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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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equa_HTML.gif)
and let denote the class of functions
of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ1_HTML.gif)
A function is said to be starlike of orderα in
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equb_HTML.gif)
A function is said to be convex of orderα in
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equc_HTML.gif)
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 ℋ be a subclass of the class . We define the radius of starlikeness of theclass ℋ by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equd_HTML.gif)
We denote by ,
, the class of functions
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Eque_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equf_HTML.gif)
In particular, we denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equg_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equh_HTML.gif)
where ,
(
) are positive real numbers satisfying the followingconditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equi_HTML.gif)
Dimkov [2] studied the class of functions F given by the formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equj_HTML.gif)
where (
) are complex numbers satisfying the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equk_HTML.gif)
Let p, n be positive integer and let a, m,M, N be positive real numbers, . Moreover, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equl_HTML.gif)
be fixed vectors, with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equm_HTML.gif)
We denote by the class of functions F given by theformula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ2_HTML.gif)
By we denote union of all classes
for which
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ3_HTML.gif)
Finally, let us denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ4_HTML.gif)
It is clear that the class contains functions F given by the formula(2) for which
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equn_HTML.gif)
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 pointω with smooth boundary given by the formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equo_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ6_HTML.gif)
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 functionmaps the disk
,
, onto a domain starlike with respect to theorigin if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ7_HTML.gif)
For a function it is easy to verify that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equp_HTML.gif)
Thus, after some calculations we get the following lemma.
Lemma 2Let,
,
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equq_HTML.gif)
Lemma 3[8]
If, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equr_HTML.gif)
Theorem 1Letm, b, cbe defined by (3) and set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ8_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ9_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ10_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ11_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ12_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ13_HTML.gif)
Moreover, set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ15_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ16_HTML.gif)
Proof Let F belong to the class and let
satisfy (5). The functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equs_HTML.gif)
belong to the class together with the functions
. Thus, by (2) the functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equt_HTML.gif)
belong to the class together with the function
. In consequence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ17_HTML.gif)
Therefore, without loss of generality we may assume that a is nonnegativereal number. Since , there exist functions
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equu_HTML.gif)
or equivalently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ18_HTML.gif)
After logarithmic differentiation of the equality (2) we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equv_HTML.gif)
Thus, using (18) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equw_HTML.gif)
By Lemma 2 and Lemma 3 we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equx_HTML.gif)
Setting and using (3) the above inequality yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equy_HTML.gif)
By Lemma 1 it is sufficient to show that the right-hand side of the lastinequality is nonnegative, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ19_HTML.gif)
If we put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equz_HTML.gif)
then we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equaa_HTML.gif)
Thus, using the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ20_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ21_HTML.gif)
The discriminant Δ of is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ22_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ23_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ24_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ25_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ26_HTML.gif)
Ad 1∘. Since , by (22), the condition
is equivalent to the inequality
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equab_HTML.gif)
where φ is defined by (13). Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equac_HTML.gif)
Then γ is the curve which is tangent to the straight lines and
at the points
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ27_HTML.gif)
where ,
, q are defined by (10), (11), (12),respectively.
Moreover, γ cuts the straight line at the points
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equad_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equae_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equaf_HTML.gif)
and consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ28_HTML.gif)
where φ is defined by (13) (see Figure 1).
Ad 2∘. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equag_HTML.gif)
It is easy to verify that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equah_HTML.gif)
where q is defined by (12) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ29_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ30_HTML.gif)
we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equai_HTML.gif)
Thus, the inequality is true if
. The inequality
may be written in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ32_HTML.gif)
It is easy to verify that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equaj_HTML.gif)
Thus we determine the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ33_HTML.gif)
where φ is defined by (13) (see Figure 2).
Ad 3∘. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equak_HTML.gif)
and let q and be defined by (12) and (29), respectively. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equal_HTML.gif)
Moreover, by (30) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equam_HTML.gif)
Thus, we conclude that the inequality is true if
. The inequality
may be written in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ35_HTML.gif)
where φ is defined by (13) (see Figure 2).The union of the sets ,
,
defined by (28), (33), and (35) gives the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equan_HTML.gif)
Thus, by (17) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ36_HTML.gif)
where is defined by (8).
Now, let . Then the inequality (21) is satisfied for every
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ37_HTML.gif)
We see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equao_HTML.gif)
where q and are defined by (12) and (29), respectively. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equap_HTML.gif)
the condition (37) is satisfied if and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ38_HTML.gif)
Let . Then by (21) we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equaq_HTML.gif)
The above inequality holds for every if
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equar_HTML.gif)
or equivalently (38). Thus, by (17) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ39_HTML.gif)
where is defined by (9). Because the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ40_HTML.gif)
belongs to the class , and for
,
,
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equas_HTML.gif)
Lemma 1 yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ41_HTML.gif)
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 2Let 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equat_HTML.gif)
The functions described by (40), with (3) are the extremalfunctions.
Theorem 3
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ42_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equau_HTML.gif)
The equality in (42) is realized by the functionFof the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ43_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equav_HTML.gif)
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)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equaw_HTML.gif)
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equax_HTML.gif)
Therefore, by (4) we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equ44_HTML.gif)
and by Theorem 2 we get (42). Putting ,
in (3) we see that
are negative real numbers. Thus, the extremalfunction (40) has the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equay_HTML.gif)
or equivalently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equaz_HTML.gif)
Consequently, using (3) we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equba_HTML.gif)
that is, we have the function (43) and the proof is completed. □
Since , by Theorem 1 we obtain the followingtheorem.
Theorem 4Let,
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equbb_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equbc_HTML.gif)
Moreover, let us put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equbd_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Eqube_HTML.gif)
The function F of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equbf_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equbg_HTML.gif)
Corollary 2
The radius of starlikeness of the class
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equbh_HTML.gif)
Corollary 3
The radius of starlikeness of the class
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2015-5/MediaObjects/13660_2014_1516_Equbi_HTML.gif)
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
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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