JJ II

volume 7, issue 5, article 160, 2006.

Received 28 September, 2006;

accepted 06 November, 2006.

Communicated by:N.E. Cho

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Journal of Inequalities in Pure and Applied Mathematics

COEFFICIENT INEQUALITIES FOR CLASSES OF UNIFORMLY STARLIKE AND CONVEX FUNCTIONS

SHIGEYOSHI OWA, YA ¸SAR POLATO ˘GLU AND EMEL YAVUZ

Department of Mathematics Kinki University

Higashi-Osaka, Osaka 577-8502 Japan

EMail:owa@math.kindai.ac.jp

Department of Mathematics and Computer Science

˙Istanbul Kültür University Bakırköy 34156, ˙Istanbul, Turkey EMail:y.polatoglu@iku.edu.tr EMail:e.yavuz@iku.edu.tr

c

2000Victoria University ISSN (electronic): 1443-5756 246-06

Coefficient Inequalities for Classes of Uniformly Starlike

and Convex Functions

Shigeyoshi Owa, Ya¸sar Polato ˘glu and Emel Yavuz

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Abstract

In view of classes of uniformly starlike and convex functions in the open unit discUwhich was considered by S. Shams, S.R. Kulkarni and J.M. Jahangiri, some coefficient inequalities for functions are discussed.

2000 Mathematics Subject Classification:Primary 30C45.

Key words: Uniformly starlike, Uniformly convex.

Contents

1 Introduction. . . 3 2 Coefficient Inequalities. . . 5

References

Coefficient Inequalities for Classes of Uniformly Starlike

and Convex Functions

Shigeyoshi Owa, Ya¸sar Polato ˘glu and Emel Yavuz

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1. Introduction

LetAbe the class of functionsf(z)of the form

(1.1) f(z) =z+

∞

X

n=2

a_nzⁿ

which are analytic in the open unit discU={z ∈C| |z|<1}.

LetS^∗(β)denote the subclass ofA consisting of functionsf(z)which sat- isfy

(1.2) Re

zf⁰(z) f(z)

> β (z ∈U)

for some β (0 5 β < 1). A function f(z) ∈ S^∗(β) is said to be starlike of order β in U. Also let K(β) be the subclass of A consisting of all functions f(z)which satisfy

(1.3) Re

1 + zf⁰⁰(z) f⁰(z)

> β (z ∈U)

for someβ (05β <1). A functionf(z)inK(β)is said to be convex of order β in U. In view of the class S^∗(β), Shams, Kulkarni and Jahangiri [3] have introduced the subclassSD(α, β)ofAconsisting of functionsf(z)satisfying

(1.4) Re

zf⁰(z) f(z)

> α

zf⁰(z) f(z) −1

+β (z ∈U)

Coefficient Inequalities for Classes of Uniformly Starlike

and Convex Functions

Shigeyoshi Owa, Ya¸sar Polato ˘glu and Emel Yavuz

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for someα= 0andβ (05β < 1). We also denote byKD(α, β)the subclass ofAconsisting of all functionsf(z)which satisfy

(1.5) Re

1 + zf⁰⁰(z) f⁰(z)

> α

zf⁰⁰(z) f⁰(z)

+β (z ∈U)

for someα=0andβ(05β <1). Then we note thatf(z)∈ KD(α, β)if and only ifzf⁰(z)∈ SD(α, β). For such classesSD(α, β)andKD(α, β), Shams, Kulkarni and Jahangiri [3] have shown some sufficient conditions forf(z)to be in the classesSD(α, β)orKD(α, β).

Coefficient Inequalities for Classes of Uniformly Starlike

and Convex Functions

Shigeyoshi Owa, Ya¸sar Polato ˘glu and Emel Yavuz

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2. Coefficient Inequalities

Our first result is contained in

Theorem 2.1. Iff(z) ∈ SD(α, β)with 0 5 α 5 β orα > ^1+β₂ then f(z) ∈ S^{∗ β−α}_1−α

.

Proof. SinceRe(w) 5 |w| for any complex numberw, f(z) ∈ SD(α, β)im- plies that

(2.1) Re

zf⁰(z) f(z)

> α Re

zf⁰(z) f(z) −1

+β,

or that

(2.2) Re

zf⁰(z) f(z)

> β−α

1−α (z ∈U).

If05α5β, then we have that

05 β−α 1−α <1,

and ifα > 1 +β

2 , then we have

−1< α−β α−1 50.

Coefficient Inequalities for Classes of Uniformly Starlike

and Convex Functions

Shigeyoshi Owa, Ya¸sar Polato ˘glu and Emel Yavuz

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Corollary 2.2. Iff(z)∈ KD(α, β)with05 α 5 βorα > ^1+β₂ , thenf(z)∈ K ^β−α_1−α

.

Next we derive

Theorem 2.3. Iff(z)∈ SD(α, β), then

(2.3) |a2|5 2(1−β)

|1−α|

and

(2.4) |a_n|5 2(1−β) (n−1)|1−α|

n−2

Y

j=1

1 + 2(1−β) j|1−α|

(n =3).

Proof. Note that, forf(z)∈ SD(α, β),

Re

zf⁰(z) f(z)

> β−α

1−α (z ∈U).

If we define the functionp(z)by

(2.5) p(z) = (1−α)^zf_f(z)^0(z) −(β−α)

1−β (z ∈U),

then p(z) is analytic inU withp(0) = 1andRe(p(z)) > 0 (z ∈ U). Letting p(z) = 1 +p₁z+p₂z²+· · ·, we have

(2.6) zf⁰(z) =f(z) 1 + 1−β 1−α

∞

X

n=1

pnzⁿ

! .

Coefficient Inequalities for Classes of Uniformly Starlike

and Convex Functions

Shigeyoshi Owa, Ya¸sar Polato ˘glu and Emel Yavuz

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Therefore, (2.6) implies that (2.7) (n−1)a_n = 1−β

1−α(pn−1+a₂pn−2+· · ·+an−1p₁).

Applying the coefficient estimates such that |p_n| 5 2 (n = 1) (see [1]) for Carathéodory functions, we obtain that

(2.8) |a_n|5 2(1−β)

(n−1)|1−α|(1 +|a₂|+|a₃|+· · ·+|an−1|).

Therefore, forn = 2,

|a₂|5 2(1−β)

|1−α| , which proves (2.3), and, forn = 3,

|a₃|5 2(1−β) 2|1−α|

1 + 2(1−β)

|1−α|

. Thus, (2.4) holds true forn = 3.

Supposing that (2.4) is true forn = 3,4,5, . . . , k, we see that

|a_k+1|5 2(1−β) k|1−α|

1 + 2(1−β)

|1−α| +2(1−β) 2|1−α|

1 + 2(1−β)

|1−α|

+· · ·+ 2(1−β) (k−1)|1−α|

k−2

Y

j=1

1 + 2(1−β) j|1−α|

)

= 2(1−β) k|1−α|

k−1

Y

j=1

1 + 2(1−β) j|1−α|

.

Coefficient Inequalities for Classes of Uniformly Starlike

and Convex Functions

Shigeyoshi Owa, Ya¸sar Polato ˘glu and Emel Yavuz

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Consequently, using mathematical induction, we have proved that (2.4) holds true for anyn =3.

Remark 1. If we takeα= 0in Theorem2.3, then we have

|a_n|5 Qn

j=2(j−2β)

(n−1)! (n=2)

which was given by Robertson [2].

Sincef(z)∈ KD(α, β)if and only ifzf⁰(z)∈ SD(α, β), we have Corollary 2.4. Iff(z)∈ KD(α, β), then

(2.9) |a₂|5 1−β

|1−α|

and

(2.10) |a_n|5 2(1−β) n(n−1)|1−α|

n−2

Y

j=1

1 + 2(1−β) j|1−α|

(n=3).

Remark 2. Lettingα= 0in Corollary2.4, we see that

|a_n|5 Qn

j=2(j −2β)

n! (n=2),

given by Robertson [2].

Further applying Theorem2.3we derive:

Coefficient Inequalities for Classes of Uniformly Starlike

and Convex Functions

Shigeyoshi Owa, Ya¸sar Polato ˘glu and Emel Yavuz

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Theorem 2.5. Iff(z)∈ SD(α, β), then

max (

0,|z| −2(1−β)

|1−α| |z|²−

∞

X

n=3

2(1−β) (n−1)|1−α|

n−2

Y

j=1

1 + 2(1−β) j|1−α|

!

|z|ⁿ )

5|f(z)|5|z|+2(1−β)

|1−α| |z|²+

∞

X

n=3

2(1−β) (n−1)|1−α|

n−2

Y

j=1

1 + 2(1−β) j|1−α|

!

|z|ⁿ

and

max (

0,1− 4(1−β)

|1−α| |z| −

∞

X

n=3

2n(1−β) (n−1)|1−α|

n−2

Y

j=1

1 + 2(1−β) j|1−α|

!

|z|ⁿ⁻¹ )

5|f⁰(z)|51+4(1−β)

|1−α| |z|+

∞

X

n=3

2n(1−β) (n−1)|1−α|

n−2

Y

j=1

1 + 2(1−β) j|1−α|

!

|z|ⁿ⁻¹.

Corollary 2.6. Iff(z)∈ KD(α, β), then

max (

0,|z| − 1−β

|1−α||z|²−

∞

X

n=3

2(1−β) n(n−1)|1−α|

n−2

Y

j=1

1 + 2(1−β) j|1−α|

!

|z|ⁿ )

5|f(z)|5|z|+ 1−β

|1−α||z|²+

∞

X

n=3

2(1−β) n(n−1)|1−α|

n−2

Y

j=1

1 + 2(1−β) j|1−α|

!

|z|ⁿ

Coefficient Inequalities for Classes of Uniformly Starlike

and Convex Functions

Shigeyoshi Owa, Ya¸sar Polato ˘glu and Emel Yavuz

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and

max (

0,1− 2(1−β)

|1−α| |z| −

∞

X

n=3

2(1−β) (n−1)|1−α|

n−1

Y

j=1

1 + 2(1−β) j|1−α|

!

|z|ⁿ⁻¹ )

5|f⁰(z)|51+2(1−β)

|1−α| |z|+

∞

X

n=3

2(1−β) (n−1)|1−α|

n−1

Y

j=1

1 + 2(1−β) j|1−α|

!

|z|ⁿ⁻¹.

Coefficient Inequalities for Classes of Uniformly Starlike

and Convex Functions

Shigeyoshi Owa, Ya¸sar Polato ˘glu and Emel Yavuz

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

[3] S. SHAMS, S.R. KULKARNI AND J.M. JAHANGIRI, Classes of uni- formly starlike and convex functions, Internat. J. Math. Math. Sci., 55 (2004), 2959–2961.