R E S E A R C H
Open Access
Some properties concerning
close-to-convexity of certain analytic
functions
Mamoru Nunokawa
1, Melike Aydo ˜gan
2*, Kazuo Kuroki
3, Ismet Yildiz
4and Shigeyoshi Owa
3 *Correspondence:melike.aydogan@isikun.edu.tr
2Department of Mathematics, I¸sık
University, Me¸srutiyet Koyu, ¸Sile Kampusu, Istanbul, 34980, Turkey Full list of author information is available at the end of the article
Abstract
Let f (z) be an analytic function in the open unit disk D normalized with f (0) = 0 and
f(0) = 1. With the help of subordinations, for convex functions f (z) in D, the order of close-to-convexity for f (z) is discussed with some example.
MSC: Primary 30C45
Keywords: analytic; starlike; convex; close-to-convex; subordination
1 Introduction
LetA be the class of functions f (z) of the form
f (z) = z +
∞
n=
anzn
which are analytic in the open unit diskD = {z ∈ C||z| < }. A function f (z) ∈A is said to be convex of orderα if it satisfies
+ Rezf
(z)
f(z) >α in D
for some realα ( α < ). This family of functions was introduced by Robertson [] and we denote it byK(α).
A function f (z)∈A is called starlike of order α in D if it satisfies Rezf
(z)
f (z) >α in D
for some realα ( α < ).
This class was also introduced by Robertson [] and we denote it byS*(α). By the
defini-tions for the classesK(α) and S*(α), we know that f (z) ∈K(α) if and only if zf(z)∈S*(α).
Marx [] and Strohhäcker [] showed that f (z)∈K() implies f (z) ∈ S*( ).
This estimate is sharp for an extremal function
f (z) = z
– z.
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Jack [] posed a more general problem: What is the largest numberβ = β(α) so that
K(α) ⊂ S*β(α).
MacGregor [] determined the exact value ofβ(α) for each α ( α < ) as the infimum over the discD of the real part of a specific analytic function. It has been conjectured that this infimum is attained on the boundary ofD at z = –.
Wilken and Feng [] asserted MacGregor’s conjecture: If α < and f (z) ∈K(α), then
f (z)∈S*(β(α)), where β(α) = ⎧ ⎨ ⎩ –α (–α)– ifα = , log ifα = . ()
Ozaki [] and Kaplan [] investigated the following functions: If f (z)∈A satisfies Ref
(z)
g(z)> inD
for some convex function g(z), then f (z) is univalent inD. In view of Kaplan [], we say that f (z) satisfying the above inequality is close-to-convex inD.
It is well known that the above definition concerning close-to-convex functions is equiv-alent to the following condition:
Rezf
(z)
g(z) > inD
for some starlike function g(z)∈A.
Let us define a function f (z)∈A which satisfies Rezf
(z)
g(z) >α in D
for some realα ( α < ) and for some starlike function g(z) in D. Then we call f (z) close-to-convex of orderα in D with respect to g(z).
It is the purpose of the present paper to investigate the order of close-to-convexity of the functions which satisfy f (z)∈K(α) and α < .
2 Preliminary
To discuss our problems, we have to give here the following lemmas.
Lemma Let p(z) = +∞n=cnznbe analytic inD and suppose that
p(z)≺ –αz
+βz inD,
where≺ means the subordination, < α < and < β < . Then we have
–α
+β < Re p(z) < +α –β.
This shows that
Rep(z) > inD.
A proof is very easily obtained.
Lemma Let p(z) = +∞
n=cnznbe analytic inD, and suppose that there exists a point
z∈ D such that
Rep(z) > c for|z| < |z|
and
Rep(z) = c, p(z)= c
for some real c ( < c < ). Then we have
Rezp (z ) p(z) ⎧ ⎨ ⎩ ––cc when c < , –(–c)c when < c <.
Proof Let us put q(z) =p(z) – c
– c , q() = . Then q(z) is analytic inD and
Req(z) > for|z| < |z|
and
Req(z) = , q(z)= .
Then, from [, Theorem ], we have
zq(z) q(z) = i, where a + a when arg q(z) = π and – a + a – when arg q(z) = –π , where q(z) =±ia and a > .
For the case arg q(z) = π, we have zq(z) q(z) = zp (z ) p(z) – c = i and so zp(z) p(z) =p(z) – c p(z) i, Rezp (z ) p(z) = Re i( – c)a c + i( – c)ai = Re c – i( – c)a c+ ( – c)a –a( – c) = –( – c)ca c+ ( – c)a – c( – c) + a c+ ( – c)a . If we put h(x) = + x c+ ( – c)x (x > ),
then it is easy to see that c < h(x) < ( – c) when c < and ( – c) < h(x) < c when < c < . This shows that
Rezp (z ) p(z) ⎧ ⎨ ⎩ ––cc when c < , –(–c)c when < c <.
For the case arg q(z) = –π, applying the same method as above, we have the same
conclu-sion Rezp (z ) p(z) ⎧ ⎨ ⎩ ––cc when c < , –(–c)c when < c <.
This completes the proof of the lemma. Our next lemma is
Lemma Let p(z) = +∞n=cnznbe analytic inD and suppose that there exists a point
z∈ D such that
and
Rep(z) = c, p(z)= c
for some real c (c < ). Then we have Rezp (z ) p(z) > – c ( – c)> . ()
Proof Let us put
q(z) =p(z) – c
– c , q() = .
Then q(z) is analytic inD. If p(z) satisfies the hypothesis of the lemma, then there exists a point z∈ D such that
Req(z) > for|z| < |z|
and
Req(z) = and q(z)= ,
then p(z) satisfies the conditions of the lemma.
For the case arg q(z) = π, applying the same method as in the proof of Lemma , we
have Rezp (z ) p(z) = – ( – c)ca c+ ( – c)a – c( – c) + a c+ ( – c)a . Putting h(x) = + x c+ ( – c)x (x > ), it follows that h(x) = (c – )x (c+ ( – c)x) < (x > ). ()
Therefore, from () we obtain () .
For the case arg q(z) = –π, applying the same method as above, we have the same
con-clusion as in the case arg q(z) =π.
3 The order of close-to-convexity
Now, we discuss the close-to-convexity of f (z) with the help of lemmas.
(i) for the case c < , + Rezf (z) f(z) > Re zg(z) g(z) – – c c inD, zf(z) g(z) = c in D and
(ii) for the case < c <,
+ Rezf (z) f(z) > Re zg(z) g(z) – c ( – c) inD, zf(z) g(z) = c in D. Then we have Rezf (z) g(z) > c inD.
This means that f (z) is close-to-convex of order c inD. Proof Let us put
p(z) =zf
(z)
g(z) , p() = .
Then it follows that +zf (z) f(z) = zg(z) g(z) + zp(z) p(z) .
(i) For the case c < , if there exists a point z∈ D such that
Rep(z) > c for|z| < |z|
and
Rep(z) = c,
then, applying Lemma and the hypothesis of Theorem , we have
p(z)= c and Rezp (z ) p(z) – – c c .
Thus, it follows that + Rezf (z ) f(z) = Rezg (z ) g(z) + Rezp (z ) p(z) Rezg(z) g(z) – – c c ,
which contradicts the hypothesis of Theorem . (ii) For the case < c < , applying the same method as above, we also have that
Rezf
(z)
g(z) > c inD.
This completes the proof of the theorem. Applying Theorem , we have the following corollary.
Corollary Let f (z) ∈A be convex of order α ( < α < ), and suppose that there exists a starlike function g(z) such that
(i) for the case α < c, + Rezf (z) f(z) > Re zg(z) g(z) – –β(c) β(c) inD, zf(z) g(z) = β(c) in D and
(ii) for the case <α < c , + Rezf (z) f(z) > Re zg(z) g(z) – β(c) ( –β(c)) inD, zf(z) g(z) = β(c) in D. Then we have Rezf (z) g(z) >β(c) > β(α) > α in D.
Remark For the case < α < c < , it is trivial that α < β(α) < β(c) < .
Example Let f (z) ∈A satisfy
+ Rezf (z) f(z) > Re – Az + Az– –β() β( ) > – inD, () where A =β( ) – β() + .
andβ() = log . If we consider the starlike function g(z) given by g(z) = z ( + Az), then we have Rezf (z) g(z) >β .,
which means that f (z) is close-to-convex of orderβ() inD. Next we show
Theorem Let f (z) ∈A and g(z) ∈ A be given by g(z) = ⎧ ⎪ ⎨ ⎪ ⎩ z (+βz)α+ββ (β = ), ze–αz (β = )
for some α ( α < ) and some β ( β < ). Further suppose that for arbitrary r
( < r < ), min |z|=r Rezf (z) g(z) = Rezf (z ) g(z) |z|=r =zf(z) g(z) and + Rezf (z) f(z) – c ( – c)+ –α +β
for c < . Then we have
Rezf
(z)
g(z) > c inD.
Proof Let us define the function p(z) by p(z) =zf
(z)
g(z) , p() =
for c < . If there exists a point z∈ D such that
Rep(z) > c for|z| < |z|
and
Rep(z) = c
for c < , then from the hypothesis of Theorem , we have Rep(z)= p(z).
Therefore, applying Lemma and Lemma , we have + Rezf (z ) f(z) = Rezp (z ) p(z) + Rezg (z ) g(z) > – c ( – c)+ –α +β. This is a contradiction, and therefore we have
Rezf
(z)
g(z) > c inD.
Remark In view of the definition for close-to-convex functions, if f (z) satisfies
Rezf
(z)
g(z) > inD,
then we can say that f (z) is close-to-convex inD. But c should be a negative real number in Theorem . Therefore, we cannot say that f (z) is close-to-convex inD in Theorem .
Competing interests
The authors declare that they have no competing interests. Authors’ contributions
All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript. Author details
1University of Gunma, Hoshikuki 798-8, Chuou-Ward, Chiba, 260-0808, Japan.2Department of Mathematics, I¸sık
University, Me¸srutiyet Koyu, ¸Sile Kampusu, Istanbul, 34980, Turkey.3Department of Mathematics, Kinki University,
Higashi-Osaka, Osaka 577-8502, Japan.4Department of Mathematics, Duzce University, Konuralp Yerleskesi, Duzce,
81620, Turkey. Acknowledgements
The authors thank the referees for their helpful comments and suggestions to improve our manuscript. Received: 1 August 2012 Accepted: 8 October 2012 Published: 24 October 2012
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doi:10.1186/1029-242X-2012-245
Cite this article as: Nunokawa et al.: Some properties concerning close-to-convexity of certain analytic functions.
