This paper studies analytic functions f defined on the open unit disk of the complex plane for which f/g and (1 + z)g/z are both functions with positive real part for some analytic function g. We determine radius constants of these functions to belong to classes of strong starlike functions, starlike functions of order α, parabolic starlike functions, as well as to the classes of starlike functions associated with lemniscate of Bernoulli, cardioid, lune, reverse lemniscate, sine function, exponential function and a particular rational function. The results obtained are sharp.

We consider three classes of functions defined using the class $${\mathcal {P}}$$ P of all analytic functions $$p(z)=1+cz+\cdots $$ p ( z ) = 1 + c z + ⋯ on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions f with $$f/g\in {\mathcal {P}}$$ f / g ∈ P and $$g/(zp)\in {\mathcal {P}}$$ g / ( z p ) ∈ P for some normalized analytic function g and $$p\in {\mathcal {P}}$$ p ∈ P . The second class is defined by replacing the condition $$f/g\in {\mathcal {P}}$$ f / g ∈ P by $$|(f/g)-1|<1$$ | ( f / g ) - 1 | < 1 while the other class consists of normalized analytic functions f with $$f/(zp)\in {\mathcal {P}}$$ f / ( z p ) ∈ P for some $$p\in {\mathcal {P}}$$ p ∈ P . We have determined radii so that the functions in these classes to belong to various subclasses of starlike functions. These subclasses includes the classes of starlike functions of order $$\alpha $$ α , parabolic starlike functions, as well as the classes of starlike functions associated with lemniscate of Bernoulli, reverse lemniscate, sine function, a rational function, cardioid, lune, nephroid and modified sigmoid function.

This paper studies analytic functions f defined on the open unit disk of the complex plane for which f /g and (1 + z)g/z are both functions with positive real part for some analytic function g. We determine radius constants of these functions to belong to classes of strong starlike functions, starlike functions of order α, parabolic starlike functions, as well as to the classes of starlike functions associated with lemniscate of Bernoulli, cardioid, lune, reverse lemniscate, sine function, exponential function and a particular rational function. The results obtained are sharp.

"A normalized function $f$ on the open unit disc is starlike (or convex) univalent if the associated function $zf'(z)/f(z)$ (or $1+zf''(z)/f'(z)$) is a function with positive real part. The radius of starlikeness or convexity is usually obtained by using the estimates for functions with positive real part. Using subordination, we examine the radius of various starlikeness, in particular, radii of Janowski starlikeness and starlikeness of order $\beta$, for the function $f$ when the function $f$ is either convex or $(zf'(z)+\alpha z^2f''(z))/f(z)$ lies in the right-half plane. Radii of starlikeness associated with lemniscate of Bernoulli and exponential functions are also considered."

For a normalised analytic function f defined on the open unit disk in the complex plane, we determine several sufficient conditions for starlikeness in terms of the quotients and the Schwarzian derivative2 /2 . These conditions were obtained by using the admissibility criteria of starlikeness in the theory of second order differential subordination.