Starlikeness of certain non-univalent functions

Abstract

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.

On a Class of Analytic Functions Related to Robertson’s Formula Involving Crescent Shaped Domain and Lemniscate of Bernoulli

“…In [17], Lecko proposed an alternative analytic characterization of starlike functions with respect to a boundary point and proved the necessity. The sufficiency was shown by Lecko and Lyzzaik [18] and in this way they confirmed this new analytic characterizations (see also [19] Chapter VII). Based on Robertson's idea, Aharanov et al [20] introduced the class of spiral-like functions with respect to a boundary point (see also [21][22][23]).…”

“…Since (7/4, 1/2) ∈ D 2 , it follows from ( 21) that H(7/4, 1/2) = 1. Therefore by applying (19), (20) and (34) we obtain…”

“…which yields (28). By using (35) and the formula for d 1 , and by applying (19) and (34), for δ ∈ C we obtain…”

Radii Constants for Functions with Fixed Second Coefficient

“…Remark 3.5. Putting b = 1 and c = 1 and q = 2 in Theorem 3.5, we get the result [18,Theorem 5,p. 11].…”

“…Among several studies available on radius problems, many results involving ratio between two classes of functions, where one of them belong to some particular subclasses of A has been a center of major focus in geometric function theory and can be seen in [31,32,22,23,24]. Recently, Lecko and Ravichandran [18] estimated certain best possible radii for the classes of function g ∈ A satisfying the conditions (i) g/h ∈ P where h/(zp) ∈ P or h/(zp) ∈ P(1/2) (ii) g/(zp) ∈ P. Motivated by above work, we define three classes of functions by making use of the classes A 6b , A 4c and P as follows: The results presented in this paper are nice extensions of radii estimates results in [18] along with some improved radii. It includes radii estimates for functions in the classes H 1 b,c , H 2 b,c and H 3 b to belong to several subclasses of normalized analytic functions A like starlike functions of order α, starlike functions associated with lemniscate of Bernoulli, parabolic starlike functions, exponential function, cardioid, sine function, lune, a particular rational function, nephroid and modified sigmoid function.…”