Coefficient Estimates for a Subclass of Bi-univalent Functions Defined

Applied Mathematical Sciences, Vol. 11, 2017, no. 35, 1725 - 1732 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.75165

Coefficient Estimates for a Subclass of Bi-univalent Functions Defined by Salagean Operator Using Quasi Subordination C. Ramachandran Department of Mathematics University College of Engineering Villupuram, Kakuppam Villupuram 605103, Anna University Tamilnadu, India D. Kavitha Department of Mathematics IFET College of Engineering Villupuram 605108 Tamilnadu, India c 2017 C. Ramachandran and D. Kavitha. This article is distributed under Copyright the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In this present investigation, we consider the subclass of analytic and bi-univalent functions associated with salagean operator consisting of the function class Σ in the unit open disk, which satisfies the qusi-subordination conditions. Also we obtain the first two Taylor-Maclaurin coefficients for functions in this new subclass.

Mathematics Subject Classification: 30C45, 30C50 Keywords: Univalent functions, starlike functions, convex functions, quasisubordination, Salagean operator

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C. Ramachandran and D. Kavitha

1. Introduction and preliminaries Let A denote the family of normalized analytic functions f of the form ∞ X ak z k , (z ∈ U) (1.1) f (z) = z + k=2

in the unit open disc U = {z : z ∈ C : |z| < 1}. Further, let S denote the subclass of functions in A which are also univalent in U. Considering the famous Koebe one-quarter theorem [7] which ensures that a disk of radius 1/4 is contained in the image of U under every univalent function f ∈ A. Hence there exist an inverse function f −1 satisfying f −1 (f (z)) = z, (z ∈ U) and f −1 (f (w)) = w, (|w| < r0 (f ), r0 (f ) ≥ 1/4) where, g(w) = f −1 (w) = w − a2 w2 + (2a22 − a3 )w3 − (5a32 − 5a2 a3 + a4 )w4 + . . . (1.2) For a function f ∈ A, if both f and f −1 are univalent in U then f is said to be bi-univalent in U. Let Σ denote the class of bi-univalent functions in U given by (1.1). For instant, given below [18] are functions in the class Σ   1 1+z z , −log(1 − z), log . 1−z 2 1−z The class Σ of bi-univalent functions was introduced by Lewin [12] in the year 1967, and he showed that |a2 | < 1.51. In 1969, Netanyahu [14] showed that maxf ∈Σ |a2 | = 4/3 and Suffridge [20] have given an example of f ∈ Σ for which |a2 | =√4/3. Later, in 1980, Brannan and Clunie [5] improved the result as |a2 | ≤ 2. In 1985, Kedzier-awski [10] proved this conjecture for a special case when the function f and f −1 are starlike. In 1984, Tan [21] proved that |a2 | ≤ 1.485 which is the best estimate for the function in the class of bi-univalent functions. Lately, many authors have introduced and gone through various subclasses of analytic and bi-univalent functions. Some of the recent analyses in these topics are [8, 9, 22, 19] for reference to the readers. Brannan and Taha [6] introduced certain subclasses of the bi-univalent function class Σ for the familiar subclasses S ∗ (α) and C(α). Ali et al. [2] widened the result of Brannan and Taha using subordination. If f (z) and g(z) are analytic in U, we say that the function f (z) is subordinate to g(z) in U, and write f (z) ≺ g(z); z ∈ U if there exits a Schwartz function ω(z), which is analytic in U with ω(0) = 0 and |ω(z)| < 1 (z ∈ U)

Coefficient estimates for a subclass of bi-univalent functions . . .

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such that f (z) = g(ω(z)), z ∈ U. Especially, if the function g is univalent in U, then We have that f ≺g

or

f (z) ≺ g(z) (z ∈ U)

if and only if f (0) = g(0) and f (U) ⊆ g(U). The idea of quasi-subordination was proposed by Robertson [16] in the year 1970. For, two analytic functions f (z) and g(z), the function f (z) is quasi-subordinate to g(z) in the open unit disc U, written by f (z) ≺q g(z), if there exist an analytic function ϕ and ω, with |ϕ(z)| ≤ 1, ω(0) = 0 and |ω(z)| < 1 such that f (z) = ϕ(z)g(ω(z)) (z ∈ U). Note that if ϕ(z) ≡ 1, then f (z) = g(ω(z)), so that f (z) ≺ g(z) in U. Additionally, if ω(z) = z, then f (z) = ϕ(z)g(z) and it is said that f (z) is majorized by g(z) and denoted by f (z)  g(z) in U. Hence it is very much certain that the quasi-subordination is a generalization of subordination as well as majorization [4, 11, 15]. By seeing in literature, the functions f is Ma-Minda bi-starlike functions and Ma-Minda bi-convex functions of complex order b (b ∈ C \ {0}) if both f and f −1 are respectively Ma-Minda type of starlike and Ma-Minda type of convex of complex order b and are denoted by SΣ∗ (b, φ) and CΣ (b, φ) respectively. Recently Obaid Algahtani [1] defined the subclasses of analytic functions involving quasi-subordination and found the bounds of |a2 | and |a3 |. To prove our main result, it is necessary to recall the following: Let φ(z) be an analytic function with positive real part on U with φ(0) = 0 1, φ (0) > 0 which maps the unit open disk U onto a region starlike with respect to 1 and its symmetric with respect to the real axis. The Taylor’s series expansion of such function is φ(z) = 1 + B1 z + B2 z 2 + B3 z 3 + . . . ,

(1.3)

where B1 > 0. Lemma 1. [13] Let ϕ be an analytic function with positive real part in U, with |ϕ(z)| ≤ 1 and let ϕ(z) = P0 + P1 z + P2 z 2 + . . . then |Pn | ≤ 1 − |P0 |2 ≤ 1, for n > 0. Let u(z) = c1 z + c2 z 2 + c3 z 3 + . . . and v(z) = d1 z + d2 z 2 + d3 z 3 + . . . are two analytic functions in U with the conditions u(0) = 0, v(0) = 0, |u(z)| < 1

C. Ramachandran and D. Kavitha

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and |v(z)| < 1. And it is well known that,  |cn | = |dn | ≤

1, n=1 1 − |c1 |2 , n ≥ 2.

(1.4)

For f ∈ A, S˘ al˘ agean [17] introduced the following differential operator: D0 f (z) = f (z),

0

D1 f (z) = zf (z),

and in general, Dn f (z) = D(Dn−1 f (z)),

n∈N

or equivalent to Dn f (z) = z +

∞ X

k n ak z k ,

n ∈ N0 = N ∪ {0}.

(1.5)

k=2

Motivation of the subclass defined in [3], we define the new subclass SCq,Σ (n, λ, γ, φ) using Salagean operator. Definition 1. A function f ∈ Σ be in the class SC q,Σ (n, λ, γ, φ) if   0 1 z[(1 − λ)Dn f (z) + λDn+1 f (z)] − 1 ≺q φ(z) − 1, z ∈ U, γ (1 − λ)Dn f (z) + λDn+1 f (z) = ϕ(z)[φ(u(z)) − 1], z ∈ U. and   0 1 w[(1 − λ)Dn g(w) + λDn+1 g(w)] − 1 ≺q φ(w) − 1, w ∈ U, γ (1 − λ)Dn g(w) + λDn+1 g(w) = ϕ(w)[φ(v(w)) − 1], w ∈ U. where 0 ≤ λ ≤ 1; n ∈ N0 ; γ ∈ C \ {0} and g(w) = f −1 (w). ∗ Note that SC q,Σ (0, λ, γ, φ) ≡ Mq,Σ (λ, γ, φ), SC q,Σ (0, 0, γ, φ) ≡ Sq,Σ (γ, φ) and SC q,Σ (0, 1, γ, φ) ≡ Cq,Σ (γ, φ) .

If ϕ(z) ≡ 1, then the above Definition 1 will be reduced to the class SC Σ (n, λ, γ, φ) and the following subordination holds.   0 1 z[(1 − λ)Dn f (z) + λDn+1 f (z)] − 1 ≺ φ(z), z ∈ U. γ (1 − λ)Dn f (z) + λDn+1 f (z) and   0 1 w[(1 − λ)Dn g(w) + λDn+1 g(w)] − 1 ≺ φ(w), w ∈ U. γ (1 − λ)Dn g(w) + λDn+1 g(w) In this paper, we shall obtain the first two coefficient estimates for the function f belongs to the class SC q,Σ (n, λ, γ, φ).

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2. coefficient estimates Theorem 1. Let f of the form (1.1) be in the class SC q,Σ (n, λ, γ, φ), then s ) ( |γ|(B1 + |B2 |) |γ|B1 ; , (2.1) |a2 | ≤ min 2n (1 + λ) 2(3n )(1 + 2λ) − 4n (1 + λ)2  |a3 | ≤ min

|γ|B1 |γ|2 B12 |γ|(B1 + |B2 |) |γ|B1 + + ; n n n 2 n 3 (1 + 2λ) 4 (1 + λ) 3 (1 + 2λ) 2(3 )(1 + 2λ) − 4n (1 + λ)2 (2.2)

Proof. Since f ∈ SC q,Σ (n, λ, γ, φ), there exists two analytic functions u, v : U → U, with u(0) = 0, v(0) = 0, such that   0 1 z[(1 − λ)Dn f (z) + λDn+1 f (z)] − 1 = ϕ(z)[φ(u(z)) − 1], z ∈ U (2.3) γ (1 − λ)Dn f (z) + λDn+1 f (z) and   0 1 w[(1 − λ)Dn g(w) + λDn+1 g(w)] − 1 = ϕ(w)[φ(v(w)) − 1], γ (1 − λ)Dn g(w) + λDn+1 g(w)

(g := f −1 ). (2.4)

Since φ(u(z)) = 1 + B1 c1 z + (B1 c2 + B2 c21 )z 2 + . . . , and φ(v(w)) = 1 + B1 d1 w + (B1 d2 + B2 d21 )w2 + . . . . Hence, ϕ(z)[φ(u(z)) − 1] = 1 + P0 B1 c1 z + [P1 B1 c1 + P0 (B1 c2 + B2 c21 )]z 2 + . . . , (2.5) and ϕ(w)[φ(v(w)) − 1] = 1 + P0 B1 d1 w + [P1 B1 d1 + P0 (B1 d2 + B2 d21 )]w2 + . . . . (2.6) It follows from (2.3),(2.4),(2.5) and (2.6) that 1 n 2 (1 + λ)a2 = P0 B1 c1 , γ 1 [2(3n )(1 + 2λ)a3 − 22n (1 + λ)2 a22 ] = P1 B1 c1 + P0 (B1 c2 + B2 c21 ), γ 1 − 2n (1 + λ)a2 = 1 + P0 B1 d1 , γ

(2.7) (2.8) (2.9)

and 1 [2(3n )(1 +2λ)(2a22 − a3 ) − 22n (1 + λ)2 a22 ] = P1 B1 d1 + P0 (B1 d2 + B2 d21 ). (2.10) γ From (2.7) and (2.9), it follows that c1 = −d1 .

(2.11)

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Now, squaring and adding of (2.7) and (2.9), we get a22 =

γ 2 P02 B12 d21 . 22n (1 + λ)2

(2.12)

On the otherhand, from (2.8) and (2.10), a22 =

γP0 [B1 (c2 + d2 ) + 2B2 d21 ] . 4(3n )(1 + 2λ) − 22n+1 (1 + λ)2

(2.13)

Using the fact that |c2 | ≤ 1 and |d2 | ≤ 1 in (2.12),(2.13), we have |a2 | ≤

|γ|B1 , + λ)

2n (1

(2.14)

and s |a2 | ≤

4(3n )(1

2|γ|(B1 + |B2 |) . + 2λ) − 22n+1 (1 + λ)2

(2.15)

Hence we get the desired estimation of |a2 |. Next to find the bounds on |a3 |, subtract (2.10) from (2.8) and using (2.12), we get γ[−2P1 B1 d1 + P0 B1 (c2 − d2 )] γ 2 P02 B12 d21 a3 = + . (2.16) 4(3n )(1 + 2λ) 22n (1 + λ)2 Also from (2.8), (2.10) and (2.13), we obtain a3 =

−2γP1 B1 d1 + γP0 B1 (c2 − d2 ) γP0 B1 (c2 + d2 )2γP0 B2 d21 + . (2.17) 4(3n )(1 + 2λ) 4(3n )(1 + 2λ) − 22n+1 (1 + λ)2

Hence we can find the estimates on |a3 | using (1.4),   |γ|B1 1 |a3 | ≤ |γ|B1 | + , (1 + 2λ)3n 22n (1 + λ)2

(2.18)

and |γ|(B1 + |B2 |) |γ|B1 + . n + 2λ) 2(3 )(1 + 2λ) − 22n (1 + λ)2 This completes the proof our Theorem 1. |a3 | ≤

3n (1

(2.19) 

Now, we are giving some remarkable results and corollaries which are the special consequences of our proof. Remark 1. For γ = 1, λ = 0 and n = 0, the inequality (2.1) and (2.2) reduced to the results of Theorem 2.5 [1]. That is, p |a2 | ≤ min {B1 ; B1 + |B2 |}, and |a3 | ≤ min {B1 (1 + B1 ); 2B1 + |B2 |}.

Coefficient estimates for a subclass of bi-univalent functions . . .

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Remark 2. For γ = 1, λ = 1 and n = 0, the inequality (2.1) and (2.2) reduced to the results of Theorem 2.8 [1]. That is, ( ) p B1 + |B2 | B1 ; , |a2 | ≤ min 2 2 and

 |a3 | ≤ min

B12 B1 5B1 |B2 | + ; + 4 3 6 2

 .

∗ Corollary 1. Let f of the form (1.1) be in the class SC q,Σ (n, 0, γ, φ) = Sq,Σ (n, γ, φ), then s ( ) |γ|B1 |γ|(B1 + |B2 |) |a2 | ≤ min , ; 2n 2(3n ) − 4n

and

 |a3 | ≤ min

|γ|B1 |γ|2 B12 |γ|B1 |γ|(B1 + |B2 |) + ; n + 3n 4n 3 2(3n ) − 4n

 .

Corollary 2. Let f of the form (1.1) be in the class SC q,Σ (n, 1, γ, φ) = Cq,Σ (n, γ, φ), then s ( ) |γ|B1 |γ|(B1 + |B2 |) |a2 | ≤ min ; , 2n+1 2(3n+1 ) − 4n+1 and

 |a3 | ≤ min

|γ|B1 |γ|2 B12 |γ|B1 |γ|(B1 + |B2 |) + n+1 ; n+1 + n+1 3 4 3 2(3n+1 ) − 4n+1

 .

Theorem 2. Let f of the form (1.1) be in the class SC q,Σ (n, λ, γ, φ) = SC Σ (n, λ, γ, φ), then s ) ( |γ|B1 |γ|(B1 + |B2 |) ; , |a2 | ≤ min 2n (1 + λ) 2(3n (1 + 2λ)) − 4n (1 + λ)2 and  |a3 | ≤ min

|γ|B1 |γ|2 (B12 ) |γ|B1 |γ|(B1 + |B2 |) + ; + n n 2 n n 2(3 )(1 + 2λ) 4 (1 + λ) 2(3 )(1 + 2λ) 2(3 )(1 + 2λ)) − 4n (1 + λ)2

Proof. By taking ϕ(z) ≡ 1 in the above theorem and the straight forward calculation provides the desired results.  References [1] O. Algahtani, Estimates of initial coefficients for certain subclasses of bi-univalent functions involving quasi-subordination, J. Nonlinear Sci. Appl., 10 (2017), 1004-1011. https://doi.org/10.22436/jnsa.010.03.12 [2] R. M. Ali, L. S. Keong, V. Ravichandran, Shamani Supramaniam, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett., 25 (2012), no. 3, 344-351. https://doi.org/10.1016/j.aml.2011.09.012 [3] O. Altindas, On a subclass of certain stralike functions with negative coefficients, Math. Japan, 36 (1991), no. 3, 489–495.

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Received: May 24, 2017; Published: June 27, 2017