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Extensions of some generalized conditions for starlikeness and convexity
Journal of Inequalities and Applications volume 2014, Article number: 431 (2014)
Abstract
In the present paper we obtain generalized sufficient conditions for the starlikeness and convexity of some normalized analytic functions; the results extend those recently obtained in the work of Uyanik and Owa (J. Inequal. Appl. 2011:87, 2011).
MSC:30C45.
1 Introduction and preliminaries
Let be the class of holomorphic functions in the unit disk , and define
Also, let
be the class of n-starlike functions in U, and
be the class of n-convex functions in U. The notations and are well known, where and represent, respectively, the class of normalized starlike and normalized convex functions in the unit disc U.
In the recent years, many authors have obtained some sufficient conditions for starlikeness and convexity, that can be found, for example, in [1–5]etc., together with many references. In the present paper, by using some simple integral inequalities, we generalize the results of [4] and we give some interesting special cases of our results. The methods used here are frequently used in complex analysis and operator theory on spaces of analytic functions; among others, for example, in obtaining estimations for point evaluation operators on spaces of analytic functions of one or several variables containing derivatives in the definition of the spaces (see, for example, [6] and [7]).
To obtain the generalized conditions for n-starlikeness and n-convexity, presented in the following sections, we shall require the following lemma.
If and
then .
2 Generalized conditions for n-starlikeness
Using the previous lemma, we will prove the next sufficient condition for n-starlikeness.
Theorem 2.1 Let and suppose that
for some integer , with
Then .
Proof Since , we have
-
(i)
First, we will prove that (2.1) implies for . Thus, if
then we have
and according to Lemma 1.1 we deduce that .
-
(ii)
Suppose that there exists an integer such that inequality (2.1) holds. Using relation (2.3), we may easily see that
Consequently, if inequality (2.1) holds for some integer , then
Using the fact that we have already proved that if (2.1) holds for , then , from the above remark it follows that our result holds for any integer .
-
(iii)
Now we will prove that the result holds for . Thus, a simple computation shows that
that is,
and using the result of the part (ii) of our proof for , it follows that .
-
(iv)
Finally, supposing that there exists an integer such that (2.1) holds, it follows that
Consequently, if inequality (2.1) holds for some integer , then
Since we have already proved that if (2.1) holds for then , from the above remark the second part of our result follows, and the theorem is completely proved. □
Remark 2.1 If we put in the above result, we get [[4], Theorem 2.1].
3 Generalized conditions for n-convexity
In order to prove our main result, first we give the following two lemmas.
Lemma 3.1 Let and suppose that
for some integer . Then .
Proof (i) First, we will show that this implication holds for . Letting , then if and only if . If
a simple computation shows that
and by Lemma 1.1 we deduce that , i.e., .
-
(ii)
Suppose that there exists an integer such that inequality (3.1) holds. Using relation (2.3), we may easily see that
Thus, we have obtained that if inequality (3.1) holds for some integer , then
Since we have already proved that if (3.1) holds for then , our result follows from the above remark. □
Lemma 3.2 Let and suppose that
for some integer , with
where is given by (2.2). Then .
Proof Suppose that there exists an integer such that inequality (3.2) holds. If we let , then if and only if . Since
we obtain that
Thus, we have obtained that if inequality (3.2) holds for some integer , then
and from Theorem 2.1 we deduce that , hence , which completes the proof of the lemma. □
Combining Lemma 3.1 and Lemma 3.2, we obtain the next conclusion for the n-convexity sufficient conditions.
Theorem 3.1 Let and suppose that
for some integer , with
where is given by (3.3). Then .
Remark 3.1 If we put in the above results, we obtain [[4], Theorem 3.1].
Using a similar method as in the proof of Theorem 2.1 and Lemma 3.1, we can easily derive the following result.
Theorem 3.2 Let and suppose that
for some integer , where is defined by (2.2), i.e.,
Then .
Proof (i) For , the result follows from Lemma 3.1.
-
(ii)
We will prove that the result holds for . A simple calculus shows that
that is,
and using the result of the part (i) of our proof for , it follows that .
-
(iii)
Supposing that there exists an integer such that (3.4) holds, it follows that
Thus, if inequality (3.4) holds for some integer , then
Since we have already proved to the part (ii) that if (3.4) holds for then , our theorem follows from the above remark. □
Remark 3.2 If we put in the above results, then we get [[4], Theorem 3.2].
It remains to determine which of the above last two theorems gives weaker sufficient conditions for n-convexity, i.e.,
implies .
-
(i)
First, let us consider the case .
A simple computation shows that if and only if
which is equivalent to , where
Moreover, it is easy to check that
and
Reversely, if and only if
that holds whenever , where
Moreover, we have
-
(ii)
Secondly, let us consider the case .
A simple calculus shows that
if and only if
which is equivalent to
Reversely,
if and only if
that holds whenever
In order to have
we obtain that , but this case is contained in the first part of (ii).
Now, from Theorem 3.1 and Theorem 3.2, we deduce the next conclusion.
Theorem 3.3 Let and suppose that
for some integer , with
where , A, B and C are given by (2.2), (3.5) and (3.6), respectively. Then .
4 Special cases
In this section we give some special cases of our main results.
-
1.
Taking the polynomial function
(4.1)
where , then , . It is easy to see that
and from Theorem 2.1 we deduce the following result.
Corollary 4.1 Let be the polynomial function of the form (4.1) and suppose that for some integer , one of the following inequalities holds:
where
Then .
Taking and in Corollary 4.1, we deduce that if one of the following inequalities holds:
then . It is easy to check that condition (4.6) is stronger than (4.7), hence we get the following.
Example 4.1 If is the polynomial function of the form (4.1) and
then .
Remark 4.1 For and , the above example reduces to the [[4], Example 2.1], that is, if , then .
According to Theorem 3.3 and using the differentiation formulas (4.2) and (4.3), we obtain the next result.
Corollary 4.2 Let be the polynomial function of the form (4.1) and suppose that for some integer , one of the following inequalities holds:
where are given by (4.4) and (4.5), and
Then .
Taking in Corollary 4.2, we deduce the following special case.
Example 4.2 If is the polynomial function of the form (4.1) and
then .
Remark 4.2 For and , the above example reduces to the following one: if , then .
-
2.
For the function
(4.8)
where , we have , and we may easily check that
If we take in Theorem 2.1 and in Theorem 3.3, we deduce, respectively, the following.
Example 4.3 If is the function defined by (4.8) and
then .
Example 4.4 If is the function of the form (4.8) and
then .
-
3.
Considering the function
(4.9)
where , then , and
Taking in Theorem 2.1 and in Theorem 3.3, we obtain, respectively, the following.
Example 4.5 If is the function defined by (4.9) and
then .
Example 4.6 If is the function of the form (4.9) and
then .
-
4.
Taking the function
(4.10)
where , then , and
If we choose in Theorem 2.1 and in Theorem 3.3, we obtain, respectively, the following.
Example 4.7 If is the function defined by (4.10) and
then .
Example 4.8 If is the function of the form (4.10) and
then .
We emphasize that all the above n-starlikeness and n-convexity criteria are very useful since the direct proofs are too difficult in each of these examples.
References
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Acknowledgements
Authors are thankful to reviewers for careful reading of the paper. The third author BS Alkahtani is grateful to King Saud University, Deanship of Scientific Research, College of Science Research Center to support this project.
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Goswami, P., Bulboacă, T. & Alkahtani, B.S. Extensions of some generalized conditions for starlikeness and convexity. J Inequal Appl 2014, 431 (2014). https://doi.org/10.1186/1029-242X-2014-431
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DOI: https://doi.org/10.1186/1029-242X-2014-431