- Open Access
On products of multivalent close-to-star functions
© Arif et al.; licensee Springer. 2015
- Received: 11 October 2014
- Accepted: 3 December 2014
- Published: 6 January 2015
In the present paper we define a class of products of multivalent close-to-starfunctions and determine the set of pairs , , such that every function from the class maps thedisk onto a domain starlike with respect to theorigin. Some consequences of the obtained result are also considered.
MSC: 30C45, 30C50, 30C55.
- analytic functions
- close-to-star functions
- generalized starlikeness
- radius of starlikeness
We denote by the class of all functions , which are convex of order α in and by we denote the class of all functions , which are starlike of order α in .We also set .
where Argw denote the principal argument of the complex number w(i.e. from the interval ). The class is the well-known class of Carathéodoryfunctions.
The class is the well-known class of close-to-star functionswith argument 0.
Aleksandrov  stated and solved the following problem.
Find conditions for the function such that for each the image domain is starlike with respect to .
and every function maps the disk onto a domain starlike with respect to the origin.The set is called the set of generalized starlikeness of theclass ℋ.
In this paper we determine the sets , and . The sets of generalized starlikeness for somesubclasses of the defined classes are also considered. Moreover, we obtain the radiiof starlikeness of these classes of functions.
We start from listing some lemmas which will be useful later on.
Thus, after some calculations we get the following lemma.
First, we discuss the case . Thus, the inequality (21) is satisfied for every if one of the following conditions is fulfilled:
2∘ , and ,
3∘ , and ,
where , , q are defined by (10), (11), (12),respectively.
where is defined by (8).
From (36) and (41) we have (15), while (39) and (41) give (16), which completes theproof. □
Since the set ℬ defined by (14) is dependent only of m, b,c, the following result is an immediate consequence ofTheorem 1.
The functions described by (40), with (3) are the extremalfunctions.
Proof Let M, N be positive real numbers and let , , and be defined by (8), (9), (12), and (13),respectively.
Since , by Theorem 1 we obtain the followingtheorem.
is the extremal function.
Using (6) and Theorems 1-4, we obtain the radii of starlikeness for the classes , , , .
Remark 1 Putting in Corollary 3 we get the radius of starlikenessof the class obtained by Dziok . Putting we get the radius of starlikeness of the class obtained by Ratti . Putting, moreover, we get the radius of starlikeness of the class obtained by MacGregor .
This work is partially supported by the Centre for Innovation and Transfer ofNatural Sciences and Engineering Knowledge, University of Rzeszów.
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