Some sufficient conditions for univalence and close-to-convexity

Abstract

Abstract In the class of analytic functions in the unit disc | z | < 1 {\lvert z\rvert<1} we prove some new sufficient conditions for functions to be univalent or to be close-to-convex in the unit disc. Also we extend Ozaki’s condition that ℜ ⁢ 𝔢 ⁢ { exp ⁡ ( i ⁢ α ) ⁢ f ( p ) ⁢ ( z ) } > 0 {\mathfrak{Re}\{\exp(i\alpha)f^{(p)}(z)\}>0} in | z | < 1 {\lvert z\rvert<1} implies that f ⁢ ( z ) {f(z)} is at most p-valent in | z | < 1 {\lvert z\rvert<1} .