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Some applications of differential subordinations in the geometric function theory
Journal of Inequalities and Applications volume 2013, Article number: 389 (2013)
Let denote the class of functions f, which are analytic in the open unit disc and normalized by . We investigate the connection of the quantity with . Obtained results are the extensions of those presented by Tuneski and Obradović (Comput. Math. Appl. 62:3438-3445, 2011). Moreover, we solve the open problems posed in the paper above.
Dedicated to Professor Hari M Srivastava
Let , , denote the class of functions of the form
which are analytic in the open unit disc on the complex plane ℂ. For two analytic functions f, g, we say that f is subordinate to g, written as , if and only if there exists an analytic function ω with property in such that . In particular, if g is univalent in , then we have the following equivalence
The idea of subordination was used for defining many of classes of functions studied in the geometric function theory. For obtaining the main result, we shall use the method of differential subordinations. The main result in the theory of differential subordinations was introduced by Miller and Mocanu in [1, 2]. A function p, analytic in , is said to satisfy a first order differential subordination if
where , is analytic in D, h is analytic and univalent in . The function q is said to be a dominant of the differential subordination (1.2) if for all p satisfying (1.2). If is a dominant of (1.2) and for all dominants q of (1.2), then we say that is the best dominant of the differential subordination (1.2).
By using a Miller-Mocanu lemma on the first order differential subordination, the authors of  proved the following theorem.
Theorem 1.1 
Let , for all and . If
and μz is the best dominant of (1.4). Furthermore,
and this conclusion is sharp, i.e., in inequality (1.5), μ cannot be replaced by a smaller number such that the implications holds.
2 Main results
It is easy to verify that under the assumptions of Theorem 1.1, the function is analytic in , and that the subordination (1.3) becomes
Thus, (1.3) can be rewritten as the Briot-Bouquet-type differential subordination
with , , , .
One of the basic results in the theory of Briot-Bouquet differential subordinations is a theorem from , which says (in its particular case) that if h is convex univalent with positive real part in and the functions s, h satisfy (2.1), then . It can sometimes be improved if we know more about the function h. A better subordination will be derived by applying the results from the book . After some adaptation, Theorem 3.2j [, p.97] becomes the following lemma.
Lemma 2.1 
Suppose that and n is a positive integer. Let us denote
and let h be a convex univalent function in with such that
Then the Briot-Bouquet differential equation
has a univalent solution analytic in . Furthermore, if the functions h and satisfy the Briot-Bouquet differential subordination
and the function is the best dominant of the subordination (2.4) in the sense that if there exists a function p such that , then .
Notice that the function is called the open door function, and it is univalent in , . Thus, by (1.1), in order to verify (2.3), it is sufficient to show that and . The set is the complex plane with slits along the half-lines and . For , these slits are placed in Figure 1 with respect to the circle centered at and the radius .
If h is a special bilinear transformation
and if we replace condition (2.3) by a simpler condition , then it will be satisfied (see [, p.108]) if and only if
when , or
when . Then the univalent solution of the Briot-Bouquet differential equation
has the form
Theorem 2.2 Let , for all . If satisfies the Briot-Bouquet differential subordination
with a convex univalent function h, such that and
where is the open door function given in (2.2), then
and is the best dominant of (2.10).
Proof The function is analytic in and
hence subordination (2.9) becomes
By Lemma 2.1, the equation
has the univalent solution of the form , where is in (2.11) (for more details see [, p.86]). Therefore, again by Lemma 2.1, the function is the best dominant of the subordination
which is equivalent to (2.10) with the best dominant . □
Theorem 2.3 Let , for all . Assume that , , and that A and B satisfy either (2.5) or (2.6) with . If satisfies the Briot-Bouquet differential subordination
and is the best dominant of (2.14).
Proof Subordination (2.13) with becomes
If we return to Lemma 2.1 with and to the remarks below Figure 1, then we can see that under assumptions (2.5) or (2.6) there exists a univalent solution of equation (2.7). Moreover, this function has the form (2.8) and is best dominant of the subordination . It also implies , which is equivalent to (2.14). □
Theorem 2.3 with , and becomes the earlier Theorem 1.1, cited in the first section. Notice that for the function on the right hand side of (2.13) maps the open unit disc onto the disc
This function is also univalent in whence, by (1.1), Theorem 2.3 can be written in the following form.
Corollary 2.4 Let , for all . If satisfies the inequality
where , , satisfy (2.5) with , then
and it is the best dominant of (2.15). Moreover, if A satisfies (2.6) with , and if satisfies condition (2.13) with , i.e.,
and is the best dominant of (2.16).
If , , , then (2.5) is satisfied, and Corollary 2.4 becomes the following one, which is a generalization of Corollary 1 in .
Corollary 2.5 Let , . Suppose that for all . If f satisfies the inequality
and it is the best dominant of (2.18).
Note that instead of (2.18), in  is the inequality
where , under the same assumption as in Corollary 2.5. In , the authors posed also the problem of finding the smallest number μ such that (2.19) holds under the assumptions of Corollary 2.5. In view of (2.18), solving this problem is equivalent to finding
To find (2.20), we use the Taylor series (2.18) with .
For all , the number (2.21) is better than the bound in (2.19), because it is smaller than , given in . This is because
Therefore, the following corollary contains the solution of the first open problem posed in .
Corollary 2.6 Let , . Suppose that for all . If f satisfies the inequality
and this bound is the best possible.
The second open problem posed in  is to find the sharp version of the following corollary.
Corollary 2.7 Let , . If f satisfies the inequality
Using the earlier results, we can improve the corollary above.
Corollary 2.8 Under the assumptions of Corollary 2.7, we have
Proof Inequality (2.22) follows that for all . Hence (2.22) is the same as (2.17), thus, from (2.18), we obtain
Therefore, for , we obtain
So the bound (2.27) is the best possible. Applying (2.27) in (2.22), we directly obtain (2.25). Making use of (2.22) and of (2.27), we get the inequality
which is (2.26). □
For all the bound (2.25) is smaller than the bound , given in (2.24). To show this, observe that
Similarly, for all , the bound (2.26) is greater than the bound , given in (2.23). To show this, observe that the inequality
that has to be proved, after some calculations, becomes
which was proved above in (2.28). The function shows that bound (2.25) delivers the sharp version of (2.23), so this is the solution of the first part of the second open problem posed in . The bound (2.26) does not seem to be the best possible. We conjecture that the best possible bound is
which is suggested by the function .
In , Robertson introduced the classes , of starlike and convex functions of order , which are defined by
If , then a function in each of these sets is univalent, if , it may fail to be univalent. In particular, we have , , the usual classes of starlike and of convex functions, respectively. In Corollary 2.8, we obtained the order of starlikeness for functions satisfying (2.22), so we have
Therefore, if , then condition (2.22) is sufficient for f to be starlike, namely,
Notice that the number is better than given in this place in . Recall that in (2.29), we conjectured that for all a function satisfying (2.22) is starlike, even more, starlike of order . Using instead of f in Corollary 2.8, we obtain the order of convexity for a function satisfying (2.22), so we have
where α is described in (2.30), which means that . For some similar conditions for starlikeness of order α, we also refer to [6, 7] and to .
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Sokół, J. Some applications of differential subordinations in the geometric function theory. J Inequal Appl 2013, 389 (2013). https://doi.org/10.1186/1029-242X-2013-389
- differential subordination
- starlike function
- convex function
- order of starlikeness
- order of convexity
- best dominant