International Journal of Analysis and Applications

Volume 18, Number 5 (2020), 849-858

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

DOI: 10.28924/2291-8639-18-2020-849

ON SOME SUBCLASSES OF STRONGLY STARLIKE ANALYTIC FUNCTIONS

EL MOCTAR OULD BEIBA∗

El Moctar Ould Beiba, Department of Mathematics and Computer Sciences, Faculty of Sciences and

Techniques, University of Nouakchott Al Aasriya, P.O. Box 5026, Nouakchott, Mauritania

∗Corresponding author: elbeiba@yahoo.fr

Abstract. The aim of the present article is to investigate a family of univalent analytic functions on the

unit disc D defined for M ≥ 1 by

<
(zf′(z)
f(z)

)
> 0,

∣∣∣∣
(
zf′(z)

f(z)

)2
−M

∣∣∣∣ < M, z ∈ D.
Some proprieties, radius of convexity and coefficient bounds are obtained for classes in this family.

1. Introduction

Let A be the set of analytic function on the unit disc D with the normalization f(0) = f′(0) − 1 = 0.

f ∈A if f is of the form

(1.1) f(z) = z +

+∞∑
n=2

anz
n, z ∈ D.

S denotes the subclass of A of univalent functions. A function f ∈S is said to be strongly starlike of order

α, 0 < α ≤ 1, if it satisfies the condition

∣∣Argzf′(z)
f(z)

∣∣ < απ
2
, ∀z ∈ D.

Received April 12th, 2020; accepted May 8th, 2020; published July 28th, 2020.

2010 Mathematics Subject Classification. 30C45.

Key words and phrases. strongly Starlike Functions; Subordination; Radius of Convexity; Coefficient Bounds.

©2020 Authors retain the copyrights

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

849

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-849


Int. J. Anal. Appl. 18 (5) (2020) 850

This class is denoted by SS∗(α) and was first introduced by D. A. Brannan and W. E. Kirwan [1] and

independently by J. Stankiewicz [9].

SS∗(1) is the well known class S∗ of starlike functions. Recall that a function f ∈S belongs to S∗ if the

image of D under f is a starlike set with respect to the origin or, equivalently, if

<
(zf′(z)
f(z)

)
> 0, z ∈ D.

A function f ∈S belongs to SS∗(α) if the image of D under zf
′(z)

f(z)
lies in the angular sector

Ωα =

{
z ∈ C,

∣∣Argz∣∣ < απ
2

}
.

Let B denotes the set of Schwarz functions, i.e. ω ∈ B if and only ω is analytic in D, ω(0) = 0 and∣∣ω(z)∣∣ < 1 for z ∈ D. Given two functions f and g analytic in D, we say that f is subordinate to g and we
write f ≺ g if there exists ω ∈B such that f = g ◦ω in D.

If g is univalent on D, f ≺ g is equivalent to f(0) = g(0) and f(D) ⊂ g(D).

We obtain from the Schwarz lemma that if f ≺ g then
∣∣f′(0)∣∣ ≤ ∣∣g′(0)∣∣. As a consequence of this statement,

we have

(1.2) f, g ∈A ,
f(z)

z
≺
g(z)

z
=⇒

∣∣a2∣∣ ≤ ∣∣b2∣∣,
where a2 and b2 are respectively the second coefficients of f and g.

W. Janowski [2] investigated the subclass

S∗(M) =

{
f ∈S,

zf′(z)

f(z)
∈DM,∀z ∈ D

}
,

where

DM =
{
w ∈ C,

∣∣w −M∣∣ < M}, M ≥ 1
J. Sókol and J. Stankiewicz [8] introduced a subclass of SS∗( 1

2
), namely, the class S∗L defined by

S∗L =
{
f ∈S,

zf′(z)

f(z)
∈L1,∀z ∈ D

}
,

where

L1 =
{
w ∈ C,<w > 0,

∣∣w2 − 1∣∣ < 1}.
L1 is the interior of the right half of the Bernoulli’s lemniscate

∣∣w2 − 1∣∣ = 1.
In the present paper we are interested to the family of subclass of S

(1.3) S∗L(M) =
{
f ∈S,

zf′(z)

f(z)
∈LM,∀z ∈ D

}
, M ≥ 1,



Int. J. Anal. Appl. 18 (5) (2020) 851

where

(1.4) LM =
{
w ∈ C,<w > 0,

∣∣w2 −M∣∣ < M}.
is the interior of the right half of the Cassini’s oval

∣∣w2 −M∣∣ = M. For the particular case M = 1, S∗L(1)
stands for the class S∗L introduced by J. Sókol and J. Stankiewicz [8]. Since LM ⊂ Ω(

1
2
), all functions in

S∗L(M) are strongly starlike of order
1
2
.

Note that all classes above correspond to particular cases of the classes of S∗(ϕ) introduced by W. Ma

and D. Minda [3],

S∗(ϕ) =
{
f ∈A,

zf′(z)

f(z)
≺ ϕ

}
.

where ϕ is Analytic univalent function with real positive part in the unit disc D, ϕ
(
D
)

is symmetric with

respect to the real axis and starlike with respect to ϕ(0) = 1 and ϕ′(0) > 0.

Let m = 1 − 1
M

and ϕm be the function

ϕm(z) =

√
1 + z

1 −mz
, z ∈ D

where the branch of the square root is chosen so that ϕm(0) = 1. We have

(1.5) S∗L(M) = S
∗(ϕm) =

{
f ∈A,

zf′(z)

f(z)
≺ ϕm

}
.

Observe that S∗L corresponds to m = 0 so that S
∗
L = S

∗(
√

1 + z).

2. Some properties of the class S∗L(M)

Let P the class of analytic functions p in D with p(0) = 1 and <p(z) > 0 in D. For M ≥ 1, let

PL(M) =

{
p ∈ P,

∣∣p2(z) −M∣∣ < M, z ∈ D}.
It is easy to see that PL(M1) ⊂ PL(M2) for M1 ≤ M2.

Remark 2.1. A function f ∈A belongs to S∗L(M) if and only if there exists p ∈ PL(M) such that

zf′(z)

f(z)
= p(z), z ∈ D.

Theorem 2.1. A function f belongs to S∗L(M) if and only if there exists p ∈ PL(M) such that

(2.1) f(z) = z exp

∫ z
0

p(ξ) − 1
ξ

dξ.

Proof. (2.1) is an immediate consequence of the Remark 2.1 �



Int. J. Anal. Appl. 18 (5) (2020) 852

Let fm ∈A be the unique function such that

(2.2)
zf
′

m(z)

fm(z)
= ϕm(z), z ∈ D

with m = 1 − 1
M

. fm belongs to S∗L(M) and we have

(2.3) fm(z) = z exp

∫ z
0

ϕm(ξ) − 1
ξ

dξ.

Evaluating the integral in (2.3), we get

(2.4) fm(z) =
4z exp

∫ϕm(z)
1

Hm(t)dt(
ϕm(z) + 1

)2 , z ∈ D,
where

Hm(t) =
2mt + 2

mt2 + 1
, m = 1 −

1

M

For M = 1, H0 is the constant function H(t) = 2 and we have

f0(z) =
4z exp

(
2
√

1 + z − 2
)

(√
1 + z + 1

)2 for z ∈ D.
f0 is extremal function for problems in the class S∗L (see [8]).

It is easy to see that

(2.5) fm(z) = z +
m + 1

2
z2 +

(m + 1)(5m + 1)

16
z3 +

(m + 1)(21m2 + 6m + 1)

96
z4 + . . .

We need the following result by St. Ruscheweyh [5]

Lemma 2.1. [ [5], Theorem 1] Let G be a convex conformal mapping of D, G(0) = 1, and let

F(z) = z exp

∫ z
0

G(ξ) − 1
ξ

dξ.

Let f ∈A. Then we have
zf′(z)

f(z)
≺ G

if and only if for all
∣∣s∣∣ ≤ 1, ∣∣t∣∣ ≤ 1

tf(sz)

sf(tz)
≺
tF(sz)

sF(tz)
.

Theorem 2.2. If f belongs to S∗L(M) then

(2.6)
f(z)

z
≺
fm(z)

z
.

Proof. From (1.5), we obtain by applying Lemma 2.1 to the convex univalent function G = ϕm,

tf(z)

f(tz)
≺
tfm(z)

fm(tz)
.

Letting t −→ 0, we obtain the desired conclusion. �



Int. J. Anal. Appl. 18 (5) (2020) 853

Corollary 2.1. Let f belongs to S∗L(M) and |z| = r < 1, then

(2.7) −fm(−r) ≤ |f(z)| ≤ fm(r);

(2.8) f
′

m(−r) ≤ |f
′
(z)| ≤ f

′

m(r).

Proof. (2.7) follows from (2.6). Now If M ≥ 1 we have 0 ≤ m < 1. Thus for 0 ≤ r < 1

(2.9) min
|z|=r

|ϕm(z)| = ϕm(−r), max
|z|=r

|ϕm(z)| = ϕm(r)

From (2.6) and (2.9) we get (2.8) by applying Theorem 2 ( [3], p. 162). �

3. Radius of convexity for the class S∗L(M)

In the sequel m = 1 − 1
M
.

For M ≥ 1, let P(M) be the family of analytic functions P in D satisfying

(3.1) P(0) = 1, |P(z) − M| < M, for z ∈ D.

We have

(3.2) f ∈S∗L(M) ⇐⇒ ∃P ∈P(M) /
zf′(z)

f(z)
=
√
P.

We need the two following lemmas by Janowski [2]:

Lemma 3.1. [ [2] , Theorem 1] For every P(z) ∈P(M) and |z| = r, 0 < r < 1, we have

(3.3) inf
P∈P(M)

<P(z) =
1 −r

1 + mr
.

The infimum is attained by

(3.4) P(z) =
1 − �z

1 + �mz
, |�| = 1.

Lemma 3.2. (Theorem 2, [2]) For every P(z) ∈P(M) and |z| = r, 0 < r < 1, we have

(3.5) inf
P∈P(M)

<
zP
′
(z)

P(z)
= −

(
1 + m

)
r(

1 −r
)(

1 + mr
).

The infimum is attained by

(3.6) P(z) =
1 − �z

1 + �mz
, |�| = 1.

Theorem 3.1. The radius of convexity of the class S∗L(M) is is the unique root in (0, 1) of the equation

(3.7) 4
(
1 + mr

)(
1 −r

)3 −(1 + m)2r2 = 0.



Int. J. Anal. Appl. 18 (5) (2020) 854

Proof. Let f ∈S∗L(M). From (3.2), there exists P ∈P(M) such that

(3.8)
zf
′
(z)

f(z)
=
√
P(z), z ∈ D.

(3.8) can be written

zf
′
(z) = f(z)

√
P(z)

which gives

1 +
zf
′′
(z)

f
′
(z)

=
z
(
zf
′)′

(z)

zf
′
(z)

=
√
P(z) +

1

2

zP
′
(z)

P(z)
.

This yields for |z| = r, 0 < r <,

(3.9) <
(
1 +

zf
′′
(z)

f
′
(z)

)
≥ inf
P∈P(M)

<
√
P(z) +

1

2
inf

P∈P(M)
<
zP
′
(z)

P(z)
.

Replacing (3.3) and(3.5 in (3.9), we obtain

(3.10) <
(
1 +

zf
′′
(z)

f
′
(z)

)
≥
√

1 −r
1 + mr

−
1

2

(
1 + m

)
r(

1 −r
)(

1 + mr
)

Let h
M

be defined by

h
M

=

√
1 −r

1 + mr
−

1

2

(
1 + m

)
r(

1 −r
)(

1 + mr
).

h
M

is decreasing in the interval (0, 1) , h
M

(0) = 1 and the limit of h
M

in 1− is −∞. Let r
M−1 be the unique

solution of h
M

(r) = 0 in (0, 1), then f is convex on the disc |z| < r
M−1 . On the other hand,

1 +
zf
′′

m(z)

f
′
m(z)

=

√
1 + z

1 −mz
+

1

2

(
1 + m

)
z(

1 −mz
)(

1 + z
)

vanishes in z = −r
M−1 . Thus rM−1 is the best value.

To finish, we observe that the equation h
M

(r) = 0 is equivalent in the interval (0, 1) to the equation

4
(
1 + mr

)(
1 −r

)3 −(1 + m)2r2 = 0.
�

Remark 3.1. As a consequence of Theorem 3.1 applying for M = 1, we find Theorem 4 [8] which gives r0

the radius of convexity of the class S∗L. r0 =
1
12

(
11 +

3
√√

44928 − 181 − 3
√√

44928 + 181

)
≈ 0.5679591

Remark 3.2. As observed above, S∗L(M) increases with M. Therefore rM−1 decreases when M increases.

Let

r∞ = lim
M→+∞

r
M−1.

Substituting in (3.7), we obtain



Int. J. Anal. Appl. 18 (5) (2020) 855

(
1 + r∞

)(
1 −r∞

)3 −r2∞ = 0.
Solving this equation in (0, 1), we get

r∞ =
1

2

(
1 −
√

2 +

√√
8 − 1

)
≈ 0.46899.

We have

r∞ ≤ rM−1 ≤ r0.

4. Coefficient bounds for S∗L(M)

Theorem 4.1. Let f(z) =
∑∞
n=0 anz

n be a function in S∗L(M) . Then

for 1 ≤ M ≤ 2 we have

(4.1)

∞∑
n=2

(
(1 −m)n2 − 2

)
|an|2 ≤ 1 + m

and for M > 2 we have

(4.2)
∑

n≥
√

2
1−m

(
(1 −m)n2 − 2

)
|an|2 ≤ 1 + m−

∑
2≤k<

√
2

1−m

(
(1 −m)k2 − 2

)
|ak|2.

with m = M−1
M

.

Proof. If f ∈S∗L(M) there exists ω ∈B such that

(4.3)
(
1 −mω(z)

)(
zf
′
(z)
)2 −f(z)2 = ω(z)f(z)2, z ∈ D.

For 0 < r < 1 we have

2π

∞∑
n=1

|an|2r2 =
∫ 2π
0

|f
(
reiθ

)
|2dθ

≥
∫ 2π
0

|ω
(
reiθ

)
||f
(
reiθ

)
|2dθ(4.4)

Replacing (4.3) in the right side of (4.5) we obtain

2π

∞∑
n=1

|an|2r2 ≥
∫ 2π
0

∣∣(1 −mω(reiθ))(reiθf′(reiθ))2 −f(reiθ)2∣∣dθ
≥

∫ 2π
0

∣∣(1 −mω(reiθ))(reiθf′(reiθ))2∣∣dθ −∫ 2π
0

∣∣f(reiθ)2∣∣dθ
≥ (1 −m)

∫ 2π
0

∣∣(reiθf′(reiθ))2∣∣dθ −∫ 2π
0

∣∣f(reiθ)2∣∣dθ
= 2π

∞∑
n=1

(1 −m)n2|an|2r2 − 2π
∞∑
n=1

|an|2r2.



Int. J. Anal. Appl. 18 (5) (2020) 856

Thus

2

∞∑
n=1

|an|2r2 ≥
∞∑
n=1

(1 −m)n2|an|2r2.

If we let r → 1−, we obtain from le last inequality

2

∞∑
n=1

|an|2 ≥
∞∑
n=1

(1 −m)n2|an|2

which gives,

(4.5) 1 + m ≥
∞∑
n=2

(
(1 −m)n2 − 2

)
|an|2.

Since (1 −m)n2 − 2 ≥ 0 for all n ≥ 2 if and only if 1 ≤ M ≤ 2 then (4.5) yields (4.1) and (4.2) according to

the case 1 ≤ M ≤ 2 or M > 2. �

The following corollary is an immediate consequence of (4.2).

Corollary 4.1. Let f(z) =
∑∞
n=0 anz

n be a function in S∗L(M) .Then

for 1 ≤ M ≤ 2 we have

(4.6) |an| ≤

√
1 + m

(1 −m)n2 − 2
, for n ≥ 2

and for M > 2 we have

(4.7) |an| ≤

√√√√1 + m−∑2≤k<√ 21−m ((1 −m)k2 − 2)|ak|2(
1 −m

)
n2 − 2

; for n ≥
√

2

1 −m
.

with m = M−1
M

.

Remark 4.1. For M = 1, (4.1) and (4.6) give respectivly Theorem 1 and Corollary 1 [6].

Theorem 4.2. Let f(z) =
∑∞
n=0 anz

n be a function in S∗L(M) .Then

(i) |a2| ≤ m+12 , for 0 ≤ m ≤ 1;

(ii) |a3| ≤ m+14 , for 0 ≤ m ≤
3
5
;

(iii) |a4| ≤ m+16 , for 0 ≤ m ≤
√
3−1
7

.

This estimations are sharp.

Proof. If f ∈S∗L(M) there exists ω(z) =
∑∞
n=1 Cnz

n ∈B such that

(4.8)
(
zf
′
(z)
)2 −f(z)2 = ω(z)(m(zf′(z))2 + f(z)2), z ∈ D.

Let f(z)2 =
∑∞
n=2 Anz

n,
(
zf
′
(z)
)2

=
∑∞
n=2 Bnz

n. (4.8) becomes

(4.9)

∞∑
n=2

(
Bn −An

)
zn =

( ∞∑
n=2

(
mBn + An

)
zn
)( ∞∑

n=1

Cnz
n

)



Int. J. Anal. Appl. 18 (5) (2020) 857

Equating coefficients for n = 2, n = 3 in both sides of (4.9), we obtain

(Sm)




B3 −A3 =
(
mB2 + A2

)
C1

B4 −A4 =
(
mB2 + A2)C2 +

(
mB3 + A3

)
C1

B5 −A5 =
(
mB2 + A2)C3 +

(
mB3 + A3

)
C2 +

(
mB4 + A4

)
C1

A little calculation yields

A2 = a1 = 1, A3 = 2a2, A4 = 2a3 + a
2
2, A5 = 2a4 + 2a2a3

and

B2 = a1 = 1, B3 = 4a2, B4 = 6a3 + 4a
2
2, B5 = 8a4 + 12a2a3.

Replacing in (Sm), we obtain


(1) 2a2 =
(
m + 1

)
C1

(2) 4a3 + 3a
2
2 =

(
m + 1)C2 +

(
4m + 2)a2C1

(3) 6a4 + 10a2a3 =
(
m + 1

)
C3 +

(
2m + 1

)(
m + 1

)
C1C2 +

(
(6m + 2)a3 + (4m + 1)a

2
2

)
C1

Since |C1| ≤ 1 then (1) implies that |a2| ≤ 1+m2 . This proves the assertion (i). On the other hand we have

from (1) and (2)

a3 =
1 + m

4
C2 +

(5m + 1)(m + 1)

16
C21.

Thus

|a3| ≤
1 + m

4

(
|C2| +

5m + 1

4
|C1|

)
.

It is well known that |C2| ≤ 1 −|C1|2. Therefore we obtain

|a3| ≤
1 + m

4

(
1 −|C1|2 +

5m + 1

4
|C1|

)
=

1 + m

4

(
1 +

5m− 3
4
|C1|

)
.(4.10)

Since 5m− 3 ≤ 0 if and only if m ≤ 3
5

then (4.10) yields the assertion (ii).

Replacing the values of a2 and a3 in the equation (3), we obtain

a4 =

(
m + 1

)
6

C3 +

(
m + 1

)(
9m + 1

)
24

C1C2 +

(
m + 1

)(
21m2 + 6m + 1

)
96

C31

=
m + 1

6

(
C3 +

9m + 1

4
C1C2 +

21m2 + 6m + 1

16
C31

)
.(4.11)

Let µ = 9m+1
4

and ν = 21m
2+6m+1
16

. Under the assumption 0 ≤ m ≤
√
3−1
7

, we have (µ,ν) ∈ D1 (see [4], p.

127). Therefore by Lemma 2 [4] we obtain∣∣∣∣C3 + 9m + 14 C1C2 + 21m
2 + 6m + 1

16
C31

∣∣∣∣ ≤ 1



Int. J. Anal. Appl. 18 (5) (2020) 858

which yields from (4.11) the assertion (iii).

The sharpness of (i) is given by the function fm. If we take in (4.8) ω(z) = z
2 and ω(z) = z3 successively,

we obtain two functions in S∗L(M):

f1,m(z) = z +
m + 1

4
z3 + . . . and f2,m(z) = z +

m + 1

6
z4 + . . .

which give respectively the sharpness of estimations (ii) and (iii). �

Remark 4.2. The estimation (i) can be obtained directly from (2.6).

Remark 4.3. If we take m = 0 in Theorem 4.2, we obtain as particular case Theorem 2 [6].

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 properties of the class S*L(M)
	3. Radius of convexity for the class  S*L(M)
	4. Coefficient bounds for  S*L(M)  
	References