COEFFICIENT INEQUALITIES FOR CERTAIN CLASSES OF RUSCHEWEYH TYPE ANALYTIC FUNCTIONS

Coefficient Inequalities S. Latha vol. 9, iss. 2, art. 52, 2008

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COEFFICIENT INEQUALITIES FOR CERTAIN CLASSES OF RUSCHEWEYH TYPE ANALYTIC

FUNCTIONS

S. LATHA

Department of Mathematics and Computer Science Maharaja’s College

University of Mysore Mysore - 570 005, INDIA.

EMail:drlatha@gmail.com

Received: 18 January, 2007

Accepted: 5 May, 2008

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 30C45.

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

Abstract: A class of univalent functions which provides an interesting transition from star- like functions to convex functions is defined by making use of the Ruscheweyh derivative. Some coefficient inequalities for functions in these classes are dis- cussed which generalize the coefficient inequalities considered by Owa, Pola- to˘glu and Yavuz.

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Contents

1 Introduction 3

2 Main Results 5

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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 functionsf obeying the condition

(1.2) <

1−2

b +2 b

D^δ+1f(z) D^δf(z)

> β

where, b 6= 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,

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) intro- duced 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 func- tions 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 or-

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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 <1and z ∈ U.

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

(1.5) <

1 + zf⁰⁰(z) f⁰(z)

> α

zf⁰⁰(z) f⁰(z) −1

+β

for some α≥0, 0≤β <1and z ∈ U.

We introduce the class VD(α, β, b, δ) as the subclass of N consisting of func- tionsf 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.

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2. Main Results

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

Theorem 2.1. Iff(z)∈ VD(α, β, b, δ) with0 ≤α ≤ β,or, α > ^1+β₂ ,thenf(z) ∈ V ^β−α_1−α, b, δ

.

Proof. Since <{ω} ≤ |ω| for any complex numberω, f(z) ∈ VD(α, β, b, δ) im- plies 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 Corollary2.2 in [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 . Theorem 2.4. If f(z)∈ VD(α, β, b, δ) then,

(2.3) |a | ≤ b(1−β)

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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

pnzⁿ.

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].

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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.

Consider,

|a_k+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|Bk(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)

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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) |a_n| ≤ 2(1−β) n(n−1)|1−α|

n−2

Y

j=1

1 + 2(1−β) j|1−α|

, (n≥3).

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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 konstan- ten von possitiven harmonischen funktionen, Rend. Circ. Palermo.,32 (1911), 193–217.

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

[3] S. OWA, Y. POLATO ˇGLU AND E. 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 AND J. M. JAHANGIRI, Classes of uniformly starlike convx functions, Internat. J. Math. and Math. Sci., 55 (2004), 2959–

2961.