1. Introduction
Let
be the class of functions
of the form
that are analytic
p-valent functions in the closed unit disk
. For functions
we consider
If
for
is considered by Bernardi [
1]. Therefore,
in (
3) is said to be the Bernardi integral operator. Further, if
and
for
is defined by Libera [
2]. Therefore,
in (
4) is called the Libera integral operator.
For
in (
2), we consider
and
with
and
From the various definitions of fractional calculus of
(that is, fractional integrals and fractional derivatives) given in the literature, we would like to recall here the following definitions for fractional calculus which were used by Owa [
3] and Owa and Srivastava [
4].
Definition 1. The fractional integral of order λ for is defined bywhere the multiplicity of is removed by requiring to be real when and Γ
is the Gamma function. With the above definitions, we know that
for
and
Using the fractional integral operator over
we consider
where
If
in (
9), then
and if
in (
9), then we see that
With the operator
given by (
9), we know
where
and
The operator
is a generalization of the Bernardi integral operator
. From the definition of
we know that
From
s different boundary points
with
we consider
where
and
is the open unit disk. For such
if
satisfies
for some real
we say that the function
belongs to the class
It is clear that a function
belongs to the class
provided that the condition
is satisfied. If we consider the function
given by
then
satisfies
Therefore,
given by (
16) is in the class
Discussing our problems for
we have to recall here the following lemma due to Miller and Mocanu [
5,
6] (refining the old one in Jack [
7].)
Lemma 1. Let the function given bybe analytic in with If attains its maximum value on the circle at a point then there exists a real number such thatand 2. Properties of Functions Concerning with the Class Tp,n (αs, β, ρ; m, λ)
We begin with a sufficient condition on a function which makes it a member of
Theorem 1. If satisfiesfor some given by (13) with such that and for some real thenthat is, Proof. We introduce the function
defined by
Then,
is analytic in
with
and
Noting that
we obtain that
and that
by employing (
21). Assume, to arrive at a contradiction, that there exists a point
such that
Then, we can write that
and
by Lemma 1. For such a point
satisfies
Since this contradicts our condition (
21), we see that there is no
such that
This shows us that
that is, that
This completes the proof of the theorem. □
Example 1. We consider a function given bywith , whereFor such , we havethat is, thatNow, we consider five boundary points such thatandFor these five boundary points, we know thatandThus is given byThis gives us thatwith For such and we take withIt follows from the above thatFor such and , satisfies Our next result reads as follows.
Theorem 2. If satisfiesfor some defined by (13) with such that and for some real thenthat is, Proof. Define a function
by (
23). Using (
24) and (
26), we have
We suppose that there exists a point
such that
Then, Lemma 1 leads us that
and
It follows from the above that
This contradicts our condition (
51) for
. Therefore, there is no
such that
This means that
□
Example 2. Consider a function given by (32) with , where Q is given by (33). For this function we haveConsider five boundary points and in Example 1. Then, we havewith With such and we take byThen, this ρ satisfiesFor such and ρ, we know that Next, we derive the following result.
Theorem 3. If satisfiesfor some defined by (13) with and for some real then Proof. Define the function
by
It follows from the above that
By the definition of
we know that
Our condition implies that
for all
Suppose that there exists a point
such that
Then, Lemma 1 says that
and
. Therefore, we have
which contradicts the inequality (
67). This means that there is no
such that
Thus we know that
This completes the proof of the theorem. □
Theorem 3 implies the following one.
Theorem 4. If satisfiesfor some given by (13) with and for some real thenor, equivalently, Proof. By means of Theorem 3, we see that if
satisfies the inequality (
71), then
Similarly, we have
Continuing this consideration, we obtain that
□
Example 3. Consider the functionwhich satisfiesIt follows from (77) thatwhere Q is given by (33). Now, we consider the five boundary points and as in Example 1. Then we seewhere With the above relation (79), we consider such thatthat is, ρ satisfiesThus, we have that Remark 1. If we take in the results of this section, then these results correspond to applications of the Libera integral operator as introduced by Libera [2]. Let us write thatfor in (11). Then Theorem 1 says that if satisfiesfor some given bywhere and for some real then For another result, we consider again the Libera integral operator with
3. Application of Carathéodory Lemma
In this section, we will apply Carathéodory Lemma for coefficients of functions
In 1907, Carathéodory [
8] gave the following result.
Lemma 2. Let a function given bybe analytic in and Then satisfiesThe inequality (88) is sharp for each Applying the above lemma, we derive the following thorem.
Theorem 5. If is in the class thenwhereThe result is sharp for given by Proof. For
we see that
If we define a function
with
by
then
is analytic in
with
and
Also,
has the following power series expansion:
Therefore, by applying Lemma 2 to
we obtain
This shows the coefficient inequalities (
89). Note that
is analytic in
and
Therefore, considering
such that
we have
This completes the proof of the theorem. □
Remark 2. If we take in Theorem 5, then we get the following result for the Libera integral operator.
If satisfiesthenwhereThe result is sharp for given by Finally, we derive
Theorem 6. If satisfiesthen where R is given by (90). Proof. For
we consider
Therefore, if
satisfies (
102), then we know
□
Remark 3. Letting in Theorem 6, we have the result concerning with the Libera integral operator
Author Contributions
Conceptualization, S.O.; Investigation, S.O. and H.Ö.G.; Methodology, S.O.; Writing—original draft, S.O.; Writing—review and editing, H.Ö.G. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Acknowledgments
The authors would like to give our thanks to T. Bulboaca, Department of Mathematics, Faculty of Mathematics and Computer Sciences, Babes-Bolyai University, Romania for his kind support for our paper and also the reviewers for valuable remarks and suggestions in order to revise and improve of our paper.
Conflicts of Interest
The authors declare no conflict of interest.
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