $ K_n $. Special elements of $ K_n $ include functions like $ b_\gamma(z) = \sum_{j=1}^\infty \frac{\gamma+1}{\gamma+j} z^j $ with $ \mathrm{Re} \gamma \geq (n-1)/2 $. These functions have interesting convolution properties and extend results by Pólya and Schoenberg. The paper also addresses open problems, such as whether $ K_\alpha \subset K_\beta $ for $ \alpha > \beta $, and whether $ K_\alpha $ is closed under Hadamard products. It conjectures that these properties hold for all $ \alpha \geq -1 $. Additionally, it explores the smallest values $ \delta_n $ such that $ \mathrm{Re}(D^{n+1}f/D^n f) > \delta_n $ guarantees univalence of $ f \in A $. It is known that $ \delta_0 = 0 $, $ \delta_1 = 1/4 $. The paper concludes with further conjectures and open questions about the geometric structure of $ K_n $.