Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 574014,7 pages doi:10.1155/2009/574014

Research Article

Subordination Results on Subclasses Concerning Sakaguchi Functions

B. A. Frasin1 and M. Darus2

1 Department of Mathematics, Al al-Bayt University, P.O. Box 130095, Mafraq, Jordan

2 School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D. Ehsan, Malaysia

Correspondence should be addressed to M. Darus, maslina@ukm.my

Received 30 July 2009; Accepted 6 October 2009

Recommended by Ramm Mohapatra

We derive some subordination results for the subclasses S(a,t), T(a,t), S0(a,t), and T0(a,t) of analytic functions concerning with Sakaguchi functions. Several corollaries and consequences of the main results are also considered.

Copyright © 2009 B. A. Frasin and M. Darus. 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.

1. Introduction and Definitions

Let denote the class of functions of the form

f (z) = z + 2 anZn (1.1)

which are analytic in the open unit disc A = {z : |z| < 1}. A function f (z) eA is said to be in the class S(a, t), if it satisfies

Re{ ff} > a « 1-t -1 (12)

for some 0 < a < 1 and for all z e A.

The class S(a, t) was introduced and studied by Owa et al. [4], where the class S(0, -1) was introduced by Sakaguchi [5]. Therefore, a function f (z) e S(a,-1) is called Sakaguchi function of order a.

We also denote by T(a,t) the subclass of A consisting of all functions f (z) such that zf '(z) e S(a, t).

We note that S (a, 0) = S*(a), the usual star-like function of order a and T (a, 0) = K(a) the usual convex function of order a.

We begin by recalling each of the following coefficient inequalities associated with the function classes S(a, t) and T(a, t).

Theorem 1.1 (see [4]). If f (z) e A satisfies

- Un\ + (1 - a)\un\}\an\ < 1 - a, (1.3)

where un = 1 + t +1 + ■■■ + tn 1 and 0 < a< 1, then f (z) e S(a, t). Theorem 1.2 (see [4]). If f (z) e A satisfies

Yn{\n - un\ + (1 - a)|un|}|an| < 1 - a, (1.4)

where un = 1 + t +1 + ■■■ + tn 1 and 0 < a < 1, then f (z) e T(a, t).

In view of Theorems 1.1 and 1.2, Owa et al. [4] defined the subclasses S0(a, t) c S(a, t) and T0(a, t) c T(a, t), where

S0(a,t) = f (z) e A : f (z) satisfies (1.3)},

T0(a,t) = f (z) e A : f (z) satisfies (1.4)}.

Before we state and prove our main results we need the following definitions and lemma.

Definition 1.3 (Hadamard product). Given two functions f,g e A, where f (z) is given by (1.1) and g(z) is defined by g(z) = z + ^^=2 bnzn the Hadamard product (or convolution) f * g is defined as

f * g) (z) = z + £anbnzn. (1.6)

Definition 1.4 (subordination principle). Let g(z) be analytic and univalent in A. If f (z) is analytic in A,f(0) = g(0), and f (A) c g(A), then we see that the function f (z) is subordinate to g(z) in A, and we write f (z) < g(z).

Definition 1.5 (subordinating factor sequence). A sequence {fc>n}^°=i of complex numbers is called a subordinating factor sequence if, whenever f (z) is analytic, univalent and convex in A, we have the subordination given by

^bnUnZn < f (z) (z e A, ai = 1). (1.7)

Lemma 1.6 (see [6]). The sequence {bn}°°=1 is a subordinating factor sequence if and only if

Re 1 + 2^bnzn\ > 0 (z e A). (1.8)

In this paper, we obtain a sharp subordination results associated with the classes S(a, t) , T(a,t),S0(a,t), and T0(a, t) by using the same techniques as in [1, 2, 7, 8].

2. Subordination Results for the Classes S0(a, t) and S(a, t)

Theorem 2.1. Let the function f (z) defined by (1.1) be in the class S0(a,t). Also let K denote the familiar class of functions f (z) e A which are also univalent and convex in A. If {n|n - un\ + (1 - a) |un|}°=2 is increasing sequence for all n > 2, then

11 - t| + (1 - a)|1 + t| -(f * g)(z) « g(z) (|t|< 1, t = 1; 0 < a < 1; z e A; g eK),

2(|1 - t| + (1 - a)(1 + |1 + t|))

Re(f(z)) >j1+m c e a). (2,)

The constant (|1 - t| + (1 - a)|1 +1|)/2(|1 -1| + (1 - a)(1 + |1 + t|)) is the best estimate. Proof. Let f (z) e S](a, t) and let g(z) = z + £°°=2 cnzn e K. Then

|1 - t| + (1 - a)|1 + t| f )(z) = I1 - t| + (1 - a)|1 + t| ( ^ 2(|1 - t| + (1 - a)(1 + |1 + t|)) f g (z) = 2(|1 - t| + (1 - a)(1 + |1 + tD)\z + ¿fnCnz

Thus, by Definition 1.5, the assertion of our theorem will hold if the sequence

|1 - t| + (1 - a) |1 + t| (2.4)

2(|1 - t| + (1 - a)(1 + |1 + t|)) ^n=1

is a subordinating factor sequence, with a1 = 1. In view of Lemma 1.6, this will be the case if and only if

4 -1, v<1 ^ i ;„> ^ > o,,<25,

RH1 - I 1.-'l<+ ^-f1 + 'I >¿a,,;'

I I1 - i| + <1 - aX1 + I1 + i|) ,=1

= Rei 1 + 11 - t| + <1 - a)|1 + z +_1_V| 1 - t|

1 |1 - t| + <1 - a)<1 + |1 + t|) |1 - t| + <1 - a)<1 + |1 + t|) |

+<1 - a) | 1 + t| a,;'

>1__^ - t + <1 -^ + t r__1_fn_u |

" |1 - t| + <1 - a)<1 + |1 + t|) |1 - t| + <1 - a)<1 + |1 + t|) n=2| n|

+ <1 - a) | Un | | an | rn

| 1 - t| + <1 - a) 11 + t| 1 - a > 1 _ t:-;-Tz-tv„-t:-^ r -

| 1 - t| + (1 - a)(1 + | 1 + t|) | 1 - t| + (1 - a)(1 + | 1 + t|) > 0, ( |z | = r< 1).

Thus (2.5) holds true in A. This proves inequality (2.1). Inequality (2.2) follows by taking the convex function g(z) = z/(1 - z) = z + ^°=2 zn in (2.1). To prove the sharpness of the constant (|1 -1| + (1 - a)|1 +1|)/(2(11 -1| + (1 - a)(1 + |1 +1|))), we consider the function f0(z) e S0(a, t) given by

fo(z) = Z-|1 - t| + (1 - a)|1 + t|z2 (0 * a< 1). (27)

Thus from <2.1), we have

p - l| + <! - a)|1 +t\ /o<=) < = <2.8)

2<|1 - t| + <1 - a)<1 + |1 + tDY 0W 1 - z'

It can easily verified that

min{Re( 2<|1 - -^ £ ,|)) fo<z))} = -2 <z S .). <2.9)

This shows that the constant <|1 - t| + <1 - a)1 + t|)/<2<|1 - t| + <1 - a)<1 + |1 + t|))) is best possible. □

Corollary 2.2. Let the function f (z) defined by (1.1) be in the class S(a,t). Also let K denote the familiar class of functions f (z) e A which are also univalent and convex in A. If {|n - u„| + (1 - a)\u„ |}°°=2 is increasing sequence for all n > 2, then (2.1) and (2.2) of Theorem 2.1 hold true. Furthermore, the constant (|1 - t| + (1 - a)|1 + t|)/(2(|1 - t| + (1 - a)(1 + |1 + t|))) is the best estimate.

Letting t = -1 in Corollary 2.2, we have the following.

Corollary 2.3. Let the function f (z) defined by (1.1) be in the class S(a, -1). Also let K denote the familiar class of functions f (z) e A which are also univalent and convex in A. Then

<f * g) (z) « g(z) (0 < a < 1; z e A; g eK) ,

3 - a (2.10)

Ref (z)) > - ^ (z e A).

The constant 1/(3 - a) is the best estimate.

Letting t = 0 in Corollary 2.2, we have the following result obtained by Ali et al. [1] and Frasin [2].

Corollary 2.4 (see [1, 2]). Let the function f (z) defined by (1.1) be in the class S(a). Also let K denote the familiar class of functions f (z) e A which are also univalent and convex in A. Then

2(3_2a) f * g) (z) « g(z) < a < 1; z e A; g e^,

( ) (2.11)

Ref (z)) > - 32-a (z e A).

The constant (2 - a)/2(3 - 2a) is the best estimate.

Letting a = 0 in Corollary 2.4, we have the following result obtained by Singh [3].

Corollary 2.5 (see [3]). Let the function f (z) defined by (1.1) be in the class S*. Also let K denote the familiar class of functions f (z) e A which are also univalent and convex in A. Then

3 f * g) (z) « g (z) (z e A; g eK),

3 (2.12)

Re( f (z)) > - - (z e A). The constant 1/3 is the best estimate.

3. Subordination Results for the Classes T0(a,t) and T(a,t)

By applying Theorem 1.2 instead of Theorem 1.1, the proof of the next theorem is much akin to that of Theorem 2.1.

Theorem 3.1. Let the function f (z) defined by (1.1) be in the class T0 (a,t). Also let K denote the familiar class of functions f (z) e A which are also univalent and convex in A. If {(|n - un| + (1 - a)\u„\) }°°=2 is increasing sequence for all n > 2, then

11 - tl + (1 - a)|1 + tl -if * g)(z) « g(z) (|t|< 1, t /1; 0 < a < 1; z e A; g eK),

2|1 - t| + (1 - a)(1 + 2|1 + t|)

Re(f(z)) > -2|1 -1| + (1 - a)(1 + 2|1 +t|) (z e a) (3 2)

f (z^ > 2(|1 -1| + (1 - a)|1 + t|) (z e A)' (3.2)

The constant (|1 - t| + (1 - a)|1 +1|)/(2|1 -1| + (1 - a)(1 + 2|1 +1|)) is the best estimate.

Corollary 3.2. Let the function f (z) defined by (1.1) be in the class T(a,t). Also let K denote the familiar class of functions f (z) e A which are also univalent and convex in A. If {n|n - u„| + (1 - a)|u„|}°°=2 is increasing sequence for all n > 2, then (3.1) and (3.2) of Theorem 3.1 hold true. Furthermore, the constant (|1 - t| + (1 - a)|1 + t|)/(2|1 - t| + (1 - a)(1 + 2|1 + t|)) is the best estimate.

Letting t = -1 in Corollary 3.2, we have the following.

Corollary 3.3. Let the function f (z) defined by (1.1) be in the class T(a, -1). Also let K denote the familiar class of functions f (z) e A which are also univalent and convex in A. Then

if * g) (z) « g(z) (0 < a < 1; z e A; g eK),

5 - a 5 (3-3) Re(f (z)) > -(z e A).

The constant 2/(5 - a) is the best estimate.

Letting t = 0 in Corollary 3.2, we have the following result obtained by Ali et al. [1], andFrasin [2] (see also [9]).

Corollary 3.4 (see [1]). Let the function f (z) defined by (1.1) be in the class T(a, 0). Also let K denote the familiar class of functions f (z) e A which are also univalent and convex in A. Then

; (f * g)(z) « g (z) (0 < a < 1; z e A; g eK)

5 - 3a

Re(f(2)) >-Ira? (=e A)-

The constant (2 - a)/(5 - 3a) is the best estimate.

Letting a = 0 in Corollary 3.4, we have the following result obtained by Ozkan [9]. Corollary 3.5 (see [9]). Let the function f (z) defined by (1.1) be in the class X. Then

5 f * g) (z) « g (z) (z e A; g eX),

, (3.5)

Ref (z)) > -4 (z e A).

The constant 2/5 is the best estimate.

Acknowledgment

The second author is under sabbatical program and is supported by the University Research Grant: UKM-GUP-TMK-07-02-107, UKM, Malaysia.

References

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[2] B. A. Frasin, "Subordination results for a class of analytic functions defined by a linear operator," Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 4, article 134, 7 pages, 2006.

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[4] S. Owa, T. Sekine, and R. Yamakawa, "On Sakaguchi type functions," Applied Mathematics and Computation, vol. 187, no. 1, pp. 356-361, 2007.

[5] K. Sakaguchi, "On a certain univalent mapping," Journal of the Mathematical Society of Japan, vol. 11, pp. 72-75,1959.

[6] H. S. Wilf, "Subordinating factor sequences for convex maps of the unit circle," Proceedings of the American Mathematical Society, vol. 12, pp. 689-693,1961.

[7] A. A. Attiya, "On some applications of a subordination theorem," Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 489-494, 2005.

[8] H. M. Srivastava and A. A. Attiya, "Some subordination results associated with certain subclasses of analytic functions," Journal of Inequalities in Pure and Applied Mathematics, vol. 5, no. 4, article 82, pp. 1-6, 2004.

[9] O. Ozkan, "Some subordination results on the classes starlike and convex functions of complex order," Applied Mathematics and Computation, vol. 187, no. 1, pp. 362-368, 2007.

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