1. Introduction
Let
be the collection of functions of the form
which are analytic in the open unit disk
and let
denote the subclass of
consisting of functions which are univalent in
.
Since the early twentieth century many mathematicians have been interested in different problems involving the coefficients of functions
f in a given subclass of
. The most important and inspiring problem known as the Bieberbach conjecture was solved by de Branges only 70 years after its formulation. Over the years, many interesting tasks connected with these coefficients appeared. The most important ones were settled by Robertson, Bombieri, Zalcman, Krzyż and Landau. In the 1960s Pommerenke defined the Hankel determinant
, for a given
f of the form (
1), as follows
where
.
The studies on Hankel determinants are concentrated on estimating
and
for different subclasses of
. These particular determinants can be written as
and
Although we know many sharp bounds of and significantly less sharp bounds of for some proper subfamilies of , the sharp results for the whole class are not known. Moreover, we are even unable to formulate a reasonable conjecture about it.
Among numerous results for the subclasses of
, we cite only the most important one. In [
1], Janteng et al. proved that
for
and
for
, where
and
are very well known classes of starlike and convex functions. The sharp results for
are difficult to obtain. It is worth citing the sharp bound
for
and the non-sharp estimate
for
obtained by Kowalczyk et al. and Kwon et al., respectively (see, [
2,
3]).
The definitions of
and
can be written in terms of subordination. Namely
and
Let us recall that for two functions f and g analytic in , we say that g is subordinated to f (), if there exists a function analytic in with , and such that . The relation results in .
Let
be an analytic function such that
for
with
. Ma and Minda [
4] defined the classes of
and
in the following way
and
From (
5) it is seen that
reduces to
if
. For other specific choices of
we obtain for example:
if , then is the class of starlike functions of order ,
if , then is the class of strongly starlike functions of order ,
if , then is the class of Janowski starlike functions.
In a similar way we defined the relative subclasses of .
Recently, Mendiratta et al. [
5] discussed
and
, i.e., the classes
and
with
. Various problems, including distortion and growth theorems, radii problems, inclusion relations and coefficient estimates, were discussed there.
In their two following papers Zhang et al. [
6] and Shi et al. [
7] broadened the range of discussed coefficient problems. They found the coefficient estimates for
,
and the bounds of the following functionals:
,
,
and, as a consequence,
. Except for the bounds of
,
and
, all results are non-sharp.
In this paper, we improve all non-sharp results mentioned above. The main idea is to express the discussed functionals depending on the second coefficient of
or
. In fact, the second coefficient of
or
can be replaced by the coefficient
of a corresponding function
P with a positive real part. This idea leads to better estimates than those in [
6,
7]. Moreover, the new bounds of
and
are sharp.
2. Auxiliary Lemmas
To prove our results, we need two lemmas concerning functions in the class
of functions
P such that
and
P has the Taylor series representation
Lemma 1 ([
8])
. Let . A function P of the form (9) is in if and only if- 1.
,
- 2.
,
for some x and y such that , .
Lemma 2 ([
4,
9])
. If is of the form (9) and , then the following sharp estimates hold- 1.
for ,
- 2.
In fact, the second inequality of Lemma 2 did not appear in [
4], but it is a reformulation of the result obtained for bounded functions.
At the end of this section, observe that both classes and possess a specific type of symmetry. They are invariant (or symmetric) under rotations. Recall that the class A is invariant under rotations when f is in A if and only if , is also in A. A functional defined for functions is called invariant under rotations in A if and for all . It can be easily checked that and , as well as the functionals , , and considered in or in , satisfy the above definitions. Due to the symmetry described above, in the considerations we can assume that one coefficient (usually the second one) is a positive real number.
3. Coefficient Problems for
It follows from the definition of
that there exists a function
with
,
such that
Define
. The function
P is in
and (
10) is equivalent to
Now, expanding both sides of (
11) in the Taylor series and comparing coefficients at
,
, we obtain (see also Formulae (15)–(18) in [
7]),
Now, we can prove the following theorem.
Theorem 1. If is given by (1) and , , then - 1.
- 2.
- 3.
.
Proof. The bound of
is clear. To obtain the bound of
and
it is enough to write
and to apply Lemma 2. □
Corollary 1. If is given by (1), then - 1.
- 2.
- 3.
- 4.
.
The first three bounds are sharp.
The function which gives equality in the bounds of
,
and
corresponds to
, so
. This means that the extremal function is of the form
so
In [
5], Mendiratta et al. proved that if
, then
. Although this inequality is sharp, we can easily generalize it by applying Lemma 2 in the following identity
.
Theorem 2. If is given by (1) and , , then . The result is sharp. For sharpness, it is enough to discuss a function
f which corresponds to
Now, we shall improve the estimate
found by Zhang et al. (see, Theorem 2 in [
6]).
Theorem 3. If is given by (1) and , , then Consequently, we get the following corollary.
Corollary 2. If is given by (1), then The result is sharp.
Proof of Theorem 3. From (
12) and Lemma 1 we obtain
with
x,
y such that
,
.
In a view of the invariance of
under rotations, assume that
and
. Then,
with
The function achieves the greatest value in when if and when if . □
Proof of Corollary 2. Define
,
and
,
. It is easy to observe that
and
which proves (
17).
Observe that the equality in (
17) holds if
and
in (
18). This means that
is of the form (
15) with
. In this case,
□
The next theorem improves the bound of
also found by Zhang et al. (see, Theorem 3 in [
6]).
Theorem 4. If is given by (1) and , , then The bound is sharp.
Consequently, we get the following corollary.
Corollary 3. If is given by (1), thenThe bound is sharp. Proof of Theorem 4. By Lemma 1,
where
,
.
We can assume that
is a non-negative real number. Applying the triangle inequality and writing
p instead of
,
, we have
where
However,
h is an increasing function of
, so
, which results in (
20).
Observe that the equality in (
20) holds only when
. This means that
P is of the form (
15). Hence,
so
□
For
we have
, so from (
11),
Hence, the corresponding function in
is of the form
Finally, we find a new bound of
for the class
. In [
6] it was proved that
… In the succeeding paper Shi et al. showed that
(see, Theorem 1 in [
7]). We improve these results essentially in the following way.
Theorem 5. If is given by (1), then Proof. Applying Theorems 1–4 and the triangle inequality in (
4) we obtain
where
and
The function is decreasing for . Moreover, attains its greatest value in , which is equal to , at . This results in the declared bound. □
4. Coefficient Problems for
Directly from the definitions of
and
it follows that for
,
Consequently, if
,
and
is given by (
1), then
. For this reason, Theorem 1 results in the two following facts.
Theorem 6. If is given by (27) and , , then - 1.
- 2.
- 3.
.
Corollary 4. If is given by (27), then - 1.
- 2.
- 3.
- 4.
.
Equalities in the bounds of the first three coefficients hold for
such that
, where
is defined by (
13). Hence,
Proceeding in the same manner as in
Section 3, we obtain results for
,
and
.
Theorem 7. If is given by (27) and , , then . The result is sharp. The proof of this theorem is obvious.
Theorem 8. If is given by (27) and , , then Consequently, we get the following corollary.
Corollary 5. If is given by (27), then The result is sharp.
Proof of Theorem 8. From (
26), (
12) and Lemma 1 we obtain
with
x,
y such that
,
.
Assume that
and
. Then,
with
The maximal value of in is achieved if for and if for . □
Proof of Corollary 5. Define
,
and
,
. It is easy to observe that
and
which leads to (
32).
The equality in (
32) holds if we put
,
and
into (
31). This means that
. Consequently, the extremal function
is such that
, where
is of the form
In the next theorem we improve the bound of
obtained by Shi et al. (see, Theorem 4 in [
7]).
Theorem 9. If is given by (27) and , , then The bound is sharp.
Consequently, we get the following corollary.
Corollary 6. If is given by (27), then The bound is sharp.
Proof of Theorem 9. From (
12) and (
26) we obtain
Applying Lemma 1, we get
where
,
.
Without loss of generality, we assume that
. Hence,
where
Clearly,
, which results in (
33).
The equality in (
33) holds only when
. In this case
P is given by (
15) and
so
□
By combining the results presented above we can derive a bound of when .
Theorem 10. If is given by (27), then Proof. From (
4) and the triangle inequality it follows that
where
and
The function is decreasing for . Moreover, attains its greatest value in , which is equal to , at . This results in the declared bound. □