1. Introduction
Let
denote the class of analytic functions in the open unit disk
and
denote the subclass of
consisting of functions of the form
Also, let
be the subclass of
consisting of all univalent functions in
Then the logarithmic coefficients
of
are defined with the following series expansion:
These coefficients play an important role for various estimates in the theory of univalent functions. Note that we use
instead of
. The idea of studying the logarithmic coefficients helped Kayumov [
1] to solve Brennan’s conjecture for conformal mappings.
Recall that we can rewrite (
2) in the series form as follows:
Now, considering the coefficients of
for
, it follows that
For two functions
f and
g that are analytic in
we say that the function
f is subordinate to
g in
and write
if there exists a Schwarz function
that is analytic in
with
and
such that
In particular, if the function g is univalent in then if and only if and .
Using subordination, different subclasses of starlike and convex functions were introduced by Ma and Minda [
2], in which either of the quantity
or
is subordinate to a more general superordinate function. To this aim, they considered an analytic univalent function
with positive real part in
.
is symmetric respecting the real axis and starlike considering
and
. They defined the classes consisting of several well-known classes as follows:
and
For example, the classes and reduce to the classes and of the well-known Janowski starlike and Janowski convex functions for , respectively. By replacing and where , we conclude the classes and of the starlike functions of order and convex functions of order , respectively. In particular, and are the class of starlike functions and of convex functions in the unit disk , respectively. The Koebe function is starlike but not convex in . Thus, every convex function is starlike but not conversely; however, each starlike function is convex in the disk of radius .
Lately, several researchers have subsequently investigated similar problems in the direction of the logarithmic coefficients, the coefficient problems, and differential subordination [
3,
4,
5,
6,
7,
8,
9,
10,
11], to mention a few. For example, the rotation of Koebe function
for each
has logarithmic coefficients
If
then by using the Bieberbach inequality for the first equation of (
3) it concludes
and by utilizing the Fekete–Szegö inequality for the second equation of (
3), (see [
12] (Theorem 3.8)),
It was shown in [
12] (Theorem 4) that the logarithmic coefficients
of every function
satisfy
and the equality is attained for the Koebe function. For
the inequality
holds but is not true for the full class
, even in order of magnitude (see [
12] (Theorem 8.4)). In 2018, Ali and Vasudevarao [
3] and Pranav Kumar and Vasudevarao [
6] obtained the logarithmic coefficients
for certain subclasses of close-to-convex functions. Nevertheless, the problem of the best upper bounds for the logarithmic coefficients of univalent functions for
is presumably still a concern.
Based on the results presented in previous research, in the current study, the bounds for the logarithmic coefficients of the general classes and were estimated. It is worthwhile mentioning that the given bounds in this paper would generalize some of the previous papers and that many new results are obtained, noting that our method is more general than those used by others. The following lemmas will be used in the proofs of our main results.
For this work, let represent the class of all analytic functions in that equips with conditions and for . Such functions are called Schwarz functions.
Lemma 1. [13] (p. 172) Assume that ω is a Schwarz function so that Then Lemma 2. [14] Let be any convex univalent functions in . If and , then where . We observe that in the above lemma, nothing is assumed about the normalization of and , and “∗” represents the Hadamard (or convolution) product.
Lemma 3. [12,15] (Theorem 6.3, p. 192; Rogosinski’s Theorem II (i)) Let and be analytic in , and suppose that where g is univalent in . Then Lemma 4. [12,15] (Theorem 6.4 (i), p. 195; Rogosinski’s Theorem X) Let and be analytic in , and suppose that where g is univalent in . Then - (i)
If g is convex, then .
- (ii)
If g is starlike (starlike with respect to 0), then .
Lemma 5. [16] If , then for any real numbers and , the following sharp estimate holds:where While the sets are defined as follows: 2. Main Results
Throughout this paper, we assume that
is an analytic univalent function in the unit disk
satisfying
such that it has series expansion of the form
Theorem 1. Let the function . Then the logarithmic coefficients of f satisfy the inequalities:
- (i)
- (ii)
If φ is starlike with respect to 1, then
All inequalities in (
5)
, (
7)
, (
8)
and are sharp such that for any , there is the function given by and the function f given by , respectively. Proof. Suppose that
. Then considering the definition of
, it follows that
which according to the logarithmic coefficients
of
f given by (
1), concludes
Now, for the proof of inequality (
5), we assume that
is convex in
. This implies that
is convex with
, and so by Lemma 4(i) we get
and concluding the result.
Next, for the proof of inequality (
6), we define
, which is an analytic function, and it satisfies the relation
as
is convex in
with
.
On the other hand, it is well known that the function (see [
17])
belongs to the class
, and for
,
Now, by Lemma 2 and from (
9), we obtain
Considering (
10), the above relation becomes
In addition, it has been proved in [
18] that the class of convex univalent functions is closed under convolution. Therefore, the function
is convex univalent. In addition, the above relation considering the logarithmic coefficients
of
f given by (
1) is equivalent to
Applying Lemma 3, from the above subordination this gives
which yields the inequality in (
6). Supposing that
, we deduce that
and it concludes the inequality (
7).
Finally, we suppose that
is starlike with respect to 1 in
, which implies
is starlike, and thus by Lemma 4(ii), we obtain
This implies the inequality in (
8).
For the sharp bounds, it suffices to use the equality
and so these results are sharp in inequalities (
5), (
6), and (
8) such that for any
, there is the function
given by
and the function
f given by
, respectively. This completes the proof. □
In the following corollaries, we obtain the logarithmic coefficients
for two subclasses
and
, which were defined by Khatter et al. in [
19], and
and
are the convex univalent functions in
. For
, these results reduce to the logarithmic coefficients
for the subclasses
and
(see [
20,
21]).
Corollary 1. For , let the function . Then the logarithmic coefficients of f satisfy the inequalitiesand These results are sharp such that for any , there is the function given by and the function f given by .
Corollary 2. For , let the function . Then the logarithmic coefficients of f satisfy the inequalitiesand These results are sharp such that for any , there is the function given by and the function f given by .
The following corollary concludes the logarithmic coefficients
for a subclass
defined by Cho et al. in [
22], in which considering the proof of Theorem 1 and Corollary 1, the convexity radius for
is given by
.
Corollary 3. Let the function where is a convex univalent function for in . Then the logarithmic coefficients of f satisfy the inequalitiesand These results are sharp such that for any , there is the function given by and the function f given by .
In the following result, we get the logarithmic coefficients
for a subclass
defined by Kanas and Wisniowska in [
23] (see also [
24,
25]), in which
where
is a convex univalent function in
and
and
is the complete elliptic integral of the first kind.
Corollary 4. For , let the function . Then the logarithmic coefficients of f satisfy the inequalities This result is sharp such that for any , there is the function given by .
The following result concludes the logarithmic coefficients
for a subclass
defined by Mendiratta et al. in [
26], in which
where
is a convex univalent function in
.
Corollary 5. Let the function . Then the logarithmic coefficients of f satisfy the inequalities This result is sharp such that for any , there is the function given by .
The following results conclude the logarithmic coefficients
for two subclasses
and
defined by Krishna Raina and Sokół in [
27] and Kargar et al. in [
28], where
and
respectively. These functions are univalent and starlike with respect to 1 in
.
Corollary 6. Let the function . Then the logarithmic coefficients of f satisfy the inequalities This result is sharp such that for any , there is the function given by .
Corollary 7. Let the function , where . Then the logarithmic coefficients of f satisfy the inequalities This result is sharp such that for any , there is the function given by .
Remark 1. 1. Lettingwhich is convex univalent in in Theorem 1, then we get the results obtained by Ponnusamy et al. [7] (Theorem 2.1 and Corollary 2.3). 2. For , where and in the above expression, then we get the results obtained by Ponnusamy et al. [7] (Theorem 2.5). 3. Takingwhich is convex univalent in , and in Theorem 1, then we get the results obtained by Ponnusamy et al. [7] (Theorem 2.6). 4. Settingwhich is convex univalent in , and in Theorem 1, then we get the results obtained by Kargar [5] (Theorems 2.2 and 2.3). 5. Lettingwhich is convex univalent in , and in Theorem 1, then we get the results obtained by Kargar [5] (Theorems 2.5 and 2.6). 6. Lettingwhich is convex univalent in , andin Theorem 1, then we get the results obtained for by Kargar et al. [29] (Theorem 3.1). Moreover, for , we get the result presented by Thomas in [30] (Theorem 1). 7. Let the function , where . It is equivalent to Then we have (see e.g., [31] (Theorem 1))where is a convex univalent function in , and Thus, applying Theorem 1, we get the results obtained by Obradović et al. [4] (Theorem 2 and Corollary 2). Theorem 2. Let the function . Then the logarithmic coefficients of f satisfy the inequalitiesand if , and are real values,where is given by Lemma 5, , and . The bounds (
11)
and (
12)
are sharp. Proof. Let
. Then by the definition of the subordination, there is a
with
so that
From the above equation, we get that
By substituting values
(n = 1, 2, 3) from (
14) in (
3), we have
Firstly, for
, by applying Lemma 1 we get
, and this bound is sharp for
. Next, applying Lemma 1 for
, we have
These bounds are sharp for and , respectively.
Finally, using Lemma 5 for
, we obtain
where
and
. Therefore, this completes the proof. □
Remark 2. 1. Lettingin Theorem 2, (for with respect to ) then we get the results obtained by Ponnusamy et al. [7] (Theorem 2.7 and Corollary 2.8). 2. Takingin Theorem 2, (for respect to ) then we get the results obtained by Ponnusamy et al. [7] (Theorem 2.10).