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On products of multivalent close-to-star functions

Journal of Inequalities and Applications20152015:5

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

  • Received: 11 October 2014
  • Accepted: 3 December 2014
  • Published:

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

Let denotethe class of functions which are analytic in , where
and let denote the class of functions of the form
(1)
A function is said to be starlike of orderα in if
A function is said to be convex of orderα in if

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 .

Let be a subclass of the class . We define the radius of starlikeness of theclass by
We denote by , , the class of functions such that and

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.

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
(2)
By we denote union of all classes for which
(3)
Finally, let us denote
(4)
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 be the class of functions that are univalent in and let be a domain starlike with respect to an inner pointω 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
(5)

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 .

We note that
(6)

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
(7)
For a function it is easy to verify that

Thus, after some calculations we get the following lemma.

Lemma 2Let , , . Then

Lemma 3[8]

If , then
Theorem 1Letm, b, cbe defined by (3) and set
(8)
(9)
where
(10)
(11)
(12)
(13)
Moreover, set
(14)
If , then a function maps the disk onto a domain starlike with respect to the origin. The result is sharpfor and for the set cannot be larger than . It means that
(15)
(16)
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
(17)
Therefore, without loss of generality we may assume that a is nonnegativereal number. Since , there exist functions and such that
or equivalently
(18)
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,
(19)
If we put
then we obtain
Thus, using the equality
(20)
we obtain
(21)
The discriminant Δ of is given by
(22)
where
(23)
(24)
Let
(25)

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
(26)
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
(27)

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

Moreover, γ cuts the straight line at the points
Since
we have
and consequently
(28)
where φ is defined by (13) (see Figure 1).
Figure 1
Figure 1

The set .

Ad 2. Let
It is easy to verify that
where q is defined by (12) and
(29)
Since
(30)
we see that
Thus, the inequality is true if . The inequality may be written in the form
(31)
The hyperbola , which is the boundary of the set of all pairs satisfying (31), cuts the boundary of the setD at the point defined by (27) and at the point , where
(32)
It is easy to verify that
Thus we determine the set
(33)
where φ is defined by (13) (see Figure 2).
Figure 2
Figure 2

The sets and .

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
(34)
The hyperbola , which is the boundary of the set of all pairs satisfying (34), cuts the boundary of the setD at the point defined by (27) and at the point , where is defined by (32). Thus, we describe the set
(35)
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
(36)

where is defined by (8).

Now, let . Then the inequality (21) is satisfied for every if
(37)
We see that
where q and are defined by (12) and (29), respectively. Since
the condition (37) is satisfied if and
(38)
Let . Then by (21) we obtain
The above inequality holds for every if and
or equivalently (38). Thus, by (17) we have
(39)
where is defined by (9). Because the function
(40)
belongs to the class , and for , , we have
Lemma 1 yields
(41)

From (36) and (41) we have (15), while (39) and (41) give (16), which completes theproof. □

Since the set defined by (14) is dependent only of m, b,c, the following result is an immediate consequence ofTheorem 1.

Theorem 2Let 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 cannot be larger than , where is defined by (8). It means that

The functions described by (40), with (3) are the extremalfunctions.

Theorem 3
(42)
where
The equality in (42) is realized by the functionFof the form
(43)

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 andc, and increasing with respect to b. Thus, from Theorems 1 and2 we have (see Figure 3)
and
Therefore, by (4) we obtain
(44)
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.  □
Figure 3
Figure 3

The sets and .

Since , by Theorem 1 we obtain the followingtheorem.

Theorem 4Let , , 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 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].

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan
(2)
Faculty of Mathematics and Natural Sciences, University of Rzeszów, Rzeszów, 35-310, Poland
(3)
Department of Mathematics, GC University Faisalabad, Faisalabad, Pakistan
(4)
Department of Mathematics, Rzeszów University of Technology, Rzeszów, 35-959, Poland

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Copyright

© Arif et al.; licensee Springer. 2015

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/4.0), which permitsunrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly credited.

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