On an extension of Ozaki’s condition

It is known that if {f(z)=z^{p}+\sum_{n=p+1}^{\infty}a_{n}z^{n}} and it is analytic in a convex domain {D\subset\mathbb{C}} and, for some real α, we have {\operatorname{\mathfrak{Re}}\{\exp(i\alpha)f^{(p)}(z)\}>0} , {z\in D} , then {f(z)} is at most p-valent in D. This Ozaki condition is a generalization of the well-known Noshiro–Warschawski univalence condition. In this paper, we consider the radius of univalence of functions {g(z)=z+\sum_{n=1}^{\infty}b_{n}z^{n}} such that {g^{\prime}(z)\prec[(1+z)^{2}/(1-z)^{2}]} and some related problems.

Univalency and Convexity Conditions for a General Integral Operator

For analytic functions f and g in the open unit disc 𝕌, a new general integral operator is introduced. The main objective of this paper is to obtain univalence condition and order of convexity for this general integral operator.