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# On products of multivalent close-to-star functions

- Muhammad Arif
^{10}Email author, - Jacek Dziok
^{11}Email author, - Mohsan Raza
^{12}Email author and - Janusz Sokół
^{13}Email author

**2015**:5

https://doi.org/10.1186/1029-242X-2015-5

© Arif et al.; licensee Springer. 2015

**Received:**11 October 2014**Accepted:**3 December 2014**Published:**6 January 2015

## Abstract

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.

## Keywords

- analytic functions
- close-to-star functions
- generalized starlikeness
- radius of starlikeness

## 1 Introduction

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 Arg*w* 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 [3] stated and solved the following problem.

**Problem 1**Let ℋ be the class of functions that are univalent in and let be a domain starlike with respect to an inner point

*ω*with smooth boundary given by the formula

Find conditions for the function such that for each the image domain is starlike with respect to .

Świtoniak and Stankiewicz [4, 5], Dimkov and Dziok [6] (see also [7]) have investigated a similar problem of generalized starlikeness.

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.

## 2 Main results

We start from listing some lemmas which will be useful later on.

**Lemma 1**[5]

Thus, after some calculations we get the following lemma.

**Lemma 3**[8]

*If*,

*then a function*

*maps the disk*

*onto a domain starlike with respect to the origin*.

*The result is sharpfor*

*and for*

*the set*ℬ

*cannot be larger than*.

*It means that*

*a*is nonnegativereal number. Since , there exist functions and such that

First, we discuss the case . Thus, the inequality (21) is satisfied for every if one of the following conditions is fulfilled:

1^{∘}
,

2^{∘}
,
and
,

3^{∘}
,
and
,

where
,
, *q* are defined by (10), (11), (12),respectively.

*φ*is defined by (13) (see Figure 1).

*D*at the point defined by (27) and at the point , where

*φ*is defined by (13) (see Figure 2).

*D*at the point defined by (27) and at the point , where is defined by (32). Thus, we describe the set

*φ*is defined by (13) (see Figure 2).The union of the sets , , defined by (28), (33), and (35) gives the set

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.

**Theorem 2**

*Let*ℬ

*be defined by*(14).

*If*,

*then a function*

*maps the disk*

*onto a domain starlike with respect to the origin*.

*The obtained resultis sharp for*

*and for*

*the set*ℬ

*cannot be larger than*,

*where*

*is defined by*(8).

*It means that*

*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.

*m*and

*c*, and increasing with respect to

*b*. Thus, from Theorems 1 and2 we have (see Figure 3)

Since , by Theorem 1 we obtain the followingtheorem.

*If*,

*then the function*

*maps the disk*

*onto a domain starlike with respect to the origin*.

*The obtained resultis sharp for*

*and for*

*the set*ℬ

*cannot be larger then*.

*It means that*

*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 [7]. Putting
we get the radius of starlikeness of the class
obtained by Ratti [9]. Putting, moreover,
we get the radius of starlikeness of the class
obtained by MacGregor [10].

## Declarations

### Acknowledgements

This work is partially supported by the Centre for Innovation and Transfer ofNatural Sciences and Engineering Knowledge, University of Rzeszów.

## Authors’ Affiliations

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## Copyright

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