(1)COEFFICIENT INEQUALITIES FOR CERTAIN CLASSES OF RUSCHEWEYH TYPE ANALYTIC FUNCTIONS S

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COEFFICIENT INEQUALITIES FOR CERTAIN CLASSES OF RUSCHEWEYH TYPE ANALYTIC FUNCTIONS

S. LATHA

DEPARTMENT OFMATHEMATICS ANDCOMPUTERSCIENCE

MAHARAJA’SCOLLEGE

UNIVERSITY OFMYSORE

MYSORE- 570 005 INDIA.

drlatha@gmail.com

Received 18 January, 2007; accepted 5 May, 2008 Communicated by S.S. Dragomir

ABSTRACT. A class of univalent functions which provides an interesting transition from starlike functions to convex functions is defined by making use of the Ruscheweyh derivative. Some coefficient inequalities for functions in these classes are discussed which generalize the coefficient inequalities considered by Owa, Polato˘glu and Yavuz.

Key words and phrases: Convolution, Ruscheweyh derivative, Uniformly starlike and Uniformly convex.

2000 Mathematics Subject Classification. 30C45.

1. INTRODUCTION

Let N denote the class of functions of the form

(1.1) f(z) =z+

∞

X

n=2

a_nzⁿ which are analytic in the open unit disc U ={z ∈C:|z|<1}.

We designate V(β, b, δ) as the subclass of N consisting of functionsfobeying the condition

(1.2) <

1−2

b +2 b

D^δ+1f(z) D^δf(z)

> β

where, b6= 0, δ >−1, 0≤β <1 and D^δf is the Rushceweyh derivative off [5] given by,

(1.3) D^δf(z) = z

(1−z)^1+δ ∗f(z) = z+

∞

X

n=2

a_nB_n(δ)zⁿ,

031-07

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where∗stands for the convolution or Hadamard product of two power series and B_n(δ) = (δ+ 1)(δ+ 2)· · ·(δ+n−1)

(n−1)! .

This class is obtained by putting k = 2 and λ= 0 in the class V_k^λ(β, b, δ) introduced by Latha and Nanjunda Rao [2]. The class V_k^λ(β, b, δ) is of special interest for it contains many well known as well as new classes of analytic univalent functions studied in literature. It provides a transition from starlike functions to convex functions. More specifically, V₂⁰(β,2,0) is the family of starlike functions of orderβand V₂⁰(β,1,1) is the class of convex functions of order β. Shams, Kulkarni and Jahangiri [6] introduced the subclass SD(α, β) of N consisting of functionsf satisfying

(1.4) <

zf⁰(z) f(z)

> α

zf⁰(z) f(z) −1

+β for some α≥0, 0≤β <1 andz ∈ U.

The class KD(α, β), another subclass of N, is defined as the set of all functionsf obeying

(1.5) <

1 + zf⁰⁰(z) f⁰(z)

> α

zf⁰⁰(z) f⁰(z) −1

+β for some α≥0, 0≤β <1 andz ∈ U.

We introduce the class VD(α, β, b, δ) as the subclass of N consisting of functionsf which satisfy

<

1−2

b +2 b

D^δ+1f(z) D^δf(z)

> α 2 b

D^δ+1f(z) D^δf(z) −1

+β where, b6= 0, α≥0, and 0≤β <1.

For the parametric values b = 2, δ = 0 and b = δ = 1 we obtain the classes SD(α, β) and KD(α, β) respectively.

2. MAINRESULTS

We prove some coefficient inequalities for functions in the class VD(α, β, b, δ).

Theorem 2.1. If f(z) ∈ VD(α, β, b, δ) with 0 ≤ α ≤ β, or, α > ^1+β₂ , then f(z) ∈ V ^β−α_1−α, b, δ

.

Proof. Since <{ω} ≤ |ω| for any complex numberω, f(z)∈ VD(α, β, b, δ) implies that

(2.1) <

1−2

b +2 b

D^δ+1f(z) D^δf(z)

> α 2 b

D^δ+1f(z) D^δf(z) − 2

b

+β.

Equivalently,

(2.2) <

1− 2

b + 2 b

D^δ+1f(z) D^δf(z)

> β−α

1−α, (z ∈ U).

If 0≤α≤β, we have, 0≤ ^β−α_1−α <1, and if α > ^1+β₂ , then we have −1< ^α−β_α−1 ≤0.

Corollary 2.2. For the parametric values b = 2 and δ = 0, we get Theorem2.1in [3] which reads as:

If f(z)∈ SD(α, β) with 0≤α ≤β, or, α > ^1+β₂ , then f(z)∈ S^∗ ^β−α_1−α .

Corollary 2.3. The parametric values b=δ= 1, yield the Corollary2.2in [3] stated as:

If f(z)∈ KD(α, β) with 0≤α≤β, or, α > ^1+β₂ , then f(z)∈ K ^β−α_1−α .

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Theorem 2.4. If f(z)∈ VD(α, β, b, δ) then,

(2.3) |a₂| ≤ b(1−β)

|1−α|

and

(2.4) |a_n| ≤ b(1−β)(δ+ 1) (n−1)|1−α|Bn(δ)

n−2

Y

j=1

1 + b(δ+ 1)(1−β) j|1−α|

, (n ≥3).

Proof. We note that for f(z)∈ VD(α, β, b, δ),

<

1− 2

b + 2 b

D^δ+1f(z) D^δf(z)

> β−α

1−α, (z ∈ U).

We define the function p(z) by

(2.5) p(z) =

(1−α)h

1−²_b +²_b^D_D^δ+1δf(z)^f(z)

i−(β−α)

(1−β) , (z ∈ U).

Then, p(z) is analytic in U with p(0) = 1 and <{p(z)}>0 and z ∈ U.

Let p(z) = 1 +p₁z+p₁z²+· · · . We have

(2.6) 1− 2

b +2 b

D^δ+1f(z)

D^δf(z) = 1 +

1−β 1−α

^∞ X

n=1

p_nzⁿ.

That is,

2(D^δ+1f(z)−D^δf(z)) =bD^δf(z) 1−β 1−α

∞

X

n=1

p_nzⁿ

! .

which implies that 2B_n(δ)(n−1)a_n

(δ+ 1)

= b(1−β)

(1−α) [pn−1+B₂(δ) +a₂pn−2 +B₃(δ)a₃pn−3+· · ·+Bn−1(δ)an−1p₁]. Applying the coefficient estimates |p_n| ≤2 for Carathéodory functions [1], we obtain, (2.7) |a_n| ≤ b(1−β)(δ+ 1)

|1−α|(n−1)Bn(δ)[1 +B₂(δ)|a₂|+B₃(δ)|a₃|+· · ·+Bn−1(δ)|an−1|]. For n = 2, |a2| ≤ ^b(1−β)_|1−α|, which proves(2.3).

For n = 3,

|a₃| ≤ b(1−β)(δ+ 1) 2|1−α|B₃(δ)

1 + b(1−β)(δ+ 1)

|1−α|

. Therefore(2.4)holds for n = 3.

Assume that(2.4)is true for n =k.

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Consider,

|ak+1| ≤ b(1−β)(δ+ 1) kB_k+1(δ)

1 + b(1−β)(δ+ 1)

|1−α|

+ b(1−β)(δ+ 1)

|1−α|B2(δ)

1 + b(1−β)(δ+ 1)

|1−α|

+· · ·+ b(1−β)(δ+ 1) (k−1)!|1−α|B_k(δ)

k−2

Y

j=1

1 + b(1−β)(δ+ 1) j(|1−α|)

)

= b(1−β)(δ+ 1) kB_k+1(δ)

k−1

Y

j=1

1 + b(1−β)(δ+ 1) j(|1−α|)

.

Therefore, the result is true for n =k+ 1. Using mathematical induction,(2.4)holds true for

all n ≥3.

Corollary 2.5. The parametric values b = 2 and δ = 0 yield Theorem2.3in [3] which states that:

If f(z)∈ SD(α, β), then

(2.8) |a₂| ≤ 2(1−β)

|1−α|

and

(2.9) |a_n| ≤ 2(1−β)

(n−1)|1−α|

n−2

Y

j=1

1 + 2(1−β) j|1−α|

, (n≥3).

Corollary 2.6. Putting α= 0 in Corollary 2.5,we get

(2.10) |a_n| ≤

Qn

j=1(j−2β)

(n−1)! , (n ≥2), a result by Robertson [4].

Corollary 2.7. For the parametric values b =δ = 1 we obtain Corollary2.5 in [3] given by:

If f(z)∈ KD(α, β) then,

(2.11) |a₂| ≤ (1−β)

|1−α|

and

(2.12) |an| ≤ 2(1−β)

n(n−1)|1−α|

n−2

Y

j=1

1 + 2(1−β) j|1−α|

, (n≥3).

Corollary 2.8. Letting α = 0 in Corollary 2.7,we get the inequality by Robertson [4] given by:

(2.13) |a_n| ≤

Qn

j=1(j−2β)

n! , (n ≥2).

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REFERENCES

[1] C. CARATHÉODORY, Über den variabilitätsbereich der Fourier’schen konstanten von possitiven harmonischen funktionen, Rend. Circ. Palermo.,32 (1911), 193–217.

[2] S. LATHAANDS. NANJUNDA RAO, Convex combinations ofnanalytic functions in generalized Ruscheweyh class, Int. J. Math, Educ. Sci. Technology., 25(6) (1994), 791–795.

[3] S. OWA, Y. POLATO ˇGLUANDE. YAVUZ, Cofficient inequalities fo classes of uniformly starlike and convex functions, J. Ineq. in Pure and Appl. Math., 7(5) (2006), Art. 160. [ONLINE:http:

//jipam.vu.edu.au/article.php?sid=778].

[4] M.S. ROBERTSON, On the theory of univalent functions, Ann. Math., 37 (1936), 374–408.

[5] S. RUSCHEWEYH, A new criteria for univalent function, Proc. Amer. Math. Soc., 49(1) (1975), 109–115.

[6] S. SHAMS, S.R. KULKARNI ANDJ. M. JAHANGIRI, Classes of uniformly starlike convx func- tions, Internat. J. Math. and Math. Sci., 55 (2004), 2959–2961.