1. Introduction
Let
be the family of all functions analytic in
having the power series expansion:
and let
denote the class of univalent starlike functions in
(for the definitions and properties of
and other classes, see [
1]). For a given real argument
and a given function
, a function
is called close-to-convex with argument
with respect to
g if:
Let
be the class of all such functions. Moreover, let:
Let
denote the family of all close-to-convex functions (see [
2,
3]). It is obvious that:
All functions in
are univalent.
In this paper, we consider the class
, where
k is the Koebe function:
The class
is sometimes denoted by
. Such functions have a well known geometrical meaning. Namely, for each function
f in this class, the set
is a domain such that
for every
. Such functions
f are convex in the positive direction of the real axis.
For a function
f analytic in
of the form (
1), we define two functionals for a fixed real
:
and:
The functionals
and
are the generalizations of two well known expressions:
and
. Both functionals are symmetric, or invariant, under rotations. The first one is a particular case of the generalized Zalcman functional. It was investigated, among others, by Ma [
4] and Efraimidis and Vukotić [
5]. The second functional is known as the second Hankel determinant, and it was studied in many papers. The investigation of Hankel determinants for analytic functions was started by Pommerenke (see [
6,
7]) and continued by many mathematicians in various classes of univalent functions (see, for example [
8,
9,
10,
11,
12,
13,
14,
15,
16]). The functional
was first studied by Hayami and Owa [
17]. They discussed an even more general functional
for the classes
and
,
, of functions
such that Re
and Re
, respectively. The functionals
and
for the classes
and
of starlike and convex functions, respectively, were discussed in [
18].
It is worth pointing out a particularly interesting property of
. The sharp estimates of this functional are often symmetric with respect to a certain point. It was shown in [
18] that such points for
and
are 8/9 and one, respectively. We have:
and:
A similar situation occurs for
and for the class
, where
; this point is 1/2 (see [
17,
19]). This situation appears even in the class
of typically real functions, which do not necessarily have to be univalent (see [
19]).
In this work, we derive bounds of and for functions in .
2. Preliminary Results
Let
denote the class of all analytic functions
h with a positive real part in
satisfying the normalization condition
. Let
have the Taylor series expansion:
We shall need here three results. The first one is known as Caratheodory’s lemma (see, for example, ref. [
1]). The second one is due to Libera and Złotkiewicz ([
20,
21]), and the third one is the result of Hayami and Owa.
Lemma 1 ([
1])
. If is given by (
6)
, then the sharp inequality holds for . Lemma 2 ([
20,
21])
. Let h be given by (
6)
and be a given real number, . Then, if and only if:and:for some complex numbers x, y such that , . Lemma 3 ([
17])
. If is given by (
6),
then: The next lemma is an improvement of Lemma 3 for .
Lemma 4 ([
22])
. If is given by (
6)
and , then:where . The inequality is sharp. The following lemma was proven by Lecko (see Corollary 2.3 in [
23]).
Lemma 5 ([
23])
. If is given by (
6),
then: We have proven the next lemma.
Lemma 6. If is given by (
6),
then:The inequality is sharp. Proof. By Lemma 2,
Applying the invariance of
under rotation, we can assume that
is a non-negative real number. Writing
and
, we get by the triangle inequality and the assumption
:
which gives the desired bound. The equality (
9) holds for:
which means that there is equality in (
9) for rotations of (
10). □
The next lemma is a special case of more general results due to Choi et al. [
24] (see also [
9]). Let
. Define:
Lemma 7. If , then:where:If , then: Applying the correspondence between the functions in
and
:
and Expansions (
1) and (
6) we get:
Moreover, by Lemma 1, with equality if and only if . The equality is possible only for the function , and then, .
Hence, we can express
and
for
as coefficients of a corresponding function
in the following way:
and:
4. Bounds of for the Class
In the main theorem of this section, we establish the sharp bounds of
for the class
. The proof is divided into six lemmas. The first one is a particular case of the result obtained in [
22] (Theorem 3.1 or Theorem 3.3 in [
22]), and the second one is obvious.
Lemma 8. Let . Then, . The result is sharp.
Lemma 9. Let and . Then, . The result is sharp.
Lemma 10. Let and . Then, . The result is sharp.
Proof. From (
13), we can write
as follows:
If
, then, taking into account Lemmas 1 and 3, we get:
If
, then we have:
Now, from Lemma 8 and the first part of this proof (i.e.,
), we obtain:
It is clear that
only when
, which means that this equality holds only for the Koebe function (
2). In other words, the Koebe function is the extremal function for
. □
Taking into account (
13) and Lemma 2, we can write
as follows:
From the above formula, we can obtain bounds of
, while
and
, but only with an additional assumption that
is a positive real number. The assumption of Lemma 2 enforces that
. Notice that if
, then
given by (
2), and we have:
If
, then
is in
, and so:
To shorten notation, we write
p instead of
. One can observe that
can be written as:
where:
From (
18), the triangle inequality,
, and Lemma 2, we get:
where
a,
b, and
c are given by (
19).
Lemma 11. Let , be a real number, and . Then, . The result is sharp.
Proof. For
, we have (
20) with:
We use Lemma 7. Clearly,
for
. Note that the inequality
from the first case of Lemma 7 is equivalent to the obviously true inequality:
The inequality
, which can be written as:
holds for all
. Hence, for
, we have:
For
, we have
, and the inequality
from the last case of Lemma 7 is equivalent to (
21). Therefore,
is also given by (
22).
Thus, from (
20) for
, Lemma 7, (
16) and (
17), we obtain:
where
and
according to the assumption. The function
g is increasing for
; therefore:
Moreover, we have by the triangle inequality:
From Lemma 9 and from (
24), we get:
and the proof is complete. Equality holds for the Koebe function (
2). □
Lemma 12. Let , be a real number, , and K be given by (
25).
Then, . The result is sharp. Proof. For
, we have (
20) with:
We use Lemma 7. Clearly,
for
. First, note that the inequality
is equivalent to the obviously true inequality:
The inequality
, which is equivalent to:
holds for
. For
, we have:
so from (
20) for
and Lemma 7, we obtain:
From Lemma 7, the inequality system consists of , and is contradictory, because the first inequality gives , while the second one yields .
Now, consider the third case of Lemma 7. Let
. The inequality
is equivalent to
, and it is not satisfied for any
. The inequality
, which can be written as
, is also not satisfied for any
. Thus, for
, we have:
From (
20) for
and Lemma 7, we obtain:
For
, we have
, and the inequality
from the last case of Lemma 7 is equivalent to the inequality in (
26).
Thus,
is given by (
27). Finally, from (
16), (
28) and (
30), we obtain:
where:
Now, let us consider the function
g for
. We have:
where
and:
for
. Hence,
for
.
Taking the above into account, one can check that the function
g is increasing for
and is decreasing for
, where
is given by (
25). Therefore,
so we have the desired result. □
Lemma 13. Let , be a real number, and .
- 1.
If , then .
- 2.
If , then .
Proof. We have:
From Lemmas 11 and 12, and the triangle inequality, we get the first part of Lemma 13, i.e.,:
Since:
from Lemma 12, Lemma 8, and the triangle inequality, we get the second part of Lemma 13, i.e.,:
□
The results presented in Lemmas 8–13 can be collected as follows.
Theorem 1. Let , be a real number, and . Then:where K is given by (
25).
The results are sharp for , , and . The equality holds for the Koebe function (
2)
in the first and the last case. The assumption is not necessary for and . 5. Bounds of for the Class
At the beginning of this section, we will quote the well known theorem of Marjono and Thomas [
14].
Theorem 2 ([
14])
. If , then: Now, we shall prove the bound for .
Theorem 3. Let and . Then, . The result is sharp.
Proof. Rearranging the components in (
14):
and writing
p instead of
, by Lemmas 1, 3, and 6, for
, we obtain:
If
, then:
From the previous part of this proof
and from Theorem 2, after using the triangle inequality, we get:
It is easy to verify that for the Koebe function (
2), we have
, so the derived estimate is sharp. □
In the next step, we shall prove that the Koebe function (
2) is the extremal function for
.
Theorem 4. Let and . Then, . The result is sharp.
Proof. At the beginning, let us discuss the case
. From (
14), it follows that:
Now, applying Lemmas 1 and 4 for
, Lemma 5 (remembering that
), Lemma 6, and the triangle inequality and writing
p instead of
, we obtain:
where:
Is it clear that
H is an increasing function for
, so:
If
, then:
From the previous part of this proof and the bound
valid for all functions in
,
Equality holds for the Koebe function. □
It is worth adding that the function
H given by (
31) is decreasing for
, so the choice
is important.
Now, we will find the exact bound of
for
close to one. Namely, we will discuss the case
, where:
In this result, we need in addition that the coefficient
should be real and
. From (
12), we get
. In the proof, we are going to apply Lemma 7.
Taking into account (
14) and Lemma 2, we can write
as follows:
where:
If
and
, then
and
, respectively, so:
We will show that these values are less than or equal to the real bound of
for all
. Now and on, we assume that
. Taking into account (
14) and Lemma 2, by the triangle inequality and the assumption
, we get:
where:
Now, we are ready to establish the main theorem of this section.
Theorem 5. Let , be a real number, , and , where . Then:Equality holds for the function F given by (15). In the proof of this theorem, we will need the two lemmas that follow. We assume that
a,
b, and
c are given by (
35).
Lemma 14. If are such that , then (
36)
holds. Lemma 15. If are such that , then the following inequalities hold: Proof of Lemma 14. At the beginning, observe that if
, then:
According to Lemma 7 from (
34), we obtain:
where:
If
, then from (
34), we get:
Because the right hand side of this inequality is constant and equal to ; hence, .
If
, then:
The first expression in (
38) is equal to:
Substituting
,
, we obtain:
Hence, the maximum value of
is achieved for:
This value is equal to .
The second expression in (
38) is equal to:
so:
It is easy to check that for and , we have and , respectively. This means that the maximum value of for is obtained if . □
Proof of Lemma 15. Let . At the beginning, we want to constrain the range of variability of p to some subset of for which .
From (
35) for
, we have:
which is equivalent to:
If
,
, then from (
35),
. Hence, points for which
lie below the curve
. For the function
,
, there is:
Consequently, is an increasing function if and a decreasing function if for , where is the only solution of in . Since and , then for . This means that and hold for , (in other words, if and , then ).
- I.
Since
and:
as a function of
, is increasing, it is enough to estimate this expression taking
as a limit value. Therefore,
- II.
The inequality
can be written as
. For
and
,
- III.
With the notation
and:
we can write:
We shall prove that
for
and
. We have:
For
, we obtain:
and:
Since
for
and
, we have:
In this way, we have proven that .
- IV.
Let us denote
and:
The function
h of a variable
increases for
. Indeed, for a fixed
,
and is greater than zero. Finally,
so
h, as well as
V are positive for
and
. □
Proof of Theorem 4. From Lemma 14, we know that if
and
, then (
36) holds. Assume now that
and
. By Lemmas 7 and 15, and Formula (
37),
This expression is the same as in the second line in (
38), and it takes the maximum value
for
. Observe that the function
increases. Hence,
, so
is not less than
. For this reason, the maximum value of
is equal to
, but this value is obtained if
.
It is easy to check that both values of
for
and
, which are given in (
33), are less than or equal to
. This completes the proof. □
Theorem 6. Let , , and be a real number, . Then: Proof. By Theorems 4 and 5,
and
. Putting
, we can write:
Applying the triangle inequality, we obtain our claim. □
The results presented in Theorems 2–6 can be collected as follows.
Corollary 1. Let be given by (
1)
, be a real number, and . Then: The results are sharp for and . The equality holds for the Koebe function (
2)
in the first and the last case. The function F given by (
15)
is an extremal function when . The assumption is not necessary for and . Observe that for
, we have
, so the sharp bound for
is less than the sharp bound for
given by (
5).
6. Concluding Remarks
In this paper, we estimated two functionals and for the family , where is a real number. This family is a subset of the class of all close-to-convex functions.
The results presented above broaden our knowledge about the behavior of the coefficient functionals defined for functions not only in , but also generally in the class of univalent functions. Unfortunately, there are no good estimates of the discussed functionals in the whole classes and . It seems that further research on the classes of the type , where f is different from k, may result in obtaining some conclusions about .
In our opinion, the most important problem to be solved now is the estimating of the second Hankel determinant, or in other words
for
. Even in the class
, the exact bound is unknown. It is only known that for
, there is
(see [
25]). On the other hand, the conjecture posed by Thomas [
26] about 30 years ago that
for
and
was disproven. This means that there are functions in
for which
. Finding (even non-sharp) estimates of
for
remains an interesting open problem.