On products of multivalent close-to-star functions
Journal of Inequalities and Applications volume 2015, Article number: 5 (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.
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 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 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 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  introduced the class of functions F given by the formula
where , () are positive real numbers satisfying the followingconditions:
Dimkov  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  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
Find conditions for the function such that for each the image domain is starlike with respect to .
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.
A functionmaps 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 2Let, , . Then
Theorem 1Letm, b, cbe defined by (3) and set
If, then a functionmaps the diskonto a domain starlike with respect to the origin. The result is sharpforand forthe 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
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
The discriminant Δ of is given by
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 ,
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
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
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 setD 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 setD 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 2Let ℬ be defined by (14). If, then a functionmaps the diskonto a domain starlike with respect to the origin. The obtained resultis sharp forand forthe set ℬ cannot be larger than, whereis defined by (8). It means that
The functions described by (40), with (3) are the extremalfunctions.
The equality in (42) is realized by the functionFof 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 andc, and increasing with respect to b. Thus, from Theorems 1 and2 we have (see Figure 3)
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
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 4Let, , and
Moreover, let us put
If, then the functionmaps the diskonto a domain starlike with respect to the origin. The obtained resultis sharp forand forthe 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 . 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 .
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This work is partially supported by the Centre for Innovation and Transfer ofNatural Sciences and Engineering Knowledge, University of Rzeszów.
The authors declare that they have no competing interests.
All authors jointly worked on the results and they read and approved the finalmanuscript.
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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