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Sufficient conditions for starlike functions associated with the lemniscate of Bernoulli
Journal of Inequalities and Applications volume 2013, Article number: 176 (2013)
Abstract
Let . The condition on β is determined so that () implies . Similarly, the condition on β is determined so that or () implies or . In addition to that, the condition on β is derived so that when . A few more problems of the similar flavor are also considered.
MSC:30C80, 30C45.
1 Introduction
Let be the class of analytic functions defined on the unit disk normalized by the condition . For two analytic functions f and g, we say that f is subordinate to g or g is superordinate to f, denoted by , if there is a Schwarz function w with such that . If g is univalent, then if and only if and . For an analytic function φ whose range is starlike with respect to and is symmetric with respect to the real axis, let denote the class of Ma-Minda starlike functions consisting of all satisfying . For special choices of φ, reduces to well-known subclasses of starlike functions. For example, when , is the class of Janowski starlike functions [1] (see [2]) and is the class of starlike functions of order α and is the class of starlike functions. For , the class reduces to the class introduced by Sokół and Stankiewicz [3] and studied recently by Ali et al. [4, 5]. A function is in the class if lies in the region bounded by the right half-plane of the lemniscate of Bernoulli given by . Analytically, . For and , a more general class of the functions f satisfying was considered by Paprocki and Sokół [6]. Clearly, . For some radius problems related with the lemniscate of Bernoulli, see [3, 5, 7, 8]. Estimates for the initial coefficients of functions in the class are available in [8].
Let p be an analytic function defined on with . Recently Ali et al. [4] determined conditions for when with or is subordinated to . Motivated by the works in [4–8], in Section 2 the condition on β is determined so that when (). Similarly, the condition on β is determined so that when , . Further, the condition on β is obtained in each case so that when , . At the end of this section, the problem implies is also considered.
Silverman [9] introduced the class by
and proved , . Further, this result was improved by Obradovič and Tuneski [10] by showing , . Tuneski [11] further obtained the condition for . Inspired by the work of Silverman [9], Nunokawa et al. [12] obtained the sufficient conditions for a function in the class to be strongly starlike, strongly convex, or starlike in . By setting , the inclusion can be written as
Recently Ali et al. [13], obtained the condition on the constants and β so that when , . In Section 3, alternate and easy proofs of results [[13], Lemmas 2.1, 2.10] are discussed. Further, this section is concluded with the condition on and β such that implies .
The following results are required in order to prove our main results.
Lemma 1.1 [[14], Corollary 3.4h, p.135]
Let q be univalent in , and let φ be analytic in a domain D containing . Let be starlike. If p is analytic in , and satisfies
then and q is the best dominant.
The following is a more general form of the above lemma.
Lemma 1.2 [[14], Corollary 3.4i, p.134]
Let q be univalent in , and let φ and ν be analytic in a domain D containing with when . Set
Suppose that
-
(1)
h is convex or is starlike univalent in and
-
(2)
for .
If
then and q is the best dominant.
Lemma 1.3 [[14], Corollary 3.4a, p.120]
Let q be analytic in , let ϕ be analytic in a domain D containing and suppose
-
(1)
and either
-
(2)
q is convex, or
-
(3)
is starlike.
If p is analytic in , with , and
then .
2 Results associated with the lemniscate of Bernoulli
In the first result, condition on β is obtained so that the subordination
implies .
Lemma 2.1 Let , . Let p be an analytic function defined on with satisfying
then .
Proof Let . A computation shows that the function
is starlike in the unit disk . Consider the subordination
Thus in view of Lemma 1.1, it follows that . In order to prove our result, we need to prove
Let . Then . The subordination is equivalent to . Thus in order to prove the result, we need only to show . For , , we have
A calculation shows that attains its minimum at . Further, the value of at π or −π comes out to be which is naturally greater than the value at the extreme point because if , then which is absurd. Thus
for . Hence , and the proof is complete now. □
Next result depicts the condition on β such that implies (). On subsequent lemmas, similar results are obtained by considering the expressions and .
Lemma 2.2 Let and . Let p be an analytic function defined on with satisfying
then .
Proof Define the function by
with . A computation shows that
and
Let , , . Then
Since (, ) and so , this shows that Q is starlike in . It follows from Lemma 1.1 that the subordination
implies . Now we need to prove the following in order to prove the lemma:
Let . Then . The subordination is equivalent to the subordination . Now in order to prove the result, it is enough to show , , . Now
Further,
for . Therefore and this completes the proof. □
Lemma 2.3 Let and . Let p be an analytic function defined on with satisfying
then .
Proof Let the function be defined by
A computation shows that
and
Let , , . Then
Since for and, similarly, for , it follows that Q is starlike in . Lemma 1.1 suggests that the subordination
implies . Now we have to prove
Let . Then . The subordination is equivalent to the subordination . Now in order to prove the result, it is enough to show , . Now
Further,
for . Therefore and this completes the proof. □
Lemma 2.4 Let and . Let p be an analytic function defined on with satisfying
then .
Proof Let the function be defined by
with . Then
and
Let , , . Then
Since (, ). Hence , this shows that Q is starlike in . An application of Lemma 1.1 reveals that the subordination
implies . Now our result is established if we prove
The rest of the proof is similar to that of Lemma 2.2, and therefore it is skipped here. □
In the next result, the condition on β is obtained so that implies . On subsequent lemmas, similar results are discussed by considering the expressions and .
Lemma 2.5 Let p be an analytic function defined on with satisfying , . Then .
Proof Define the function by with . Since is the right half of the lemniscate of Bernoulli, is a convex set, and hence q is a convex function. Let us define , then
Consider the function Q defined by
Further,
Thus the function Q is starlike, and the result now follows by an application of Lemma 1.3. □
Lemma 2.6 Let p be an analytic function defined on with satisfying
Then .
Proof As before, let q be given by with . Then q is a convex function. Let us define . Since is the right half of the lemniscate of Bernoulli, so
Consider the function Q defined by
Further,
Thus the function Q is starlike, and the result now follows by an application of Lemma 1.3. □
Lemma 2.7 Let p be an analytic function defined on with satisfying
Then .
Proof Let q be given by with . Then q is a convex function. Let us define and
Consider the function Q defined by
Further,
Thus the function Q is starlike, and the result now follows by an application of Lemma 1.3. □
In the next result, the condition on β is obtained such that implies that .
Lemma 2.8 Let , and
Let p be an analytic function defined on with satisfying
Then .
Proof Define the function by , . Consider the subordination
Thus, in view of Lemma 1.2, the above subordination can be written as (1.1) by defining the functions ν and φ as and (). Clearly, the functions ν and φ are analytic in ℂ and . Let the functions and be defined by
and
A computation shows that is starlike univalent in . Further,
Let , . Then
Thus by Lemma 1.2, it follows that . In order to prove our result, we need to prove that
The subordination is equivalent to the subordination . Now in order to prove the result, it is enough to show , . Now
implies
Further,
for . This completes the proof. □
3 Sufficient condition for Janowski starlikeness
The following first two results (Lemmas 3.1, 3.2) are essentially due to Ali et al. [[13], Lemmas 2.1, 2.10]. However, an alternate proof of the same result, which is much easier than that given by Ali et al. [13], is presented below.
Lemma 3.1 Assume that , and . Let p be an analytic function defined on with satisfying
Then .
Proof Define the function by
Then q is convex in with . Further computation shows that
and Q is starlike in . It follows from Lemma 1.1 that the subordination
implies . In view of the above result, it is sufficient to prove
Let . Then and
Let , . Thus
for . Hence , that is, , this completes the proof. □
It should be noted that Ali et al. [13] made the assumption in order to prove the result [[13], Lemma 2.10], whereas in the following lemma this condition has been dropped.
Lemma 3.2 Assume that , and . Let p be an analytic function defined on with satisfying
Then .
Proof As above, define the function by
Then q is convex in with . A computation shows that
and Q is starlike in . It follows from Lemma 1.1 that the subordination
implies . Now we need to prove
Let . Then and
Let , . Thus
for . Hence , that is, , this completes the proof. □
Lemma 3.3 Assume that , and . Let p be an analytic function defined on with satisfying
Then .
Proof Define the function by
Then q is convex in with . A computation shows that
and
As before, a computation shows Q is starlike in . It follows from Lemma 1.1 that the subordination
implies . To prove result, it is enough to show that
The remaining part of the proof is similar to that of Lemma 3.1, and therefore it is skipped here. □
Remark 3.4 When , Lemma 3.3 reduces to [[13], Lemma 2.6] due to Ali et al.
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
The research work is supported by a grant from University of Delhi and also by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012-0002619).
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Kumar, S.S., Kumar, V., Ravichandran, V. et al. Sufficient conditions for starlike functions associated with the lemniscate of Bernoulli. J Inequal Appl 2013, 176 (2013). https://doi.org/10.1186/1029-242X-2013-176
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DOI: https://doi.org/10.1186/1029-242X-2013-176