Applied and Computational Mathematics
Volume 4, Issue 5, October 2015, Pages: 342-345

On a Subclass of Close-to-Convex Functions Associated with Fixed Second Coefficient

Selvaraj Chellian1, Stelin Simpson2, Logu Sivalingam1

1Department of Mathematics, Presidency College (Autonomous), Chennai, India

2Department of Mathematics, Tagore Engineering College, Vandalur, Chennai, India

Email address:

(S. Chellian)
(S. Simpson)
(L. Sivalingam)

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 z0,  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

  1. O.P. Ahuja, "The influence of second coefficient on spirallike and Robertson functions", Yokohama Math. J. 34(1-2) (1986) 3-1.
  2. H.S. Al-Amiri, "On close-to-star functions of ordera", Proc. Amer. Math. Soc. 29 (1971) 103-108.
  3. V.V. Anh, "Starlike functions with a fixed coefficient", Bult. Austral. Math. Soc. 39(1) (1989) 145-158.
  4. P.L. Duren, "Univalent functions", Springer-Verlag, N.Y. Berlin, Heidelberg, Tokyo, 1983.
  5. M. Finkelstein, "Growth estimates of convex functions", Proc. Amer. Math. Soc. 18 (1967), 412-418.
  6. R.M. Goel, "The radius of convexity and starlikeness for certain classes of analytic functions with fixed coefficient", Ann. Univ. Mariar Euric Sklodowska Sect. A, 25 (1971) 33-39.
  7. A.W. Goodman, "Univalent functions", Vol. I, II, Mariner Tampa, Florida, 1983.
  8. T.H. Gronwall, "On the distortion in conformal mapping when the second coefficient in the mapping function has an assigned value", Prof. Nat. Acad. Proc. 6 (1920) 300-302.
  9. W.K. Hayman, "Multivalent functions", Cambridge University Press, 1958.
  10. W. Kaplan, "Close-to-convex functions", Mich. Math. J. 1 (1952) 169-185.
  11. R.J. Libera, "Some radius of convexity problems", Duke Math. J. 31 (1964) 143-158.
  12. A.E. Livingston, "On the radius of univalence of certain analytic functions", Proc. Amer. Math. Soc. 17 (1965) 352-357.
  13. K.I. Noor, "Radius problem for a subclass of close-to-convex univalent functions", Int. J. Math. Sci. 14(4) (1992) 719-726.
  14. M.S. Robertson, "On the theory of univalent functions", Ann. Math. 37 (1936) 374-408.
  15. C. Selvaraj, "A subclass of close-to-convex functions", Southeast Asian Bull. Math., 28 (2004) 113-123.
  16. C. Selvaraj and N. Vasanthi, "A certain subclass of close-to-convex functions defined in the unit disk", Far East J. Math. Sci. 24(2) (2010) 241-253.
  17. H. Silverman, "On a close-to-convex functions", Proc. Amer. Math. Soc. 36(2) (1972) 477-484.

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