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 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 [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 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 .

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 ℋ.

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 functionmaps 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 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

(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 .

Full size image

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 .

Full size image

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 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.

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 .

Full size image

Since , by Theorem 1 we obtain the followingtheorem.

Theorem 4Let, , and

where

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 [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].