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# On products of multivalent close-to-star functions

*Journal of Inequalities and Applications*
**volume 2015**, Article number: 5 (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.

## 1 Introduction

Let denotethe class of functions which are *analytic* in , where

and let denote the class of functions of the form

A function is said to be *starlike of order**α* in if

A function is said to be *convex of order**α* in if

We denote by the class of all functions , which are convex of order *α* inand by we denote the class of all functions, which are starlike of order *α* in.We also set .

Let ℋ be a subclass of the class . We define *the radius of starlikeness* of theclass ℋ by

We denote by , , the class of functions such that and

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.

We say that a function belongs to the class if there exists a function such that

In particular, we denote

The class is the well-known class of close-to-star functionswith argument 0.

Silverman [1] introduced the class of functions *F* given by the formula

where , () are positive real numbers satisfying the followingconditions:

Dimkov [2] studied the class of functions *F* given by the formula

where () are complex numbers satisfying the condition

Let *p*, *n* be positive integer and let *a*, *m*,*M*, *N* be positive real numbers, . Moreover, let

be fixed vectors, with

We denote by the class of functions *F* given by theformula

By we denote union of all classes for which

Finally, let us denote

It is clear that the class contains functions *F* given by the formula(2) for which

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.

**Problem 2** Let . Determine the set of all pairs , such that

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.

We note that

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]

*A function**maps the disk*, , *onto a domain starlike with respect to theorigin if and only if*

For a function it is easy to verify that

Thus, after some calculations we get the following lemma.

**Lemma 2***Let*, , . *Then*

**Lemma 3**[8]

*If*, *then*

**Theorem 1***Let**m*, *b*, *c**be defined by* (3) *and set*

*where*

*Moreover*, *set*

*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*

*Proof* Let *F* belong to the class and let satisfy (5). The functions

belong to the class together with the functions . Thus, by (2) the functions

belong to the class together with the function . In consequence, we have

Therefore, without loss of generality we may assume that *a* is nonnegativereal number. Since , there exist functions and such that

or equivalently

After logarithmic differentiation of the equality (2) we obtain

Thus, using (18) we have

By Lemma 2 and Lemma 3 we obtain

Setting and using (3) the above inequality yields

By Lemma 1 it is sufficient to show that the right-hand side of the lastinequality is nonnegative, that is,

If we put

then we obtain

Thus, using the equality

we obtain

The discriminant Δ of is given by

where

Let

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

Ad 1^{∘}. Since , by (22), the condition is equivalent to the inequality. Then

where *φ* is defined by (13). Let

Then *γ* is the curve which is tangent to the straight lines and at the points

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

Moreover, *γ* cuts the straight line at the points

Since

we have

and consequently

where *φ* is defined by (13) (see Figure 1).

Ad 2^{∘}. Let

It is easy to verify that

where *q* is defined by (12) and

Since

we see that

Thus, the inequality is true if . The inequality may be written in the form

The hyperbola , which is the boundary of the set of all pairs satisfying (31), cuts the boundary of the set*D* at the point defined by (27) and at the point, where

It is easy to verify that

Thus we determine the set

where *φ* is defined by (13) (see Figure 2).

Ad 3^{∘}. Let

and let *q* and be defined by (12) and (29), respectively. Then

Moreover, by (30) we have

Thus, we conclude that the inequality is true if . The inequality may be written in the form

The hyperbola , which is the boundary of the set of all pairs satisfying (34), cuts the boundary of the set*D* at the point defined by (27) and at the point, where is defined by (32). Thus, we describe the set

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

Thus, by (17) we have

where is defined by (8).

Now, let . Then the inequality (21) is satisfied for every if

We see that

where *q* and are defined by (12) and (29), respectively. Since

the condition (37) is satisfied if and

Let . Then by (21) we obtain

The above inequality holds for every if and

or equivalently (38). Thus, by (17) we have

where is defined by (9). Because the function

belongs to the class , and for , , we have

Lemma 1 yields

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*.

**Theorem 3**

*where*

*The equality in* (42) *is realized by the function**F**of the form*

*Proof* Let *M*, *N* be positive real numbers and let, , and be defined by (8), (9), (12), and (13),respectively.

It is easy to verify that

Moreover, the function is decreasing with respect to *m* and*c*, and increasing with respect to *b*. Thus, from Theorems 1 and2 we have (see Figure 3)

and

Therefore, by (4) we obtain

and by Theorem 2 we get (42). Putting , in (3) we see that are negative real numbers. Thus, the extremalfunction (40) has the form

or equivalently

Consequently, using (3) we obtain

that is, we have the function (43) and the proof is completed. □

Since , by Theorem 1 we obtain the followingtheorem.

**Theorem 4***Let*, , *and*

*where*

*Moreover*, *let us put*

*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*

*The function*
*F*
*of the form*

*is the extremal function*.

Using (6) and Theorems 1-4, we obtain the radii of starlikeness for the classes, , , .

**Corollary 1**
*The radius of starlikeness of the classes*
*and*
*is given by*

**Corollary 2**
*The radius of starlikeness of the class*
*is given by*

**Corollary 3**
*The radius of starlikeness of the class*
*is given by*

**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].

## References

Silverman H:

**Products of starlike and convex functions.***Ann. Univ. Mariae Curie-Skłodowska, Sect. A*1975,**29:**109–116.Dimkov G:

**On products of starlike functions. I.***Ann. Pol. Math.*1991,**55:**75–79.Aleksandrov IA:

**On the star-shaped character of the mappings of a domain by functions regular and univalent in the circle.***Izv. Vysš. Učebn. Zaved., Mat.*1959,**4**(11)**:**9–15.Stankiewicz J, Świtoniak B:

**Generalized problems of convexity and starlikeness.**In*Complex Analysis and Applications*. Publ. House Bulgar. Acad. Sci., Sofia; 1986:670–675. ’85 (Varna, 1985)Świtoniak B:

**On a starlikeness problem in the class of functions**.*Folia Sci. Univ. Tech. Resov.*1984,**14:**17–27.Dimkov G, Dziok J:

**Generalized problem of starlikeness for products of***p***-valentstarlike functions.***Serdica Math. J.*1998,**24:**339–344.Dziok J:

**Generalized problem of starlikeness for products of close-to-starfunctions.***Ann. Pol. Math.*2013,**107:**109–121. 10.4064/ap107-2-1Nunokawa M, Causey WM:

**On certain analytic functions bounded argument.***Sci. Rep. Fac. Educ., Gunma Univ.*1985,**34:**1–3.Ratti JS:

**The radius of univalence of certain analytic functions.***Math. Z.*1968,**107:**241–248. 10.1007/BF01110013MacGregor TH:

**The radius of univalence of certain analytic functions.***Proc. Am. Math. Soc.*1963,**14:**514–520. 10.1090/S0002-9939-1963-0148891-3

## Acknowledgements

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

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Arif, M., Dziok, J., Raza, M. *et al.* On products of multivalent close-to-star functions.
*J Inequal Appl* **2015**, 5 (2015). https://doi.org/10.1186/1029-242X-2015-5

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

### Keywords

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