New criteria for univalent functions

Abstract:

The classes ${K_n}$ of functions $f(z)$ regular in the unit disc $\mathfrak {U}$ with $f(0) = 0,f’(0) = 1$ satisfying \[ \operatorname {Re} [{({z^n}f)^{(n + 1)}}/{({z^{n - 1}}f)^{(n)}}] > (n + 1)/2\] in $\mathfrak {U}$ are considered and ${K_{n + 1}} \subset {K_n},n = 0,1, \cdots$, is proved. Since ${K_0}$ is the class of functions starlike of order 1/2 all functions in ${K_n}$ are univalent. Some coefficient estimates are given and special elements of ${K_n}$ are determined.