International Journal of Analysis and Applications

Volume 18, Number 4 (2020), 614-623

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

DOI: 10.28924/2291-8639-18-2020-614

ON JANOWSKI CLOSE-TO-CONVEX FUNCTIONS ASSOCIATED WITH CONIC

REGIONS

AFIS SALIU∗, KHALIDA INAYAT NOOR

Department of Mathematics, COMSATS University Islamabad, Park Road, Tarlai Kalan, Islamabad 45550,

Pakistan

∗Corresponding author: saliugsu@gmail.com

Abstract. In this work, we introduce and investigate a class of analytic functions which is a subclass

of close-to-convex functions of Janowski type and related to conic regions. Length of the image curve

|z| = r < 1 under the generalized Janowski close-to-convex function is derived. Furthermore, rate of growth

of coefficients and Hankel determinant for this class are obtained. Relevant connections of our results with

the earlier known results are also pointed out.

1. Introduction

Let E = {z : |z | < 1} and H be the class of functions f(z) defined as

f(z) = z +

∞∑
n=2

anz
n (1.1)

which are analytic in E. A function f(z) is subordinate to another function g(z) (written as f(z) ≺ g(z)) if

there exists an analytic function w(z) in E with w(0) = 1 and |w(z)| < 1 for z ∈ E such that f(z) = g(w(z)).

Let Pm(α) be the class of analytic functions p(z) in E satisfying the condition p(0) = 1 and∫ 2π
0

∣∣∣Re p(z) −α
1 −α

∣∣∣dθ ≤ mπ, (1.2)
Received March 5th, 2020; accepted April 13th, 2020; published May 14th, 2020.

2010 Mathematics Subject Classification. 30C45, 30C50, 30C55.

Key words and phrases. analytic functions; open unit disk; Janowski functions; subordination; univalent functions; conic

domains.

©2020 Authors retain the copyrights

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

614

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-614


Int. J. Anal. Appl. 18 (4) (2020) 615

where m ≥ 2,z = reiθ, 0 ≤ α < 1, see [12]. The case α = 0 gives the class Pm introduced by Pinchuk [13].

For α = 0, m = 2, we obtain the well-known class P of Carathéodory functions and for m = 2, P2(α) ≡ P(α)

is the class of functions whose real parts are greater than α. It is known in [12] that p ∈ Pm(α) has the

integral representation

p(z) =
1

2π

∫ 2π
0

1 + (1 − 2α)ze−it

1 −ze−it
dv(t), (1.3)

where v(t) is a function of bounded variation on [0, 2π] such that∫ 2π
0

dv(t) = 2π and

∫ 2π
0

|dv(t)| ≤ mπ. (1.4)

It is seen from (1.3) and (1.4) that p ∈ Pm(α) has a representation

p(z) =
m + 2

4
p1(z) −

m− 2
4

p2(z), (1.5)

where pi ∈ P(α) for i = 1, 2.

Denote by CV ,S∗,K,CV (α),S∗(α), K(α), are the subclasses of S (the class of univalent functions in E)

which consist of functions that are convex, starlike, close-to-convex, convex of order α, starlike of order α

and close-to-convex of order α (0 ≤ α < 1) respectively. We have the following class of analytic functions in

E :

Vm(α) =
{
f ∈ H :

(zf′)′

f′
∈ Pm(α), z ∈ E, m ≥ 2, 0 ≤ α < 1

}
, see [12] (1.6)

and note that V2(α) ≡ CV (α) and V2(0) ≡ CV .

Recently, Noor [11] extended the conic domain Ωk,k ≥ 0 introduced by Kanas and Wisniowska [2, 3] to

that of Janowski type, Ωk[A,B], −1 ≤ B < A ≤ 1 and defined it as

Ωk[A,B] =
{
u + iv = [(B2 − 1)(u2 + v2) − 2(AB − 1)u + (A2 − 1)]2

>k2[(−2(B + 1)(u2 + v2) + 2(A + B + 2)u− 2(A + 1))2 + 4(A−B)2v2]
}
. (1.7)

Denoted by k − P(A,B), the class of functions p(z) that map E onto Ωk[A,B]. Equivalently, we say

p ∈ k −P(A,B) if and only if

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

, k ≥ 0, −1 ≤ B < A ≤ 1, (1.8)

where the definition of pk is given in [2]. Also, it is worthy mentioning that p ∈ k−P(A,B) ⊂ P(γ1) which

implies that p(z) = (1 −γ1)h1(z) + γ1, (see [11]) where h1 ∈ P and γ1 is given by

γ1 =
2k + 1 −A
2k + 1 −B

. (1.9)

If in (1.5), p1,p2 ∈ k−P(A,B), we say p ∈ k−Pm(A,B) and if Pm(α) in (1.6) is replaced with k−Pm(A,B),

we say f belongs to the class k −UVm(A,B). We note that k −Pm(A,B) ⊂ Pm(γ1), where γ1 is given by

(1.9). Thus,

k −UVm(A,B) ⊂ Vm(γ1).



Int. J. Anal. Appl. 18 (4) (2020) 616

We introduce the following class of functions.

Definition 1.1. Let f ∈ H,−1 ≤ B < A ≤ 1, −1 ≤ D < C ≤ 1, k ≥ 0 and m1,m2 ≥ 2. Then

f ∈ k −Hm1m2 (A,B,C,D) if there exists g ∈ k −UVm2 (C,D) such that
f′(z)
g′(z)

∈ k −Pm1 (A,B).

In particular,

(i) for k = 0, m1 = m = m2,A = 1,B = −1,C = 1 − 2α,D = −1, k − Hm1m2 (1,−1, 1 − 2α,−1) ≡

Hmm(α) is the class of analytic functions studied by Noor [9],

(ii) for k = 0, m1 = m = m2,A = 1,B = −1,C = 1,D = −1,

k −Hm1m2 (1,−1, 1,−1) ≡ Kmm is the class of analytic functions investigated by Noor [8],

(iii) for k = 0, m1 = 2 = m2,A = 1,B = −1,C = 1,D = −1,

k−H22(1,−1, 1,−1) ≡ K is the class of close to convex functions first introduced and examined by

Kaplan [4]

(iv) for k = 0, m1 = 2 = m2,H22(A,B,C,D) ≡ k − UK(A,B,C,D)) is the class of analytic functions

examined by Mahmood et al [5].

We note that k −Hm1m2 (A,B, 1,−1) ≡ Hm1m2 (γ1,σ), where σ =
k
k+1

.

2. Some Preliminary Lemmas

We need the following lemmas to investigate our results.

Lemma 2.1. [10] let p ∈ Pm(γ), 0 ≤ γ < 1,m ≥ 2. Then for z = reiθ,

(i)

1

2π

∫ 2π
0

|p(z) |2 dθ ≤
1 +

(
m2(1 −γ)2 − 1

)
r2

1 −r2
, (2.1)

(ii)

1

2π

∫ 2π
0

|p ′(z) | dθ ≤
m(1 −γ)

1 −r2
. (2.2)

Lemma 2.2. [12]

(i) f ∈ Vm(α) if and only if there exist f1,f2 ∈ S∗ such that

f′(z) =

(
f1(z)
z

)( m+2
4

)(1−α)

(
f2(z)
z

)( m−2
4

)(1−α)
. (2.3)

(ii) Let f ∈ Vm(α). Then

r

(
(1 −r)(

m−2
4

)

(1 + r)(
m+2

4
)

)(1−α)
≤ |zf′(z) | ≤ r

(
(1 + r)(

m−2
4

)

(1 −r)(
m+2

4
)

)(1−α)
. (2.4)



Int. J. Anal. Appl. 18 (4) (2020) 617

We will need the hypergeometric function

Γ(a)Γ(c−a)
Γ(c)

G(a,b,c; z) =

∫ 1
0

ua−1(1 −u)c−a−1(1 −zu)−b du. (2.5)

Unless otherwise stated, we assume, m1,m2 ≥ 2,k ≥ 0 − 1 ≤ B < A ≤ 1, and

− 1 ≤ D < C ≤ 1.

3. Main Results

Theorem 3.1. Let f ∈ k −Hm1m2 (A,B,C,D). Then for 0 < r < 1,

L(r,f) ≤ π

{
C(m2,k,γ2,C,D)M(r) log

1

1 −r
+

2b+1γ1
a

[
[G(a,b,c,−1) − 2G(a, 1 + b,c− 1)] (3.1)

+ra1 [2G(a, 1 + b,c,−r1) −G(a,b,c,−r1)]

]}
, (3.2)

where M(r) = max
θ
|f(reiθ)|, C(m2,k,γ2,C,D) is a constant depending on m2,k,γ2,C and D,

a =
(m2

2
− 1
)

(1 −γ2), b = 2(γ2 − 1), c = a + 1 and r1 =
1 −r
1 + r

,

where

γ1 =
2k + 1 −A
2k + 1 −B

, γ2 =
2k + 1 −C
2k + 1 −D

. (3.3)

Proof. Let z = reiθ. Then

L(r,f) =

∫ 2π
0

|zf′(z) |dθ

=

∫ 2π
0

|zg′(z)p(z) |dθ, where g ∈ k −Vm2 (C,D) and p ∈ k −Pm1 (A,B)

≤

r∫
0

2π∫
0

(zg′(z))′p(z) |dθdρ +

r∫
0

2π∫
0

|zg′(z)p ′(z) |dθdρ

=J1(r) + J2(r). (3.4)

Let

(zg′)′(z)

g′(z)
= H(z) = 1 +

∞∑
n=1

dnz
n ∈ k −Pm2 (C,D).

Then by Schwarz inequality and Perseval’s theorem, we have

J1(r) ≤2π




r∫
0

2π∫
0

|f′(z) |2 dθdρ




1
2



r∫
0

2π∫
0

|H(z) |2 dθdρ




1
2

=2π

(∫ r
0

∞∑
n=1

n2|an |2ρ2n−2dρ

)1
2
(∫ r

0

∞∑
n=0

|dn |2ρ2ndρ

)1
2

.



Int. J. Anal. Appl. 18 (4) (2020) 618

It is easy to see that

|dn | ≤
m2(C −D)|δk |

4
,

where δk has its definition given in [11]. Therefore,

J1(r) ≤
√

2πm2(C −D)|δk |
4

(
1

r

∞∑
n=1

n2

2n− 1
|an |2r2n

)1
2
(

log
1 + r

1 −r

)1
2

≤
√

2πm2(C −D)|δk |
4

M(r)
(1
r

log
1 + r

1 −r

)1
2

, (3.5)

where we used the fact that A(r) = π
∞∑
n=1

n|an |2r2n is the area of the image of |z | < r bounded by w = f(z)

and A(r) ≤ πM2(r).

Next, we estimate J2(r). Since p ∈ k −Pm1 (A,B) ⊂ Pm1 (γ1), then using (1.3), we get

p ′(z) =
(1 −γ1)

π

∫ 2π
0

eit

(1 −zeit)2
dv(t) and

∫ 2π
0

1 −ρ2

|1 −zeit |2
dv(t) =

2π(Rep(z) −γ1)
1 −γ1

.

Therefore,

J2(r) ≤
(1 −γ1)

π

r∫
0

2π∫
0

2π∫
0

|zg′(z) |
|1 −ze−it |2

dv(t)dθdρ

=2

r∫
0

2π∫
0

|zg′(z) |(Rep(z) −γ1)
1

1 −ρ2
dθdρ

=2

r∫
0

2π∫
0

Re
(
zg′(z)e−i arg zg

′(z)p(z)
) 1

1 −ρ2
dθdρ− 2γ1

r∫
0

2π∫
0

|zg′(z) |dθdρ.

Integration by parts, application of (1.2) and Lemma 2.2(ii) give

J2(r) ≤2π(m2(1 −γ2) + 2γ2)
∫ r
0

M(ρ)

1 −ρ2
dρ− 4πγ1

∫ r
0

ρ
(1 −ρ)(

m−2
4

)(1−γ2)−1

(1 + ρ)(
m+2

4
)(1−γ2)+1

dρ

≤π(m2(1 −γ2) + 2γ2)M(r) log
1 + r

1 −r
+ 4πγ1(L1(r) −L2(r)), (3.6)

where

L1(r) =

∫ r
0

(1 −ρ)(
m−2

4
)(1−γ2)−1

(1 + ρ)(
m+2

4
)(1−γ2)+1

dρ and L2(r) =

∫ r
0

(1 −ρ)(
m−2

4
)(1−γ2)−1

(1 + ρ)(
m+2

4
)(1−γ2)

dρ.

Let u = 1−ρ
1+ρ

, so that dρ = − 2
(1+u)2

. Then

L1(r) =

(
1

2

)2(2−γ2)−1 [∫ 1
0

u(
m2
2
−1)(1−γ2)−1(1 + u)2(1−γ2)du−

∫ r1
0

u(
m2
2
−1)(1−γ2)−1(1 + u)2(1−γ2)du

]
=

1

a
G(a,b,c,−1) −

∫ r1
0

u(
m2
2
−1)(1−γ2)−1(1 + u)2(1−γ2)du, (3.7)

where a,b,c and r1 are given in Theorem 3.1. For the second integral in (3.7), we let u = r1v. Then

L1(r) =
2b−1

a
[G(a,b,c,−1) −ra1G(a,b,c,−r1)] (3.8)



Int. J. Anal. Appl. 18 (4) (2020) 619

In a similar way, we obtain

L2(r) =
2b

a
[G(a, 1 + b,c,−1) −ra1G(a, 1 + b,c,−r1)] . (3.9)

Using (3.8), (3.9) in (3.6), we get

J2(r) ≤π(m2(1 −γ2) + 2γ2)M(r) log
1 + r

1 −r
+
π2b+1γ1

a

{
[G(a,b,c,−1) − 2G(a, 1 + b,c− 1)]

+ ra1 [2G(a, 1 + b,c,−r1) −G(a,b,c,−r1)]

}
. (3.10)

The estimates for J1(r) and J2(r) yield the required result. �

Corollary 3.1. Let f ∈ Km1m2, Then for 0 ≤ r < 1,

L(r,f) ≤ C(m2)M(r) log
1

1 −r
,

where M(r) = max
θ
|f(reiθ)|, C(m2) is a constant depending on m2.

Corollary 3.2. Let f ∈ Hmm(α), Then for 0 ≤ r < 1,

L(r,f) ≤ C(m,α)M(r) log
1

1 −r
,

where M(r) = max
θ
|f(reiθ)|, C(m,α) is a constant depending on m and α.

Corollary 3.3. Let f ∈ K. Then for 0 ≤ r < 1,

L(r,f) ≤ CM(r) log
1

1 −r
,

where M(r) = max
θ
|f(reiθ)|, C is a constant.

Theorem 3.2. Let f(z) be of the form (1.1) and f ∈ k −Hm1m2 (A,B,C,D). Then

|an | ≤
π

n

(
C1(m2,k,γ2,C,D)M

(n− 1
n

)
log n +

2b+1γ1
a

{
[G(a,b,c,−1) − 2G(a, 1 + b,c− 1)]

+ ra1

[
2G

(
a, 1 + b,c,−

1

2n− 1

)
−G

(
a,b,c,−

1

2n− 1

)]})
,

where γ1,γ2,a,b and c are given as in Theorem 3.1.

Noonan and Thomas [6] define for q ≥ 1,n ≥ 1, the qth Hankel determinant of f(z) ∈ H as follows :

Hq(n) =

∣∣∣∣∣∣∣∣∣∣∣∣∣

an an+1 . . . an+q−1

an+1 an+2 . . . an+q−2
...

...
...

...

an+q−1 an+q−2 . . . an+2q−2

∣∣∣∣∣∣∣∣∣∣∣∣∣
. (3.11)



Int. J. Anal. Appl. 18 (4) (2020) 620

To estimate the growth rate of Hankel determinant for f ∈ k − Hm1m2 (A,B,C,D), we need the following

results due to Noonan and Thomas [6].

Lemma 3.1. let f ∈ H and suppose the qth Hankel determinant of f(z) for

q ≥ 1,n ≥ 1 is given by (3.11). Then writing ∆j(n) = ∆j(n,z1,f), we have

Hq(n) =

∣∣∣∣∣∣∣∣∣∣∣∣∣

∆2q−2(n) ∆2q−3(n + 1) . . . ∆q−1(n + q − 1)

∆2q−3(n + 1) ∆2q−4(n + 2) . . . ∆q−2(n + q − 2)
...

...
...

...

∆q−1(n + q − 1) ∆q−2(n + q − 2) . . . ∆q(n + 2q − 2)

∣∣∣∣∣∣∣∣∣∣∣∣∣
, (3.12)

where with ∆0(n,z1,F) = an, we define for j ≥ 1,

∆j(n,z1,f) = ∆j−1(n,z1,f) −z1∆j−1(n + 1,z1,f). (3.13)

Lemma 3.2. With x = ( n
n+1

)y,u ≥ 0 an integer,

∆j(n + u,u,x,zf
′(z)) =

j∑
i=0

(
j

i

)
yi(u− (i− 1)n)

(n + 1)i
· ∆j−i(n + u + i,y,f). (3.14)

Remark 3.1. Consider any determinant of the form

D =

∣∣∣∣∣∣∣∣∣∣∣∣∣

y2q−2 y2q−3 . . . yq−1

y2q−3 y2q−4 . . . yq−2
...

...
...

...

yq−1 yq−2 . . . y0

∣∣∣∣∣∣∣∣∣∣∣∣∣
. (3.15)

with 1 ≤ i, j ≤ q and αij = y2q−(i+j), D = det(αij). Thus

D =
∑
v1∈Sq

(sgn v1)

q∏
j=1

(y2q − (v1(j) + j) ,

where Sq is the symmetric group on q elements and sgn v1 is either +1 or −1. Thus, in the expansion of D,

each summand has q factor and the sum of the subscripts of the factor of each summand is q2 −q.

Now let n be given and Hq(n) is as Lemma 3.1, then each summand in the expression of Hq(n) is of the

form
q∏
i=1

∆v1(i) (n + 2q − 2 −v1i) ,

where v1 ∈ Sq and
q∑
i=1

v1(i) = q
2 −q; 0 ≤ v1(i) ≤ 2q − 2.



Int. J. Anal. Appl. 18 (4) (2020) 621

Theorem 3.3. Let f ∈ Hm1m2 (γ1,σ) and (m2 + 2)(1−γ2) ≥ 4j. If the qth Hankel determinant of f(z) for

q ≥ 1,n ≥ 1 is given by (3.11), then

Hq(n) = O(1)


 n

(
m2
2

+1)( 1k+1 )−1, q = 1,

n(
m2
2

+1)( 1k+1 )q−q
2

, q ≥ 2, m2 ≥ 8(k + 1)(q − 1) − 2
, (3.16)

where O(1) is a constant that depends on m1,m2,j,γ1,k only, with γ1 given by (3.3).

Proof. Since f ∈ Hm1m2 (γ1,σ), then

f′(z) = p(z)g′(z),

where g′(z) ∈ k −Vm2 (1,−1) ⊂ Vm2 (σ) and p(z) ∈ k −Pm1 (A,B) ⊂ Pm1 (γ1). Setting

F(z) = (zf′(z))′, and
(zg′(z))′

g′(z)
= h(z),

then

F(z) = g′(z)(h(z)p(z) + zp′(z)).

Now, for j ≥ 0,z1 any nonzero complex number, consider ∆j(n,z1,F(z)) as defined by (3.13). Then

∆j(n,z1,F(z)) ≤
1

2πrn+j

∣∣∣∣∣
∫ 2π
0

(z −z1)jF(z)e−i(n+j)θ dθ

∣∣∣∣∣
≤

1

2πrn+j

∫ 2π
0

|z −z1 |j|g′(z) ||h(z)p(z) + zp ′(z) |dθ.

Using Lemma 2.2(i) and the distortion theorems for starlike function, then for (m2 + 2)(1 − σ) ≥ 4j, we

obtain

∆j(n,z1,F(z)) ≤
1

2πrn+j

∫ 2π
0

|(z −z1)f1(z)|j
|f1(z)|(

m2+2
4 )(1−σ)−j

|f2(z)|(
m2−2

4 )(1−σ)
|h(z)p(z) + zp′(z)|dθ

≤
1

2πrn+j−σ

2π∫
0

|(z −z1)f1(z)|j
(

r

(1 −r)2

)( m2+2
4

)(1−σ)−j (
(1 + r)2

r

)( m2−2
4

)(1−σ)

×|h(z)p(z) + zp ′(z) |dθ.

Using the result of Golusin [1] and Schwarz inequality, we arrive at

|∆j(n,z1,F(z)) | ≤
2(

m2−2
2 )(1−σ)+j

rn−1

( 1
1 −r

)( m2−22 )(1−σ)−j
×

{( 1
2π

∫ 2π
0

|h(z) |2
)1

2
( 1

2π

∫ 2π
0

|p(z) |2
)1

2

+
1

2π

∫ 2π
0

|zp ′(z) |

}
dθ.



Int. J. Anal. Appl. 18 (4) (2020) 622

In view of Lemma 2.1, we get

|∆j(n,z1,F(z)) | ≤
2(

m2−2
2 )(1−σ)+j

rn−1

(
1

1 −r

)( m2+22 )(1−σ)−j {[1 + (m22(1 −σ)2 − 1)r2
1 −r2

]1
2

×

[
1 +

(
m21(1 −γ1)2 − 1

)
r2

1 −r2

]1
2

+
rm1(1 −γ1)

1 −r2

}

≤
2(

m2−2
2 )(1−γ2)+j

(
m1(1 −γ1) + (m21(1 −γ1)2 + 1)

1
2 (m22(1 −σ)2 + 1)

1
2

)
rn−1

×
( 1

1 −r

)( m2−22 )(1−σ)−j+1
.

Applying Lemma 3.2 with z1 =
(

n
1+n

)2
eiθn, (n →∞), r = 1 − 1

n
, we have for (m2 + 2)(1 −γ2) ≥ 4j,

∆j(n,e
iθn,F(z)) = O(1)n(

m2+2
2 )(1−σ)−j+1,

where O(1) is a constant that depends on m1,m2,γ1 and σ. We estimate the rate of growth of Hq(n) for

f ∈ Hm1m2 (γ1,σ). Then, for q=1, H1(n) = an = ∆0(n) and

H1(n) = O(1)n(
m2+2

2 )(
1

1+k
)−1.

For q ≥ 2, we use similar arguments from Noonan and Thomas [6] along with Lemma 3.1 and Remark 3.1

to arrive at

Hq(n) = O(1)n(
m2+2

2 )(
1

1+k
)q−q2, m2 ≥ 8(k + 1)(q − 1) − 2.

�

Corollary 3.4. [8] If f ∈ Kmm, then

Hq(n) = O(1)


 n

m2
2 , q = 1,

n(
m2
2

+1)q−q2, q ≥ 2, m2 ≥ 8(q − 1) − 2
,

where O(1) is a constant that depends on m and j, only.

Corollary 3.5. If f ∈ 1 −Hm1m2(A,B, 1,−1), then

Hq(n) = O(1)


 n

m2
4
−1

2 , q = 1,

n(
m2
4

+ 1
2 )q−q

2

, q ≥ 2, m2 ≥ 16(q − 1) − 2
,

where O(1) is a constant that depends on γ1,m1 and j, only.



Int. J. Anal. Appl. 18 (4) (2020) 623

4. Conclusion

Arc length and rate of growth of Hankel determinant problems have always been the main interests of many

researchers in Geometric function theory. Many studies associated to these problems revolved around classes

of normalized analytic univalent functions. In this particular work, length of the image curve |z| = r < 1

under the generalized Janowski close-to-convex function was proved; rate of growth of coefficients and Hankel

determinant for this class were also obtained.

Acknowledgements

The authors would like to thank Prof. Dr. Raheel Qamar, Rector, COMSATS University Islamabad,

Pakistan, for providing excellent research and academic environments.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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	1. Introduction
	2. Some Preliminary Lemmas
	3. Main Results
	4. Conclusion
	Acknowledgements
	References