International Journal of Analysis and Applications

Volume 16, Number 5 (2018), 689-701

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-689

SOME PROPERTIES OF ANALYTIC FUNCTIONS ASSOCIATED WITH CONIC

TYPE REGIONS

KHALIDA INAYAT NOOR1, NAZAR KHAN1,2, MASLINA DARUS3,∗, QAZI ZAHOOR

AHMAD2 AND BILAL KHAN2

1Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad,

Pakistan

2Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad, Pakistan

3School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,

43600 Bangi, Selangor, Malaysia

∗Corresponding author: maslina@ukm.edu.my

Abstract. The main purpose of this investigation is to define new subclasses of analytic functions with

respect to symmetrical points. These functions map the open unit disk onto certain conic regions in the

right half plane. We consider various corollaries and consequences of our main results. We also point out

relevant connections to some of the earlier known developments.

1. Introduction and Definitions

Let H denote the class of functions analytic in the unit disk E = {z : |z| < 1}. Let A denote the class of

analytic functions in the open unit disk E and satisfying the following conditions

f(0) = f′ (0) − 1 = 0.

Received 2018-05-09; accepted 2018-07-10; published 2018-09-05.

2010 Mathematics Subject Classification. Primary 05A30, 30C45; Secondary 11B65, 47B38.

Key words and phrases. Analytic functions; symmetric points; conic type regions.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

689

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-689


Int. J. Anal. Appl. 16 (5) (2018) 690

Therefore, for f ∈A, one has

f(z) = z +

∞∑
n=2

anz
n (∀z ∈ E) . (1.1)

Also let S be the subclass of A which consists of univalent functions in E.

Moreover the class of starlike functions in E will be denoted by S∗, which consists of normalized functions

f ∈A that satisfy the following inequality:

<
(
zf′ (z)

f (z)

)
> 0 (∀ z ∈ E) . (1.2)

Similarly the class C of convex functions in E consists of normalized functions f ∈ A that satisfy the

following inequality:

<
(

(zf′ (z))
′

f′ (z)

)
> 0 (∀ z ∈ E) . (1.3)

For two functions f and g, analytic in E, we say that f is subordinate to g, denoted by

f(z) ≺ g(z) or f ≺ g,

if there exists a Schwarz function w which is analytic in E with

w (0) = 0 and |w(z)| ≤ |z|

such that

f(z) = g (w(z)) .

Furthermore if the function g is univalent in E, then one can find that

f(z) ≺ g(z) ⇐⇒ 0 = g (0) and f (E) ⊆ g (E) .

We next denote by P the class of analytic functions p, which are normalized by

p (z) = 1 +

∞∑
n=1

pnz
n (1.4)

such that

<(p (z)) > 0.

Definition 1.1. A function f ∈A is said to belongs to the class S∗s , if and only if

zf′(z)

f(z) −f(−z)
≺

1 + z

1 −z
(∀ z ∈ E) .

The class S∗s , of starlike functions with respect to symmetrical points, was introduced by Sakaguchi in 1959,

( see [23]).

Remark 1.1. For function f ∈ A the idea of Alexander’s theorem [7] was used by Das and Singh [6] for

defining the class Cs of convex functions with respect to symmetrical points, in the following way:

f(z) ∈Cs ⇐⇒ zf′(z) ∈S∗s .



Int. J. Anal. Appl. 16 (5) (2018) 691

Definition 1.2. A given function h with h (0) = 1 is said to belong to the class P [A,B] if and only if

h (z) ≺
1 + Az

1 + Bz
, −1 ≤ B < A ≤ 1.

The analytic functions class P [A,B] was introduced by Janowski [9], who showed that h (z) ∈P [A,B] if

and only if there exist a function p ∈P such that

h (z) =
(A + 1) p (z) − (A− 1)
(B + 1) p (z) − (B − 1)

, −1 ≤ B < A ≤ 1.

Geometrically a function h belongs to P [A,B] if and only if it maps the open unit disk E onto the disk

defined by the domain

Ω [A,B] =

{
t ∈ C :

∣∣∣∣t− 1 −AB1 −B2
∣∣∣∣ < A−B1 −B2

}
.

Historically speaking, the conic domain Ωk, k ≥ 0, was first introduced by Kanas and Wísniowska

(see [11] and [12]) as

Ωk =

{
u + iv : u > k

√
(u− 1)2 + v2

}
.

Moreover for fixed k this domain represents the right half plane (k = 0), a parabola (k = 1), the right branch

of hyperbola (0 < k < 1) and an ellipse (k > 1), see also [17], [18] and recently [21]. Indeed the extremal

functions for these conic regions are

pk(z) =




1
1−k2 cosh

{(
2
π

arccos k
)

log
1+
√
z

1−
√
z

}
− k

2

1−k2 (0 ≤ k < 1)

1 + 2
π2

(
log

1+
√
z

1−
√
z

)2
(k = 1)

1
k2−1 sin

(
π

2K(κ)

∫ u(z)√
κ

0
dt√

1−t2
√

1−κ2t2

)
+ k

2

k2−1 (k > 1) ,

(1.5)

where

u(z) =
z −
√
κ

1 −
√
κz

(∀ z ∈ E)

and κ ∈ (0, 1) is chosen such that k = cosh (πK′(κ)/(4K(κ))). Here K(κ) is Legendre’s complete elliptic

integral of first kind and K′(κ) = K(
√

1 −κ2) i.e. K′ (t) is the complementary integral of K (t), [1], [2].

Assume that

pk(z) = 1 + T1(k)z + T2(k)z
2 + . . . (∀ z ∈ E) .



Int. J. Anal. Appl. 16 (5) (2018) 692

Then it was shown in [13] that for (1.5) one can have

T1 := T1(k) =




2A2

1−k2 (0 ≤ k < 1)

8
π2

(k = 1)

π2

4Ķ2(t)2(1+t)
√
t

(k > 1) ,

(1.6)

T2 := T2(k) = D(k)T1(k),

where

D(k) =




A2+2
3

(0 ≤ k < 1)

8
π2

(k = 1)

(4K(κ))2(t2+6t+1)−π2

24K(κ)2(1+t)
√
t

(k > 1)

(1.7)

with A = 2
π

arccos k.

Noor et al. [16] combine the concepts of Janowski functions and the conic regions and define the following:

Definition 1.3. A function h ∈H is said to be in the class k-P [A,B], if and only if

h(z) ≺
(A + 1)pk(z) − (A− 1)
(B + 1))pk(z) − (B − 1)

k ≥ 0, (1.8)

where pk(z) is defined by (1.5) and −1 ≤ B < A ≤ 1.

Geometrically, each function h ∈ k-P[A,B] takes all values in the domain Ωk[A,B],−1 ≤ B < A ≤ 1,k ≥

0 which is defined as

Ωk[A,B] =

{
w : <

(
(B − 1)w − (A− 1)
(B + 1))w − (A + 1)

)
> k

∣∣∣∣ (B − 1)w − (A− 1)(B + 1))w − (A + 1) − 1
∣∣∣∣
}
,

or equivalently Ωk[A,B] is a set of numbers w = u + iv such that[(
B2 − 1

)(
u2 + v2

)
− 2 (AB − 1) u +

(
A2 − 1

)]2
> k

[(
−2(B + 1))

(
u2 + v2

)
+ 2 (A + B + 2) u− 2(A + 1)

)2
+ 4(A−B)2v2

]
.

This domain represents the conic type regions for detail (see [16]). One can observe that

0-P [A,B] = P [A,B] ,

introduced by Janowski (see [9]) and

k-P [1,−1] = P (pk) ,

introduced by Kanas and Wísniowska (see [11]).



Int. J. Anal. Appl. 16 (5) (2018) 693

In the recent years, several interesting subclasses of analytic functions have been introduced and investi-

gated, see for example [3], [4], [5], [19] and [22]. Motivated and inspired by the recent research going on and

from the above mentioned work, we now introduce some new subclasses of analytic functions as following:

Definition 1.4. A function f ∈ S is said to be in the class k-USs [A,B] , k ≥ 0, −1 ≤ B < A ≤ 1, if and

only if

2zf′(z)

f(z) −f(−z)
≺

(A + 1)pk(z) − (A− 1)
(B + 1))pk(z) − (B − 1)

(∀ z ∈ E) . (1.9)

Remark 1.2. First of all, it is easily seen that

0-USs [1,−1] = S∗s ,

the class of starlike functions with respect to symmetric points introduced and studied by Sakaguchi (see [23]).

Secondly, we have

0-USs [A,B] = S∗s [A,B] ,

the class of Janowski starlike functions with respect to symmetric points introduced by Goel and Mehrok in

1982 (see [8]). Thirdly, we have

k-USs [1,−1] = k-STs,

introduced and studied by Noor (see [20]).

Definition 1.5. A function f ∈ S is said to be in the class k-UCs [A,B] , k ≥ 0,−1 ≤ B < A ≤ 1, if and

only if

2 (zf′(z))
′

(f(z) −f(−z))′
≺

(A + 1)pk(z) − (A− 1)
(B + 1))pk(z) − (B − 1)

(∀ z ∈ E) .

Remark 1.3. From Definiton 1.5 it is readily observe that

0-UCs [1,−1] = Cs,

the class of convex functions with respect to symmetric points introduced and studied by Das and Singh,

(see [6]). Secondly we have

0-UCs [A,B] = C∗s [A,B] ,

the class of Janowski convex functions with respect to symmetric points introduced by Janteng and Halim

in 2008 (see [10]) . And finally

k-UCs [1,−1] = k-UCVs,

introduced and studied by Noor (see [20]).



Int. J. Anal. Appl. 16 (5) (2018) 694

2. A Set of Lemmas

Each of the following lemmas will be needed in our present investigation.

Lemma 2.1. [14] If a function w ∈H is of the form

w(z) = c1z + c2z
2 + . . . and |w(z)| ≤ |z| (∀ z ∈ E) , (2.1)

then for every complex number s, we have

∣∣c2 −sc21∣∣ ≤ 1 + (|s|− 1) ∣∣c21∣∣ .
Lemma 2.2. Let k ∈ [0,∞) be a fixed and

qk(z) =
(A + 1)pk(z) − (A− 1)
(B + 1))pk(z) − (B − 1)

.

Then

qk(z) = 1 + H1(k)z + H2(k)z
2 + . . . (∀ z ∈ E) (2.2)

and

H1 := H1(k) =
A−B

2
T1(k) (2.3)

H2 := H2(k) =
(A−B)T1(k)

4
{2D(k) − (B + 1)T1(k)} (2.4)

where T1(k) and D(k) are defined by (1.6) and (1.7).

Proof. We have

(A + 1)pk(z) − (A− 1) = {(B + 1)pk(z) − (B − 1)}
{

1 + H1z + H2z
2 + . . .

}
.

Therefore, we obtain

2 + (A + 1)
{
T1z + T2z

2 + . . .
}

=
[
2 + (B + 1)

{
T1z + T2z

2 + . . .
}][

1 + H1z + H2z
2 + . . .

]
(2.5)

Comparing the coefficients at z gives

(A + 1)T1 = (B + 1)T1 + 2H1,

so we obtain the first equality (2.3). Similarly, comparing the coefficients at z2 gives

(A + 1)T2 = 2H2 + (B + 1)H1T1 + (B + 1)T2,

so we have

(A−B)T2 − (B + 1)H1T1 = 2H2.



Int. J. Anal. Appl. 16 (5) (2018) 695

Applying (2.3) gives

H2 =
(A−B)

2
T2 −

(A−B)(B + 1)
4

T 21

=
(A−B)

2
D(k)T1 −

(A−B)(B + 1)
4

T 21

=
(A−B)T1

4
{2D(k) − (B + 1)T1} .

so, we obtain the second equality (2.4). This completes the proof. �

3. Main Results

In this section, we will prove our main results. Throughout our discussion, we assume that

−1 ≤ B < A ≤ 1 and k ≥ 0.

Theorem 3.1. Let f ∈ k-USs [A,B]. Then the function

ϕ(z) =
1

2
(f(z) −f(−z)) , (3.1)

belongs to k-US[A,B] in E, where k-US[A,B] is the class of Janowski starlike functions g(z) ∈A such that

zg′(z)

g(z)
∈ k-P[A,B].

Proof. Taking logarithmic differentiation of (3.1), we have

zϕ′(z)

ϕ(z)
=
z (f(z))

′
+ z (f(−z))′

(f(z) −f(−z))
.

Then we find after some simplification that

zϕ′(z)

ϕ(z)
=

1

2

[
2z (f(z))

′

(f(z) −f(−z))
+

2z (f(−z))′

(f(−z) −f(z))

]
=

1

2
[p1(z) + p2(z)] , p1,p2 ∈ k-P [A,B] (∀ z ∈ E) .

Moreover one can find that k-P [A,B] is a convex set ( see [16]), it follows that

zϕ′(z)

ϕ(z)
∈ k-P [A,B]

and thus ϕ(z) ∈ k-US [A,B]. �

Remark 3.1. The above Theorem shows that the class k-USs [A,B] is a subclass of the class of close-to-

convex functions.



Int. J. Anal. Appl. 16 (5) (2018) 696

Theorem 3.2. Let 0 ≤ k < ∞ be fixed. Assume that a function qk defined in Lemma 2.2, has the form

(2.2). If the function h(z) = 1 + b1z + b2z
2 + . . . is a member of the function class k-P [A,B], then for

−∞ < u < ∞,

∣∣b2 −ub21∣∣ ≤




A−B
2
T1(k)

{
u

(A−B)
2

T1(k) − 12 {[2D(k) − (B + 1)]T1(k)}
}

(u > α1)

A−B
2
T1(k), (α1 ≤ u ≤ α2)

A−B
2
T1(k)

{
1
2
{[2D(k) − (B + 1)]T1(k))

}
−u(A−B)

2
T1(k) (u < α2) ,

(3.2)

where

α1 =
[2 + 2D(k) − (B + 1)]T1(k)

(A−B)T1(k)
,

α2 =
[2D(k) − (B + 1)]T1(k) − 2

(A−B)T1(k)
,

and T1, D(k) are defined by (1.6) and (1.7).

Proof. If f ∈ k-P [A,B] then it follows that

h(z) ≺ qk(z) = 1 +
A−B

2
T1(k)z

+
(A−B) [2D(k) − (B + 1)] T1(k)

4
T1(k)z

2 + . . . (∀z ∈ E) . (3.3)

Now by the definition of subordination there exists a function w analytic in E with

w (0) = 0 and |w(z)| < 1 such that w(z) = c1z + c2z2 + · · ·

and

h(z) = 1 +
A−B

2
T1(k)w(z)

+
(A−B) [2D(k) − (B + 1)] T1(k)

4
T1(k)w

2(z) + . . . . (3.4)

Now from (2.1), (3.3) and (3.4), we have

b1 =
A−B

2
T1(k)c1,

b2 =
A−B

2
T1(k)

{
c2 +

[2D(k) − (B + 1)] T1(k)
2

c21

}
.

Therefore, we obtain

b2 −ub21 =
A−B

2
T1(k)

{
c2 +

{
[2D(k) − (B + 1)] T1(k)

2
−u

A−B
2

T1(k)

}
c21

}
. (3.5)

This gives

∣∣b2 −ub21∣∣ = A−B2 T1(k)
∣∣∣∣c2 − c21 +

{
1 +

[2D(k) − (B + 1)] T1(k)
2

−u
A−B

2
T1(k)

}
c21

∣∣∣∣ .



Int. J. Anal. Appl. 16 (5) (2018) 697

Suppose that u > α1, then using the estimate
∣∣c2 − c21∣∣ ≤ 1 from Lemma 2.1 and the well known estimate

|c1| ≤ 1 of the Schwarz Lemma, we obtain∣∣b2 −ub21∣∣ ≤ A−B2 T1(k)
{
u

(A−B)
2

T1(k) −
(2D(k) − (B + 1))T1(k))

2

}
.

This is the first inequality in (3.2).

On the other hand if u < α2, then (3.5) gives∣∣b2 −ub21∣∣ ≤ A−B2 T1(k)
{
|c2| +

{
(2D(k) − (B + 1))T1(k))

2
−u

(A−B)
2

T1(k)

}
|c1|

2

}
.

Applying the estimates |c2| ≤ 1 −|c1|
2

of Lemma 2.1 and |c1| ≤ 1, we have∣∣b2 −ub21∣∣ ≤ A−B2 T1(k)
{

1 +

{
(2D(k) − (B + 1))T1(k))

2
−u

(A−B)
2

T1(k) − 1
}
|c1|

2

}
≤
A−B

2
T1(k)

{
(2D(k) − (B + 1))T1(k))

2
−u

(A−B)
2

T1(k)

}
.

This is the last inequality in (3.2). Finally if α1 < u < α2, then∣∣∣∣(2D(k) − (B + 1))T1(k))2 −u(A−B)2 T1(k)
∣∣∣∣ ≤ 1.

Therefore (3.5), yields

∣∣b2 −ub21∣∣ ≤ A−B2 T1(k)
{
|c2| + |c1|

2
}
≤
A−B

2
T1(k)

{
1 −|c1|

2
+ |c1|

2
}

=
A−B

2
T1(k).

We get the middle inequality in (3.2). This completes the proof. �

Remark 3.2. In above Theorem if we set

A = 1 and B = −1

we have the result given in [15].

Theorem 3.3. Let the function f given by (2.1) be in the class k-USs [A,B]. Then

∣∣µa22 −a3∣∣ ≤ 12




A−B
2
T1(k)

{
µ(A−B)

4
T1(k) −

(2D(k)−(B+1))T1(k))
2

}
(u > δ1)

A−B
2
T1(k) (δ1 ≤ u ≤ δ2)

A−B
2
T1(k)

{
(2D(k)−(B+1))T1(k))

2
− µ(A−B)

4
T1(k)

}
(u < δ2) ,

where

δ1 =
2 (2 + 2D(k) − (B + 1))T1(k))

(A−B)T1(k)
,

δ1 =
2 (2D(k) − (B + 1))T1(k) − 2)

(A−B)T1(k)
.

and T1, D(k) are defined by (1.6) and (1.7).



Int. J. Anal. Appl. 16 (5) (2018) 698

Proof. By definition of the class k-USs [A,B], there exists a function h ∈S, represented by h(z) = 1 + b1z +

b2z
2 + . . . and subordinate to qk, where qk is given by (2.2), such that

2zf′(z)

f(z) −f(−z)
= h(z) (∀ z ∈ E) .

Substituting the corresponding series expansions and by equating coefficients of z and z2, we obtain

a2 =
1

2
b1,

a3 =
1

2
b2.

Therefore ∣∣µa22 −a3∣∣ ≤ 12
∣∣∣∣µb212 − b2

∣∣∣∣ .
An application of Theorem 3.2, with u = µ

2
, we obtain the result asserted by Theorem 3.3. �

Theorem 3.4. A function f ∈ k-USs [A,B], if and only if

1

z

{
f(z) ∗

[
z −Mz2

(1 −z)2 (1 + z)

]}
6= 0 (∀ z ∈ E) , (0 < θ < 2π) , (3.6)

where

M =
(A + B + 2)pk(e

iθ) − (A + B − 2)
(A−B)pk(eiθ) + (A−B)

. (3.7)

Proof. If f ∈ k-USs [A,B], then we have

F(z) :=
2zf′(z)

f(z) −f(−z)
≺

(A + 1)pk(z) − (A− 1)
(B + 1)pk(z) − (B − 1)

(∀ z ∈ E) . (3.8)

For 0 ≤ k ≤ 1 the function pk(z) has a pole at z = 1 and the curve pk(eiθ), θ ∈ (0, 2π), is the imaginary axis,

a hyperbola or an ellipse. For k > 1 the function pk(z) is analytic on the unit disk. In each of the cases, if

f(z) ∈ k −USs [A,B], then F(|z| < 1) lies on the right with respect this curve, or

2zf′(z)

f(z) −f(−z)
6=

(A + 1)pk(e
iθ) − (A− 1)

(B + 1)pk(eiθ) − (B − 1)
(∀ z ∈ E and 0 < θ < 2π) .

A simple computations gives

1

z


 zf

′(z)
[
(B + 1)pk(e

iθ) − (B − 1)
]
− 1

2
[f(z) −f(−z)]×[

(A + 1)pk(e
iθ) − (A− 1) + (B + 1)pk(eiθ) − (B − 1)

]

 6= 0, (3.9)

for z ∈ E, θ ∈ (0, 2π). Using the convolution properties

f(z) ∗
z

(1 −z)2
= zf′(z) and f(z) ∗

z

1 −z2
=

1

2
[f(z) −f(−z)] (∀ z ∈ E) ,

we have that

1

z




f(z) ∗
[
z[(B+1)pk(eiθ)−(B−1)]

(1−z)2

−
z[(A+1)pk(eiθ)−(A−1)+(B+1)pk(eiθ)−(B−1)]

1−z2

]

 6= 0.



Int. J. Anal. Appl. 16 (5) (2018) 699

Hence it follows that

1

z


f(z) ∗ z −

(B+1)pk(e
iθ)−(B−1)+(A+1)pk(eiθ)−(A−1)

(B+1)pk(eiθ)−(B−1)−(A+1)pk(eiθ)−(A−1)
z2

(1 −z)2(1 + z)


 6= 0 (3.10)

for z ∈ E, θ ∈ (0, 2π), which is the required conditions (3.6) and (3.7).

Conversely, suppose that the condition (3.6) holds. Therefore we have

2zf′(z)

f(z) −f(−z)
6=

(A + 1)pk(e
iθ) − (A− 1)

(B + 1)pk(eiθ) − (B − 1)
(∀ z ∈ E) . (3.11)

Suppose that

H(z) =
(A + 1)pk(z) − (A− 1)
(B + 1))pk(z) − (B − 1)

(∀ z ∈ E) .

Now from relation (3.11), it is clear that

H (∂E) ∩F (E) = ∅.

Therefore the simply connected domain F (E) is contained in a connected component of C\H (∂E). The

univalence of the function H together with the fact H (0) = h (0) = 1 shows that F ≺ H which shows that

f ∈ k-USs [A,B]. �

In its special case when k = 0, Theorem 3.4 yields the following known result.

Corollary 3.1. For λ = 0,−1 ≤ B < A ≤ 1. A function f ∈ k-USλs [A,B] , if and only if

1

z


f(z) ∗


z + [(B+A+2)pk(eiθ)−(B+A−2)](B−A)(pk(eiθ)−1) z2

(1 −z)2 (1 + z)




 6= 0 (∀ z ∈ E and 0 ≤ θ < 2π) .

If, in Theorem 3.4, we set

−B = 1 = A and k = 0,

we obtain the following result.

Corollary 3.2. A function f ∈ 0-USs [1,−1], if and only if

1

z

{
f(z) ∗

[
z
(
1 −ze−iθ

)
(1 −z)2 (1 + z)

]}
6= 0 (∀ z ∈ E) , 0 < θ < 2π.

Theorem 3.5. If f ∈S, then f ∈ k-UCs [A,B], if and only if

1

z

{
f(z) ∗

1 + 2z3 + [M − 3] z2 − 3Mz4

z (1 −z)3 (1 + z)2

}
6= 0 (∀ z ∈ E) , 0 < θ < 2π,

where M is given by (3.7).



Int. J. Anal. Appl. 16 (5) (2018) 700

Proof. Let

g(z) =
z + Mz2

(1 −z)2 (1 + z)
,

then

zg′(z) =
z + Mz4 + (M + 2) z3 + (2M + 1) z2

(1 −z)3 (1 + z)2
.

Now using the Alexander type relation between k-USs [A,B] and k-UCs [A,B], the identity

zf′(z) ∗g(z) = f(z) ∗zg′(z),

and Theorem 3.4, we obtain the required result. �

4. Acknowledgements

The work here is supported by UKM Grant: GUP-2017-064.

Conflict of interest: There is no conflict of intrests between the authors, financial or whatsoever.

Declaration: All authors agreed with the contents of the manuscript.

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	1. Introduction and Definitions
	2. A Set of Lemmas
	3. Main Results
	4.  Acknowledgements
	References