APPLICATIONS OF CERTAIN FUNCTIONS ASSOCIATED WITH LEMNISCATE BERNOULLI | Halim | Journal of the Indonesian Mathematical Society 115 317 2 PB

J. Indones. Math. Soc.
Vol. 18, No. 2 (2012), pp. 93–99.

APPLICATIONS OF CERTAIN FUNCTIONS
ASSOCIATED WITH LEMNISCATE BERNOULLI
Suzeini Abdul Halim1 and Rashidah Omar2
1,2

Institute of Mathematical Sciences,
Faculty of Science, University of Malaya, Malaysia,
suzeini@um.edu.my
2
Faculty of Computer and Mathematical Sciences,
MARA University of Technology Malaysia,
Ashidah@hotmail.com

Abstract. For J(α, f (z)) = (1 − α)

zf ′ (z)
f (z)

[
+α 1+

zf ′′ (z)
f ′ (z)

]

(α ≥ 0), denote SL(α)

and SLc as classes of α-convex and convex
functions which
respectively satisfy

[
]


zf ′′ (z) 2
− 1 < 1. Using established

conditions | [J(α, f (z))]2 − 1| < 1 and  1 + f ′ (z)
βzp′ (z)

βzp′ (z)

, 1 + p(z) ≺ 1+Dz
and 1 + p2 (z) ≺ 1+Dz
results, namely 1 + βzp′ (z) ≺ 1+Dz
1+Ez
1+Ez
1+Ez
√
imply p(z) ≺ 1 + z where p(z) is an analytic function defined on the open unit
disk D with p(0) = 1. This article obtains conditions so that analytic functions f
belong to the classes SL(α) and SLc .

Key words: Convex functions, differential subordination, lemniscate Bernoulli.

Abstrak.

Untuk J(α, f (z)) = (1 − α)

zf ′ (z)
f (z)

[
+ α 1+

zf ′′ (z)
f ′ (z)

]

(α ≥ 0), ny-

atakan SL(α) dan SLc sebagai kelas-kelas dari konveks-α and fungsi-fungsi kon-

yang secaraberturut-turut
memenuhi kondisi | [J(α, f (z))]2 − 1| < 1 dan


veks
]2

[
′′
 1 + zf′ (z) − 1 < 1. Dengan menggunakan hasil-hasil yang telah diperoleh se

f (z)
βzp′ (z)

βzp′ (z)

, 1 + p(z) ≺ 1+Dz
dan 1 + p2 (z) ≺ 1+Dz
belumnya, yaitu 1 + βzp′ (z) ≺ 1+Dz
1+Ez
1+Ez
1+Ez
√
1 + z dimana p(z) adalah sebuah fungsi analitis yang

mengakibatkan p(z) ≺

didefinisikan pada cakram buka D dengan p(0) = 1. Artikel ini memperoleh kondisikondisi sehingga fungsi-fungsi analitis f berada dalam kelas-kelas SL(α) dan SLc .
Kata kunci: Fungsi Konveks, subordinasi diferensial, lemniscate Bernoulli.

2000 Mathematics Subject Classification: 30C45, 30C50.
Received: 25-10-2011, revised: 25-07-2012, accepted: 26-07-2012.
93

S.A. Halim and R. Omar

94

1. Introduction
Let A denote the class of all analytic functions f in the open unit disk D :=
{z ∈ C : |z| < 1} and normalised by f (0) = 0, f ′ (0) = 1. An analytic function f is
subordinate to an analytic function g, written f (z) ≺ g(z)(z ∈ D), if there exists
an analytic function w in D such that w(0) = 0 and |w(z)| < 1 for |z| < 1 and
f (z) = g(w(z)). In particular, if g is univalent in D, then f (z) ≺ g(z) is equivalent
to f (0) = g(0) and f (D) ⊂ g(D). For f in A and z ∈ D, the classes of starlike S ⋆ ,

convex C and α-convex functions are defined respectively by the conditions
}
{
zf ′ (z)
⋆
>0
S = z ∈ D : Re
f (z)
{
}
zf ′′ (z)
C = z ∈ D : Re ′
> −1
f (z)
Mα = {z ∈ D : Re [J(α, f (z))] > 0}
[
]
′′
′
(z)

(z)
+ α 1 + zff ′ (z)
, α ≥ 0. Note that M0 = S ⋆ and
where J(α, f (z)) = (1 − α) zff (z)
M1 = C.
Sok´
ol and Stankiewicz [5] introduced the class SL⋆ consisting
 of normalised
[ zf ′ (z) ]2

analytic functions f in D satisfying the condition  f (z)
− 1 < 1 for z ∈ D.
′

(z)
is in the interior of the right half of
Geometrically, a function f ∈ SL⋆ if zff (z)
the lemniscate of Bernoulli (x2 + y 2 )2 − 2(x2 − y 2 ) = 0. A function in the class
SL⋆ is called a Sok´
ol-Stankiewicz starlike function. Alternatively, we can also write

√
zf ′ (z)
⋆
f ∈ SL ⇔ f (z) ≺ 1 + z . Some properties of functions in class SL⋆ have been
studied by [1], [4], [6], [7], [8] and [9]. Next, we denote S ⋆ [A, B] as the class of
Janowski starlike functions defined in Janowski [2] consisting of functions f ∈ A
′
(z)
1+Az
satisfying zff (z)
≺ 1+Bz
(−1 ≤ B < A ≤ 1).

Let SL(α) and SLc denote the classes of α-convex
and convex functions
[
]2


′′

2
zf
(z)
− 1 < 1 (z ∈
which respectively satisfy | [J(α, f (z))] − 1| < 1 and  1 + f ′ (z)
√
D). It is obvious that f ∈ SL(α) ⇔ J(α, f (z)) ≺ 1 + z and f ∈ SLc ⇔
′′
√
(z)
≺ 1 + z . Recently in [4] the authors determined condition on β so
1 + zff ′ (z)
′

′

(z)
(z)
1+Dz
and 1 + βzp

that 1 + βzp′ (z), 1 + βzp
p2 (z) are subordinated to 1+Ez imply p(z)
√ p(z)
is subordinated to 1 + z where p(z) is analytic in D with p(0) = 1. Properties
of functions in the class SL(α) and SLc are obtained using results given in [4]. In
proving the lemmas, the following proposition is needed.

Proposition 1.1. [3] Let q be univalent in D and let ϕ be analytic in a domain
containing q(D) . Let zq ′ (z)ϕ[q(z)] be starlike. If p is analytic in D, p(0) = q(0)
and satisfies zp′ (z)ϕ[p(z)] ≺ zq ′ (z)ϕ[q(z)] then p ≺ q and q is the best dominant.

Applications of Certain Functions

95

Given that |E| < 1, |D| ≤ 1 and D ̸= E, we then have the following lemmas.
Lemma 1.2. Let β0 =
√
1+z .

√
2 2|D−E|
(1−|E|) .

Proof. By letting q(z) =
s(z) =

√

1+Dz
1+Ez

If 1 + βzp′ (z) ≺

(β ≥ β0 ) then p(z) ≺

1 + z and using Proposition 1.1, it suffices to show

1 + Dz
βz
= h(z).
≺ 1 + βzq ′ (z) = 1 + √
1 + Ez
2 1+z

Quite straightforward it can be established that |s−1 [h(z)]| ≥ 1 for β ≥
Thus the result.
Lemma 1.3. Let β0 =

4|D−E|
(1−|E|) .

Lemma 1.4. Let β0 =
√
1+z .

′

(z)
If 1+β zpp(z)
≺

√
4 2|D−E|
(1−|E|) .

1+Dz
1+Ez

√
2 2|(D−E)|
(1−|E|) .

(β ≥ β0 ) then p(z) ≺

′

(z)
If 1 + β zp
p2 (z) (β ≥ β0 ) ≺

1+Dz
1+Ez

√

1 + z.

then p(z) ≺

With appropriate choices of ϕ in Proposition 1.1, and using similar approach as
above Lemma 1.3 and Lemma 1.4 is easily verified. Details of proving these lemmas
can be found in [4].
2. Main Results
√

Theorem 2.1. Let β0 = 2 2(|D−E|)
, |E| < 1 , |D| ≤ 1 , D ̸= E , β ≥ β0
1−|E|
and f ∈ A. If f satisfies
{
( ′′
)
(
)}
zf ′ (z) zf (z) zf ′ (z)
1 + Dz
zf ′′ (z) z[f ′′ (z)]′
zf ′′ (z)
1+β (1 − α)
≺
−
+1 +α ′
− ′
+1
′
′′
f (z)
f (z)
f (z)
f (z)
f (z)
f (z)
1 + Ez
then f ∈ SL(α)
Proof. With p(z) = (1 − α)
′

p (z) = (1 − α)

{

[

zf ′ (z)
f (z)

]

[
+α 1+

zf ′′ (z)
f ′ (z)

f (z)[zf ′′ (z) + f ′ (z)] − z[f ′ (z)]2
[f (z)]2

f ′ (z)
= (1 − α)
f (z)

{

′′

zf ′ (z)
zf (z)
+1−
′
f (z)
f (z)

}

}

+α

]

, we have

+α

{

f ′′ (z)
f ′ (z)

f ′ (z)[z(f ′′ (z))′ + f ′′ (z)] − z[f ′′ (z)]2
[f ′ (z)]2

{

z[f ′′ (z)]′
zf ′′ (z)
+1− ′
′′
f (z)
f (z)

and from Lemma 1.2, the result implies f ∈ SL(α).
For the special case α = 1, Theorem 2.1 gives the result for the class SLc .

}

}

S.A. Halim and R. Omar

96

Corollary 2.2. Let β0 =
and f ∈ A. If f satisfies

√
2 2(|D−E|)
1−|E|

zf ′′ (z)
1+β ′
f (z)

{

, |E| < 1 , |D| ≤ 1 , D ̸= E , β ≥ β0

z[f ′′ (z)]′
zf ′′ (z)
−
+1
f ′′ (z)
f ′ (z)

}

≺

1 + Dz
1 + Ez

then f ∈ SLc .
Proof. With p(z) = 1 +

zf ′′ (z)
f ′ (z) ,

zp′ (z) =

it follows that

zf ′′ (z) z 2 [f ′′ (z)]2
z 2 [f ′′ (z)]′
+ ′
−
′
f (z)
f (z)
[f ′ (z)]2

zf ′′ (z)
= ′
f (z)

and applying Lemma 1.2 gives 1 +

Theorem 2.3. Let β0 =

1+β

[


′

(1
−
α)zf
(z)
1−


4|D−E|
(1−|E|)

zf ′ (z)
f (z)

{

z[f ′′ (z)]′
zf ′′ (z)
+
1
−
f ′′ (z)
f ′ (z)

zf ′′ (z)
f ′ (z)

1+Dz
1+Ez ,

,

hence f ∈ SLc .

and β ≥ β0 . Suppose f ∈ A and

′′

+

zf (z)
f ′ (z)

[

′ (z) + αf (z) 1 +

(1
−
α)zf


zf ′′ (z)

then f ∈ SL(α).

≺

}

f ′ (z)

]

]+

[
αzf (z) 1 −
′′

(1 −

α)f ′ (z)

zf ′′ (z)
f ′ (z)

zf ′ (z)
f (z)

+

+

z[f ′′ (z)]′
f ′′ (z)

α[f ′ (z)

+

]






zf ′′ (z)] 



≺

1 + Dz
1 + Ez

Applications of Certain Functions

Proof. Let p(z) = (1 − α)
′

p′ (z)
=
p(z)

(z)
(1 − α) ff (z)

{

′′

[

zf (z)
f ′ (z)

zf ′ (z)
f (z)

]

[
+α 1+
zf ′ (z)
f (z)

zf ′′ (z)
f ′ (z)

}

′′

]

(z)
+ α ff ′ (z)
[ ′ ]
[
(z)
(1 − α) zff (z)
+α 1+

+1−

97

.
{

z[f ′′ (z)]′ (z)
f ′′ (z)

zf ′′ (z)
f ′ (z)

+1−

zf ′′ (z)
f ′ (z)

]

}

{ ′′ ′
}
}
{
′′
′
′
(z)]
(z)
(z)
+ 1 − zff ′ (z)
αf ′′ (z)f (z) z[ff ′′ (z)
(1 − α)f ′ (z) zf ′′ (z) + f ′ (z) − zf (z)f
f (z)
}
)} +
{
(
{
=
′′ (z)
′ (z)
+ α[f ′ (z) + zf ′′ (z)]
f ′ (z) (1 − α)zf ′ (z) + αf (z) 1 + zff ′ (z)
f (z) (1 − α)f ′ (z) zff (z)
=

(1 − α)f ′ (z)

{

zf ′′ (z)
f ′ (z)

+

f ′ (z)
f ′ (z)

−

[
(1 − α)zf ′ (z) + αf (z) 1 +

z[f ′ (z)]2
f ′ (z)f (z)
zf ′′ (z)
f ′ (z)

]

}

+

αf ′′ (z)f (z)
f (z)

{

z[f ′′ (z)]′
f ′′ (z)

+1−

zf ′′ (z)
f ′ (z)

(z)
(1 − α)f ′ (z) zff (z)
+ α[f ′ (z) + zf ′′ (z)]
′

}
}
{ ′′ ′
′
′′
(z)]
(z)
(z)]
+ 1 − zff (z)
+ 1 − zff ′ (z)
αf ′′ (z) z[ff ′′ (z)
[
]+
=
′ (z)
′′ (z)
+ α[f ′ (z) + zf ′′ (z)]
(1 − α)f ′ (z) zff (z)
(1 − α)zf ′ (z) + αf (z) 1 + zff ′ (z)
(1 − α)f ′ (z)

{

zf ′′ (z)
f ′ (z)

Applying Lemma 1.3 gives f ∈ SL(α).
For α = 1 we have the result for the class SLc .
4|D−E|
(1−|E|)

, β ≥ β0 and f ∈ A. If
{
}
zf ′′ (z)
z[f ′′ (z)]′
zf ′′ (z)
1 + Dz
1+β ′
− ′
+1 ≺
′′
′′
[f (z) + zf (z)]
f (z)
f (z)
1 + Ez

Corollary 2.4. Let β0 =

then f ∈ SLc .
Proof. For p(z) = 1 +

zf ′′ (z)
f ′ (z) ,

hence

[
][
] [ 2 ′′
]
′][
zf ′′ (z) f ′ (z) − zf ′′ (z)
f ′ (z)
z [f (z)]
f ′ (z)
zp′ (z)
= ′
+
p(z)
f (z)
f ′ (z)
f ′ (z) + zf ′′ (z)
f ′ (z)
f ′ (z) + zf ′′ (z)
=

zf ′′ (z)
[zf ′′ (z)]2
z 2 [f ′′ (z)]′
−
+
[f ′ (z) + zf ′′ (z)] f ′ (z)[f ′ (z) + zf ′′ (z)] [f ′ (z) + zf ′′ (z)]

zf ′′ (z)
= ′
[f (z) + zf ′′ (z)]

{

z[f ′′ (z)]′
zf ′′ (z)
+
1
−
f ′′ (z)
f ′ (z)

}

}

S.A. Halim and R. Omar

98

and based on Lemma 1.3, f ∈ SLc .
√

2|D−E|
, β ≥ β0 and f ∈ A.
Theorem 2.5. Let β0 = 4 (1−|E|)
]
[ ′′

[ ′′ ′
z[f (z)]
zf ′ (z)
zf (z)

2 ′
′′
′
3

 (1 − α)[f (z)] f (z) f ′ (z) + 1 − f (z) + α[f (z)] f (z)f (z) f ′′ (z) + 1 −
1+βz

[(1 − α)f ′ (z)zf ′ (z) + αf (z)[f ′ (z) + zf ′′ (z)]]2



1 + Dz
⇒ f ∈ SL(α).
1 + Ez
[ ′ ]
[
(z)
Proof. Using Lemma 1.4 with p(z) = (1 − α) zff (z)
+α 1+
desired result.
≺

]

gives the

√
4 2|D−E|
(1−|E|)

Corollary 2.6. Let β0 =
1+β

zf ′′ (z)
f ′ (z)

zf ′ (z)f ′′ (z)
[f ′ (z) + zf ′′ (z)]

2

{

, β ≥ β0 and f ∈ A.
}
1 + Dz
z[f ′′ (z)]′
zf ′′ (z)
≺
+
1
−
⇒ f ∈ SLc .
f ′′ (z)
f ′ (z)
1 + Ez

′′

(z)
Proof. Set p(z) = 1 + zff ′ (z)
and in a similar manner the result is easily obtained.

3. Concluding Remark
For α = 0, Theorem 2.1, Theorem 2.3 and Theorem 2.5 reduce to the results
for f ∈ SL⋆ .
Acknowledgement This research was supported by University of Malaya grants:
UMRG108/10APR.

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functions. Submitted.

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f ′ (z)

]







Applications of Certain Functions

99

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