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IJMMS 2004:55, 2959–2961

PII. S0161171204402014

http://ijmms.hindawi.com

© Hindawi Publishing Corp.

CLASSES OF UNIFORMLY STARLIKE AND CONVEX FUNCTIONS

SAEID SHAMS, S. R. KULKARNI, and JAY M. JAHANGIRI

Received 1 February 2004

Some classes of uniformly starlike and convex functions are introduced. The geometrical

properties of these classes and their behavior under certain integral operators are investi-

gated.

2000 Mathematics Subject Classification: 30C45, 30C50.

1. Introduction. LetAdenotetheclassoffunctionsoftheformf(z) = z+?∞

which are analytic in the open unit disk U = {z : |z| < 1}. A function f in A is said

to be starlike of order β, 0 ≤ β < 1, written as f ∈ S∗(β), if Re[(zf?(z))/(f(z))] > β.

A function f ∈ A is said to be convex of order β, or f ∈ K(β), if and only if zf?∈ S∗(β).

Let SD(α,β) be the family of functions f in A satisfying the inequality

n=2anzn

Re

?zf?(z)

f(z)

?

> α

????

zf?(z)

f(z)

−1

????+β,z ∈ U, α ≥ 0, 0 ≤ β < 1.

(1.1)

We note that for α > 1, if f ∈ SD(α,β), then zf?(z)/f(z) lies in the region G ≡ G(α,β) ≡

{w : Rew > α|w−1|+β}, that is, part of the complex plane which contains w = 1 and is

bounded by the ellipse (u−(α2−β)/(α2−1))2+(α2/(α2−1))v2= α2(1−β)2/(α2−1)2

with vertices at the points ((α+β)/(α+1),0), ((α−β)/(α−1),0), ((α2−β)/(α2−1),

(β−1)/√α2−1), and ((α2−β)/(α2−1),(1−β)/√α2−1). Since β < (α+β)/(α+1) <

1 < (α−β)/(α−1), we have G ⊂ {w : Rew > β} and so SD(α,β) ⊂ S∗(β). For α = 1

if f ∈ SD(α,β), then zf?(z)/f(z) belongs to the region which contains w = 2 and is

bounded by parabola u = (v2+1−β2)/2(1−β).

Using the relation between convex and starlike functions, we define KD(α,β) as the

class of functions f ∈ A if and only if zf?∈ SD(α,β). For α = 1 and β = 0, we obtain the

class KD(1,0) of uniformly convex functions, first defined by Goodman [1]. Rønning [3]

investigated the class KD(1,β) of uniformly convex functions of order β. For the class

KD(α,0) of α-uniformly convex function, see [2]. In this note, we study the coefficient

bounds and Hadamard product or convolution properties of the classes SD(α,β) and

KD(α,β). Using these results, we further show that the classes SD(α,β) and KD(α,β)

are closed under certain integral operators.

2. Main results. First we give a sufficient coefficient bound for functions in SD(α,β).

Theorem 2.1. If?∞

n=2[n(1+α)−(α+β)]|an| < 1−β, then f ∈ SD(α,β).

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SAEID SHAMS ET AL.

Proof.

????

For the right-hand side and left-hand side of (2.1) we may, respectively, write

????

=

?

n=2

|z|

??f(z)??

and similarly

By definition, it is sufficient to show that

????

zf?(z)

f(z)

−(1+β)−α

zf?(z)

f(z)

−1

????

????<

????

zf?(z)

f(z)

+(1−β)−α

????

zf?(z)

f(z)

−1

????

????.

(2.1)

R =

zf?(z)

f(z)

1

??f(z)??

??f(z)??

+(1−β)−α

??zf?(z)+(1−β)f(z)−αeiθ??zf?(z)−f(z)????

(2−β)|z|−

?

n=2

????

zf?(z)

f(z)

−1

????

????

≥

1

∞

?

(n+1−β)??an

(n+1−β+nα−α)??an

??|z|n−α

??

∞

?

n=2

(n−1)??an

??|z|n

?

>

2−β−

∞

?

?

,

(2.2)

L =

????

zf?(z)

f(z)

−(1+β)−α

????

zf?(z)

f(z)

−1

????

????<

|z|

??f(z)??

?

β+

∞

?

n=2

(n−1−β+nα−α)|an|

?

(2.3)

.

Now, the required condition (2.1) is satisfied, since

R−L >

|z|

??f(z)??

?

2(1−β)−2

∞

?

n=2

?n(1+α)−(α+β)???an

??

?

> 0.

(2.4)

The following two theorems follow from the above Theorem 2.1 in conjunction with a

convolution result of Ruscheweyh and Sheil-Small [5] and the already discussed relation

between the classes SD(α,β) and KD(α,β).

Theorem 2.2. If?∞

Theorem 2.3. The classes SD(α,β) and KD(α,β) are closed under Hadamard prod-

uct or convolution with convex functions in U.

n=2n[n(1+α)−(α+β)]|an| < 1−β, then f ∈ KD(α,β).

From Theorem 2.3 and the fact that

F(z) =1+λ

zλ

?z

0

tλ−1f(t)dt = f(z)∗

∞

?

n=1

1+λ

n+λzn,

Reλ ≥ 0,

(2.5)

we obtain the following corollary upon noting that?∞

in U.

n=1((1+λ)/(n+λ))znis convex

Corollary 2.4. If f is in SD(α,β) or KD(α,β), so is F(z) given by (2.5).

Similarly, the following corollary is obtained for

G(z) =

?z

0

f(t)−f(µt)

t(1−µ)

dt = f(z)∗

?

z+

∞

?

n=2

1−µn

n(1−µ)zn

?

,

|µ| ≤ 1, µ ?= 1.

(2.6)

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CLASSES OF UNIFORMLY STARLIKE AND CONVEX FUNCTIONS

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Corollary 2.5. If f is in SD(α,β) or KD(α,β), so is G(z) given by (2.6).

We observed that if α > 1 and if f ∈ SD(α,β), then (zf?(z)/f(z))z∈U⊂ E, where

E is the region bounded by the ellipse (u−(α2−β)/(α2−1))2+(α2/(α2−1))v2=

α2(1−β)2/(α2−1)2with the parametric form

w(t) =α2−β

α2−1

Thus for α > 1 and z in the punctured unit disk U −{0}, we have f ∈ SD(α,β) if and

only if zf?(z)/f(z) ?= w(t) or zf?(z)−w(t)f(z) ?= 0. By Ruscheweyh derivatives (see

[4]), we obtain f ∈ SD(α,β), if and only if f(z)∗[z/(1−z)2−w(t)(z/(1−z))] ?= 0,

z ∈ U −{0}. Consequently, f ∈ SD(α,β), α > 1, if and only if f(z)∗h(z)/z ?= 0, z ∈ U

where h is given by the normalized function

?

α2−1+α(1−β)

cost+i(1−β)

√α2−1sint,

0 ≤ t < 2π.

(2.7)

h(z) =

1

1−w(t)

z

(1−z)2−w(t)

z

1−z

?

(2.8)

and w is given by (2.7). Conversely, if f(z)∗h(z)/z ?= 0, then zf?(z)/f(z) ?= w(t),

0 ≤ t < 2π. Hence (zf?(z)/f(z))z∈Ulie completely inside E or its compliment Ec. Since

(zf?(z)/f(z))z=0= 1 ∈ E, (zf?(z)/f(z))z∈U⊂ E, which implies that f ∈ SD(α,β). This

proves the following theorem.

Theorem 2.6. The function f belongs to SD(α,β), α > 1, if and only if f(z)∗h(z)/

z ?= 0, z ∈ U where h(z) is given by (2.8).

References

[1]

[2]

A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), no. 1, 87–92.

S. Kanas and A. Wi´ sniowska, Conic domains and starlike functions, Rev. Roumaine Math.

Pures Appl. 45 (2000), no. 4, 647–657.

F. Rønning, On starlike functions associated with parabolic regions, Ann. Univ. Mariae Curie-

Skłodowska Sect. A 45 (1991), 117–122.

St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109–

115.

St. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the Pólya-

Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119–135, Corrigendum in

Comment. Math. Helv. 48 (1973), 194.

[3]

[4]

[5]

Saeid Shams: Department of Mathematics, Fergusson College, Pune - 411004, India

S. R. Kulkarni: Department of Mathematics, Fergusson College, Pune - 411004, India

E-mail address: kulkarni_ferg@yahoo.com

Jay M. Jahangiri: Department of Mathematics, Kent State University, Burton, OH 44021-9500,

USA

E-mail address: jay@geauga.kent.edu

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