#### Abstract

We consider a new class of bounded doubly close-to-convex functions. Coefficient bounds, distortion theorems, and radius of convexity for the class are investigated. A corresponding class of doubly close-to-starlike functions is also considered.

#### 1. Introduction

Let denote the class of functions of the form which are analytic in the unit disk .

Let , , and denote the well-known classes of starlike, convex, and close-to-convex functions, respectively.

A function is said to be in the class of close-to-convex functions of order (see [1]) if there exist and such that It is clear that and .

Denote by the class of analytic functions in with and such that for all .

Suppose that and are two analytic functions in . The function is said to be subordinate to the function , denoted by , if there exists a function such that , .

Let be the well-known class of analytic functions normalized by and having positive real part in .

For a fixed let denote the subclass of defined by The class has been investigated by Goel [2] and also by Libera and Livingston [3].

It is easy to observe that when , the class reduces to the class .

For , the function maps the unit disk onto the domain . It follows that a function is in the class if and only if .

If then it is easy to check that

Some properties of the functions belonging to the class are listed in the next lemma.

Lemma 1 (see [2, 3]). *Let be in the class . Then
**
for and . All the inequalities are sharp.*

Let denote the class of all functions for which there exists a function such that , where is given by (4), or equivalently

It is easy to see that when the class reduces to the class of close-to-convex functions.

A slightly different class than the class was investigated in [2].

Recently, in [4] the authors considered a new class of analytic functions defined in a similar way to the class . For fixed and a function is called doubly close-to-convex if there exist and such that

Motivated by the ideas from [4] we define a new class of bounded doubly close-to-convex functions.

*Definition 2. *Let and be fixed. A function is said to be in the class if there exists a function such that , where is given by (4) with instead of , or equivalently

From the above definition and the definition of the class , it follows that if there exist a function and a function such that and , or equivalently

In the next lemma we prove that the new class is nonempty.

Lemma 3. *Let and . Then, there exists a function .*

*Proof. *Define the following three functions:
with and , . Since (see [5]) and
it follows that . The equality
together with shows that the function defined by (14) belongs to .

In this paper we obtain distortion theorems, radius of convexity, and coefficient bounds for the class . In the last section of the paper a corresponding class of bounded doubly close-to-starlike functions is also considered.

#### 2. Distortion Theorems

In this section distortion theorems for the class are obtained.

Theorem 4. *Let and . If , then
**
for and .*

*Proof. *Let . Then there exists such that , where belongs to the class defined by (3) with instead of . Since , making use of inequalities (8) from Lemma 1, we obtain
for and .

The function belongs to the class and thus, there exists a function such that , where . Using once more the inequalities (8) from Lemma 1, we have
for and .

Moreover, implies that (see [5, 6])
for and .

Combining the inequalities (19), (20), and (21), we obtain the desired inequality (18).

Theorem 5. *Let and . If , then
**
if and
**
if .*

*Proof. *Let . Integrating along the straight line segment from origin to the right-hand side of inequality (18) we obtain
which leads to inequalities (22) and (24).

To prove the lower bound of we proceed in the following way. Let be the radius of the open disk contained entirely in . Consider with such that . The minimum increases with and is less than . Hence, the linear segment which connects the origin with the point will be covered entirely by the values of . Denote by the arc in which is mapped by in . Making use of the left-hand side of inequality (18) we get

After simple calculations we obtain the inequalities (23) and (25). Thus, the proof of our theorem is completed.

#### 3. Radius of Convexity

In this section we obtain the radius of the disk which is mapped onto a convex domain by the functions belonging to .

Theorem 6. *Let and . Suppose that . Then, the function maps the disk onto a convex domain, where is the smallest positive root of the equation
*

*Proof. *Let . Then, there exists such that
where . Since , there exists a function such that
where . Moreover, since it follows that there exists a function (see [5, 6]) such that
Combining the equalities (29), (30), and (31), we get
By taking logarithmic derivative in (32), we obtain
which leads to
For , we have (see [7])
with and .

Since and , making use of inequality (9) from Lemma 1, we obtain
with and .

Substituting (35) and (36) in (34) we have

It follows that the function is convex whenever the expression in the right-hand side of (37) is positive. The numerator of this expression can be written as , where
We observe that and . It follows that the smallest root of and also of lies between and and, thus, the theorem is proved.

#### 4. Coefficient Estimates

In order to find coefficient estimates for the class , we will find first the coefficient estimates for the class .

Theorem 7. *Let . If the function given by (1) is in the class , then
*

*Proof. *Since we have
where
Equating the coefficients of on both sides of (40), we find the following relation between the coefficients:
For we have , (see [5, 6]). In virtue of inequality (7) of Lemma 1 we have , .

Making use of (42), we find
and, thus, we get the desired inequality (39).

When we find the well-known coefficient estimates of close-to-convex functions (see [5, 6]).

In the next theorem we obtain the coefficient estimates for the class .

Theorem 8. *Let and . If the function given by (1) is in the class , then
*

*Proof. *Let . Then, there exist
such that
Comparing the coefficients of on both sides of the above equality, we obtain the next relation:
Since and , from (39) and (7), we get
From (47) in connection with (48), we obtain
which leads to inequality (44).

#### 5. Maximum Value of

The problem of finding sharp upper bounds for the functional for a family of analytic functions is known as the Fekete-Szegö problem. For the classes and , the following estimates are known (see, e.g., [8–11]):

In this section, the case of the Fekete-Szegö problem will be considered, first for the class and then, for the class .

In order to prove our results we need the following lemma due to Keogh and Merkes [9].

Lemma 9. *Let be in the class and let . Then
**
Equality may be attained for and .*

Theorem 10. *Let . If is of the form (1), then
*

*Proof. *Let . Then there exists such that , where is given by (4). Let . Define
Since , there exists such that
where and are given by (6). Combining (54) and (55), after simple calculations, we get
Substituting (6) and (57) in (56) we obtain
so that
Since , making use of (50) with , we have
In virtue of Lemma 9 and taking into account that , we get
Combining (59), (60), and (61), we obtain
and, thus, the proof is completed.

It is easy to observe that when inequality (53) reduces to which is the same with (51) for .

Theorem 11. *Let and . If of the form (1) belongs to the class , then
*

*Proof. *Since , there exists such that , where is given by (4) with instead of . Let and defined by
From it follows that there exists such that .

Using the same method as in the proof of Theorem 10, we obtain
Since , from (53), we have
Moreover, for , we get from Lemma 9 that

Combining (65), (66), and (67), the inequality (63) follows.

#### 6. Bounded Doubly Close-to-Starlike Functions

Let . Consider the class of all functions for which there exists a function such that , with given by (4), or equivalently

It is easy to observe that when the class reduces to the class of close-to-starlike functions defined by Reade [12].

For and we denote by the class of functions for which there exists a function such that , with given by (4), or equivalently

Connections between the classes and and also between and are given in the next theorem.

Theorem 12. *Let and . Then, the following relationships hold:
*

*Proof. *It is well known that a function if and only if .

The definition of the class implies that if and only if there exists such that .

The relation (70) follows from

In the same way, if and only if there exists such that . We have
and taking into account (70), the relation (71) follows.

The proofs of (72) and (73) are similar and will be omitted.

The condition (71) of Theorem 12 together with Lemma 3 and (14) shows that the function where , belongs to the class and, thus, this class is nonempty.

Combining Theorem 12 with Theorems 4 and 8 the next properties of the class can be easily obtained.

Corollary 13. *Let and . If , then
**
for and .*

Corollary 14. *Let and . If is given by (1), then
*

Making use of Theorem 12, we can also obtain an upper bound of for functions in the class .

Corollary 15. *Let and . If is given by (1), then
*

*Proof. *Let . Then, from (73), the function given by
belongs to the class . Comparing the coefficients of and on both sides of the above equality, we obtain
so that
Now, the inequality (79) follows as an application of Theorem 11.

Once again making use of Theorem 12, we have that if and only if for some . Therefore, a radius of convexity for will correspond to a radius of starlikeness for .

The next result follows easily from Theorem 6.

Corollary 16. *Let and . Suppose that . Then, the function maps the disk onto a starlike domain, where is the smallest positive root of (28) in Theorem 6.*

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The author gratefully thanks the referee for his/her comments.