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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 682413, 12 pages
On the Largest Disc Mapped by Sum of Convex and Starlike Functions
1School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800 Penang, Malaysia
2Department of Mathematics, University of Delhi, Delhi 110007, India
Received 5 July 2013; Accepted 17 October 2013
Academic Editor: Ferhan M. Atici
Copyright © 2013 Rosihan M. Ali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For a normalized analytic function defined on the unit disc , let be a function of positive real part in , need not have that property in , and . For certain choices of and , a sharp radius constant is determined, , so that maps onto a specified region in the right half-plane.
Letbe the class of functionsanalytic inand normalized by. Letbe its subclass consisting of univalent functions. For two analytic functionsand, the functionis subordinate to, written, if there is an analytic self-mapwithsatisfying. Given an analytic functionwithandin, denote byandthe subclasses ofconsisting, respectively, ofsatisfyingand.
For various choices of, these classes reduce to well-known subclasses of starlike and convex functions. For instance, with,, thenandare, respectively, the subclasses consisting of starlike functions of orderand convex functions of order. The classesandare the familiar subclasses ofof starlike and convex functions. For,, is the class of functionssatisfying studied by Uralegaddi et al. . Various subclasses ofhave been investigated in [2–5]. For,, the classis the class of strongly starlike functions of order. The classintroduced by Sokół and Stankiewicz  consists of functionssatisfying
In investigating the classof uniformly convex functions, Rønning  introduced a classof parabolic starlike functions. These are functionssatisfying
It is important to keep in mind that the qualifier “parabolic” refers to the geometry of the image ofunder the map; that is, the domain necessarily lies in a parabolic region of the-plane. It does not convey the interpretation that the functionmaps the diskonto a parabolic region. This terminology of parabolic starlike functions is however widely accepted and used by authors. Ali and Ravichandran  recently surveyed works on uniformly convex and parabolic starlike functions.
This paper finds radius estimates for classes of functions in. The radius of a propertyin a given set of functions[12, page 119] is the largest numbersuch that every function in the sethas the propertyin each discfor every. For example, the Koebe function, which mapsonto the domain, is starlike but not convex. However,maps the disconto a convex domain for every. Indeed, every univalent functionmapsonto a convex domain for[13, Theorem 2.13, page 44]. This number is known as the radius of convexity for.
It is known that. The functionis convex and therefore starlike of order; it is clear that the function has real part greater than 1/2. Now the function takes values in, and therefore does not have positive real part for all. Their sum takes values inand therefore the sumdoes not have positive real part in. This motivates us to determine the largest radiussuch that
More generally, letandbe functions satisfyingin, whileneed not necessarily be positive in the whole unit discFor certain choices ofand, a sharp radius constantis determined,, so that whenever, the sumtakes values in specified regions in the complex plane. The results obtained are shown to reduce those of Singh and Paul  in certain special cases.
2. Main Results
Forand, with∈, several radius results for the sumto be in certain regions in the complex plane are obtained in the following result.
Theorem 1. Let; letbe defined by
Then (a), , whereis given by
(b), whereis given by
(c), , whereis given by
(d), , whereis given by
(e),whereis the root of the equation in:
(f)Also, whereis the root of the following equation in:
andis the root of the equation in:
Each radius constantis sharp.
For two analytic functions, their convolution or Hadamard product, denoted by, is defined by. The following results are needed in the sequel.
Lemma 2 ([15, Lemma 2.7, page 126; Lemma 3.5, page 130]). Ifand, orandbelong to, then for any functionanalytic in, wheredenotes the closed convex hull of.
Lemma 3 ([7, Lemma, page 6559]). For, letbe given by
and for, letbe given by
Proof of Theorem 1. Letbe defined by
First, for each, will be shown to, respectively, satisfy, and. Then, using Lemma 2,is deduced to satisfy the required condition.
As in , let so that
(a) By (21), (22), and (23), it follows that
Case (i). Suppose thatWe assert thatfor, where the minimum is taken over all. Let. Then if,, and that for,
On the other hand, if, then it can be shown that
Case (ii). For, thenin,. Indeed for, as in the case (i),
The previously mentioned two cases show thatin. Figure 1 illustrates sharpness of the radiusin the case.
(b) Forgiven by (21), a calculation shows that
By Lemma 3, the functionsatisfies provided that is,
This inequality holds if. Figure 2 illustrates sharpness of the radius.
(c) From (30), it follows that provided holds, which occurs whenever. Sharpness of the radiusin the caseis illustrated in Figure 3.
(d) Inequality (30) also yields provided that is, when. Figure 4 illustrates sharpness of the radiusin the case.
(e) For the functiongiven by (21), it follows from (22) and (23) that . A calculation shows thatwhere
Evidently, (38) and (41) give provided
Figure 5 illustrates sharpness of the radiusin the case.
(f) The inequality holds if or, with,. Then,
Let. Since there exists a uniquesuch thatand.
Thus,for. When, hence,forFigure 6 illustrates sharpness of the radius.
Next, consider,. Then,
Lemma 2, together with (50) and the corresponding inequality for the function, shows that each functionsatisfies the required condition. For sharpness, consider the function. Then, Sharpness of the numbersis now evident in view of the definition.
For, Theorem 1(a) reduces to the following corollary.
Corollary 4 ([14, Theorem, page 724]). If, then inThe numberis sharp.
Theorem 5. Letandbe defined by
Then (a),, whereis given by
(b),, whereis the root of the equation
(c)Also, whereis given by the equation in andis given by the equation in
Each radius constantis sharp.
Each, , is shown to, respectively, satisfy, , and. Then, it follows from Lemma 2 thatsatisfies the required condition.
(a) We claim thatin. By (22) and (23),
Case (i). SupposeIn this case, it is shown thatforover allin. Let. It can be verified that if,, and that for,
On the other hand, if, then
Sinceis a decreasing function in,
Case (ii). For, we prove thatin,Let. As in Case (i), then
It is evident from the previous two cases thatinFigure 7 shows that, for, the radiusis sharp.
(b) Let. Then,
By (67), it follows that
Letbe defined by (The caseis similar.) A calculation shows that
Then,,for, andfor. Thus,
Now (68) and (72) show that provided that is,
Thus,inFigure 8 shows that, for, the radiusis sharp.
(c) Proceeding similarly as in part (a),
Letbe defined by
LetSince there exists a uniquesuch thatand.
Evidently,forand henceinFigure 9 shows that the radiusis sharp.
Lemma 2, together with (83) and the corresponding inequality for the function, shows that each functionsatisfies the required condition. For sharpness, consider the function. Clearly hence the fact that the numberis sharp follows from the definition of.
For, Theorem 5(a) reduces to the following corollary.
Corollary 6 ([14, Theorem, page 722]). If, then in. The numberis sharp.
Theorem 7. Let,be defined by
Then,(a),, where(b),, whereis the root of the equation in andis the root of the equation In particular, The radii are sharp.
(a) We claim thatin. By (22) and (23), it follows that
A calculation shows thatif,, and that over allprovided
This inequality reduces toThus,inFigure 10 shows that, for, the radiusis sharp.
(b) Let. Then
By (94), it follows that
A calculation shows that there existssuch thatand. Thus
By (95), (96), and (97), evidently if that is,
Thus,inFigure 11 shows that, for, the radiusis sharp.
To conclude the proof, let. Then,
As in the earlier proofs, Lemma 2 together with (101) and the corresponding inequality for the functionshows that the functionsatisfies the required condition. For sharpness, consider. Then,
For, Theorem 7(a) reduces to the following result.
Corollary 8 ([14, Theorem, page 723]). If, then inThe result is sharp.
The work presented here is supported by a research university grant from Universiti Sains Malaysia, by a Senior Research Fellowship from the Council of Scientific and Industrial Research, New Delhi, and also by a grant from University of Delhi. The authors are thankful to the referee for the helpful comments.
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