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 investigated.
2000 Mathematics Subject Classification: 30C45, 30C50.
∞
1. Introduction. Let A denote the class of functions of the form f (z) = z+ n=2 an zn
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
Re
zf (z)
f (z)
zf (z)
+ β,
> α
−
1
f (z)
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 : Re w > 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))v 2 = 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 : Re w > 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 = (v 2 + 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).
2960
SAEID SHAMS ET AL.
Proof. By definition, it is sufficient to show that
zf (z)
zf (z)
zf (z)
zf (z)
f (z) − (1 + β) − α f (z) − 1 < f (z) + (1 − β) − α f (z) − 1.
(2.1)
For the right-hand side and left-hand side of (2.1) we may, respectively, write
zf (z)
zf (z)
+ (1 − β) − α
− 1
R=
f (z)
f (z)
1
iθ =
f (z) zf (z) + (1 − β)f (z) − αe zf (z) − f (z)
∞
∞
n
n
1
≥
(n + 1 − β) an |z| − α
(n − 1) an |z|
(2 − β)|z| −
f (z)
n=2
n=2
∞
|z|
2−β−
(n + 1 − β + nα − α)an ,
>
f (z)
n=2
(2.2)
and similarly
∞
zf (z)
zf (z)
|z|
− (1 + β) − α
− 1 < (n − 1 − β + nα − α)|an | .
L=
f (z) β +
f (z)
f (z)
n=2
(2.3)
Now, the required condition (2.1) is satisfied, since
∞
|z|
an > 0.
2(1 − β) − 2
R −L > n(1
+
α)
−
(α
+
β)
f (z)
(2.4)
n=2
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 n=2 n[n(1 + α) − (α + β)]|an | < 1 − β, then f ∈ KD(α, β).
Theorem 2.3. The classes SD(α, β) and KD(α, β) are closed under Hadamard product or convolution with convex functions in U .
From Theorem 2.3 and the fact that
F (z) =
1+λ
zλ
z
0
t λ−1 f (t)dt = f (z) ∗
∞
1+λ n
z ,
n
+λ
n=1
we obtain the following corollary upon noting that
in U.
Re λ ≥ 0,
∞
n=1 ((1 + λ)/(n + λ))z
(2.5)
n
is 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)
1 − µn n
dt = f (z) ∗ z +
z ,
t(1 − µ)
n(1 − µ)
n=2
|µ| ≤ 1, µ = 1.
(2.6)
CLASSES OF UNIFORMLY STARLIKE AND CONVEX FUNCTIONS
2961
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))v 2 =
α2 (1 − β)2 /(α2 − 1)2 with the parametric form
w(t) =
α2 − β α(1 − β)
i(1 − β)
sin t,
+
cos t + √ 2
α2 − 1
α2 − 1
α −1
0 ≤ t < 2π .
(2.7)
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
h(z) =
z
z
1
−
w(t)
1 − w(t) (1 − z)2
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∈U lie completely inside E or its compliment E c . 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]
[3]
[4]
[5]
A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), no. 1, 87–92.
S. Kanas and A. Wiśniowska, 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 CurieSkłodowska 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 PólyaSchoenberg conjecture, Comment. Math. Helv. 48 (1973), 119–135, Corrigendum in
Comment. Math. Helv. 48 (1973), 194.
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