On a Subclass of Close-to-Convex Functions Associated with Fixed Second Coefficient
Selvaraj Chellian^{1}, Stelin Simpson^{2}, Logu Sivalingam^{1}
^{1}Department of Mathematics, Presidency College (Autonomous), Chennai, India
^{2}Department of Mathematics, Tagore Engineering College, Vandalur, Chennai, India
Email address:
To cite this article:
Selvaraj Chellian, Stelin Simpson, Logu Sivalingam. On a Subclass of Close-to-Convex Functions Associated with Fixed Second Coefficient. Applied and Computational Mathematics. Vol. 4, No. 5, 2015, pp. 342-345. doi: 10.11648/j.acm.20150405.12
Abstract: We consider a subclass of univalent functions f (z) for which there corresponds a convex function g(z) of order α such that Re(zf'(z) / g(z)) ≥ β. We investigate the influence of the second coefficient of g(z) on this class. We also prove distortion, covering, and radius of convexity theorems.
Keywords: Analytic Function, Univalent Function, Convex Function of Order α, Close-to-Convexity, Fixed Second Coefficient, Radius of Convexity
1. Introduction
Let A be the class of regular functions on the unit disk Let S be the class of univalent functions of the form
(1.1)
normalized by the conditions Let denote the subclass of S consisting of functions of the form
(1.2)
such that and Cb(a) is the class of convex functions of order a. It is known that 0 £ b £ 1 [2]. Moreover, g(z) Î C1(a) if and only if | e | = 1.
Definition 1.1: A normalized regular function of the form (1.1) is said to belong to the class if there exists a function of the form (1.2) such that
It is clear that for and we have and
Note that In [14] the author has defined a subclass consisting of functions of the form (1.1) for which where C being the class of convex functions and obtained the inclusion relation K* Ì C¢ Ì K Ì S*, K* is the subclass known as the class of quasi-convex functions introduced and studied by K.I. Noor [12]. Thus and hence every member of is univalent. By specializing a and b we obtain some important subclasses. If then is close-to-convex; if then that is in as in [13]; if then so that is convex of order a. If then so
In this paper we prove distortion, covering and radius of convexity theorems for the class
In what follows, we assume is in with its associated function in
First we give an example for by using the following.
Lemma: [16] Let be analytic for z Î C with Then if and only if where G(z) is analytic,
G(0) = 0, and for
Example: Let
and
where
Since
(1.3)
has real part ³ b, it suffices to show that
If then we have
Solving for G(z), we obtain
Since a + b £ 1, h(z) maps E ® E, and for z Î E. Since satisfies the conditions of Lemma, So,
is convex for by Livingston [12]. Thus existence of is asserted.
2. Distortion Theorem for
Theorem 2.1: Let Then
(2.1)
and
(2.2)
where the integrand on the right hand side of (2.2) is taken to be 1 for a = 1.
Equality holds in (2.1) for the function
and equality holds in (2.2) for the function
Proof: From (1.3), by Lemma we obtain
(2.3)
where G(0) = 0 and for z Î E. Since G(z) satisfies the conditions of Schwarz’s lemma, (1.6) yields
(2.4)
That is,
We have [2, p. 105],
(2.5)
Combining (1.7) and (1.8), the result follows. Clearly and with
and
Theorem 2.2. Let Then
Equality holds on the right-hand side for in Theorem 2.1 and on the left-hand side for in Theorem 2.1.
Proof: Integrating along the straight line segment from the origin to and applying Theorem 2.1 we obtain
which proves the right-hand inequality.
To prove the left-hand inequality, for every r we choose z_{0}, such that
If is the pre-image of segment then
This completes the proof.
3. Covering Theorem for
Theorem 3.1: Let with
If for z Î E, then
Proof: First we establish that
Let Then, for
we have [9, p. 15],
(3.1)
Since f(z) does not assume the value w,
is in the class S. Therefore
(3.2)
Now, using (3.1) and (3.2),
and this completes the proof.
4. A Radius of Convexity Theorem for
Lemma 4.1: If then maps the disk onto a convex domain, where r1 is the least positive root of the equation where
Proof: Let Then for with
From it follows that
(4.1)
So, the radius of convexity of is at least equal to the smallest positive root of
For a = 0, from the inequalities in [2, p. 104] and [14, p. 384] we obtain
(4.2)
Now, let where P(z) is analytic, P(0) = 1, and Re{P(z)} > 0 in E.
Then Using the lemma of Libera [11, p. 150] we obtain
(4.3)
Using (4.2) and (4.3) in (4.1) we get
Hence, is convex in where r1 is the least positive root of the equation g(r, b, b) = 0 for given b, b. This lemma improves the result obtained in [16].
Theorem 4.2: If then maps the disk onto a convex domain, where R is the least positive root of the equation where
Proof: Let Then for with
From it follows that
(4.4)
So, the radius of convexity of is at least equal to the smallest positive root of
Using the inequalities in [2, p. 104] and [14, p. 384] we obtain
(4.5)
Now, let where P(z) is analytic, P(0) = 1, and Re{P(z)} > 0 in E.
Then Using the lemma of Libera [11, p. 150] we obtain
(4.6)
Using (4.5) and (4.6) in (4.4) we get
Hence, is convex in where R is the least positive root of the equation for given b, a, b.
References