International Journal of Analysis and Applications

Volume 17, Number 4 (2019), 652-658

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

DOI: 10.28924/2291-8639-17-2019-652

UNIVALENT FUNCTIONS FORMULATED BY THE SALAGEAN-DIFFERENCE

OPERATOR

RABHA W. IBRAHIM1,∗, AND MASLINA DARUS2

1IEEE:94086547

2Centre of Modelling and Data Science,

Faculty of Sciences and Technology, Universiti Kebangsaan Malaysia,43600 Bangi, Selangor, Malaysia

Email address: maslina@ukm.edu.my

∗Corresponding author: rabhaibrahim@yahoo.com

Abstract. We present a class of univalent functions Tm(κ,α) formulated by a new differential-difference

operator in the open unit disk. The operator is a generalization of the well known Salagean’s differential

operator. Based on this operator, we define a generalized class of bounded turning functions. Inequalities,

extreme points of Tm(κ,α), some convolution properties of functions fitting to Tm(κ,α), and other properties

are discussed.

1. Introduction

Let Λ be the class of analytic function formulated by

f(z) = z +

∞∑
n=2

anz
n, z ∈ U = {z : |z| < 1}.

Received 2019-03-02; accepted 2019-04-01; published 2019-07-01.

2010 Mathematics Subject Classification. 30C45.

Key words and phrases. Fractional calculus; fractional differential equation; fractional operator.

c©2019 Authors retain the copyrights

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

652

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-652


Int. J. Anal. Appl. 17 (4) (2019) 653

We symbolize by T(α) the subclass of Λ for which <{f′(z)} > α in U. For a function f ∈ Λ, we present the

following difference operator

D0κf(z) = f(z)

D1κf(z) = zf
′(z) +

κ

2
(f(z) −f(−z) − 2z) , κ ∈ R

...

Dmκ f(z) = Dκ(D
m−1
κ f(z))

= z +

∞∑
n=2

[n +
κ

2
(1 + (−1)n+1)]m anzn.

(1.1)

It is clear that when κ = 0, we have the Salagean’s differential operator [1]. We call Dmκ the Salagean-

difference operator. Moreover, Dmκ is a modified Dunkl operator of complex variables [2] and for recent

work [3]. Dunkl operator describes a major generalization of partial derivatives and realizes the commutative

law in Rn. In geometry, it attains the reflexive relation, which is plotting the space into itself as a set of

fixed points.

Example 1. (see Figs 1 and 2)

• Let f(z) = z/(1 −z) then

D11f(z) = z + 2z
2 + 4z3 + 4z4 + 6z5 + 6z6 + ...

• Let f(z) = z/(1 −z)2 then

D11f(z) = z + 4z
2 + 12z3 + 16z4 + 30z5 + 36z6 + ...

We proceed to define a generalized class of bounded turning utilizing the the Salagean-difference operator.

Let Tm(κ,α) denote the class of functions f ∈ Λ which achieve the condition

<{(Dmκ f(z))
′} > α, 0 ≤ α ≤ 1, z ∈ U, m = 0, 1, 2, ... .

Clearly, T0(κ,α) = T(α) (the bounded turning class of order α). The Hadamard product or convolution of

two power series is denoted by (∗) achieving

f(z) ∗h(z) =
(
z +

∞∑
n=2

anz
n
)
∗
(
z +

∞∑
n=2

ηnz
n
)

= z +

∞∑
n=2

anηnz
n.

(1.2)

The aim of this effort is to present several important properties of the class Tm(κ,α). For this purpose,

we need the following auxiliary preliminaries.



Int. J. Anal. Appl. 17 (4) (2019) 654

Figure 1. D11(z/(1 −z))

Figure 2. D11(z/(1 −z)2)

Lemma 1. Let {an}∞n=0 be a convex null sequence (a0 − a1 ≥ a1 − a2, ... ≥ 0). Then the function ρ(z) =

a0/2 +
∑∞
n=1 an z

n, is analytic and <ρ(z) > 0 in U.

Lemma 2. If ρ(z) is analytic in U, ρ(0) = 1 and <ρ(z) > 1/2,z ∈ U, then for any function % analytic in

U, the function ρ∗% assigns its credits in the convex hull of %(U).



Int. J. Anal. Appl. 17 (4) (2019) 655

Lemma 3. [4] For all z ∈ U the sum

<
(∑
n=2

zn−1

n + 1

)
> −

1

3
.

There are different techniques of studying the class of bounded turning functions, such as using partial

sums or applying Jack Lemma [5]- [7].

2. Results

In this section, we illustrate our results.

Theorem 4. Tm+1(κ,α) ⊂ Tm(κ,α).

Proof. Let f ∈ Tm+1(κ,α), then we have

<{1 +
∞∑
n=2

n[n +
κ

2
(1 + (−1)n+1)]m+1 anzn−1} > α.

Dividing the last inequality by 1 −α and adding +1 we obtain the inequality

<{1 +
1

2(1 −α)

∞∑
n=2

n[n +
κ

2
(1 + (−1)n+1)]m+1 anzn−1} >

1

2
.

By employing the definition of the convolution, we have the construction

(Dmκ f(z))
′ = 1 +

∞∑
n=2

n[n +
κ

2
(1 + (−1)n+1)]m anzn−1

=
(

1 +
1

2(1 −α)

∞∑
n=2

n[n +
κ

2
(1 + (−1)n+1)]m+1 anzn−1

)
∗
(

1 + 2(1 −α)
∞∑
n=2

zn−1

n +
κ

2
(1 + (−1)n+1)

)
.

In view of Lemma 1, with a0 = 1 and an = 1/(n +
κ

2
(1 + (−1)n+1),n = 1, 2, ..., we have

<
(

1 + 2(1 −α)
∞∑
n=2

zn−1

n +
κ

2
(1 + (−1)n+1)

)
> α.

In virtue of Lemma 2, we arrive at the required result. �

Theorem 5. Tm+1(κ,α) ⊂ Tm(κ,β), β ≤ α, 0 ≤ κ ≤ 1/2.

Proof. Let f ∈ Tm+1(κ,α) then we have

<{1 +
∞∑
n=2

n[n +
κ

2
(1 + (−1)n+1)]m+1 anzn−1} > α.



Int. J. Anal. Appl. 17 (4) (2019) 656

Also, we have the convolution

(Dmκ f(z))
′ = 1 +

∞∑
n=2

n[n +
κ

2
(1 + (−1)n+1)]m anzn−1

=
(

1 +

∞∑
n=2

n[n +
κ

2
(1 + (−1)n+1)]m+1 anzn−1

)
∗
(

1 +

∞∑
n=2

zn−1

n +
κ

2
(1 + (−1)n+1)

)
.

It is clear that

n +
κ

2
(1 + (−1)n+1) ≤ n + 2κ ≤ n + 1, 0 ≤ κ ≤ 1/2.

By applying Lemma 3 on the second term of the above convolution, we obtain

<
(

1 +

∞∑
n=2

zn−1

n +
κ

2
(1 + (−1)n+1)

)
> 2/3.

Thus, we attain that

<(Dmκ f(z))
′ >

2

3
α.

By considering

β :=
2

3
α ≤ α, α ∈ [0, 1],

we attain the requested result. �

Theorem 6. Let f ∈ Tm(κ,α) and h ∈ C, the set of convex univalent functions (C ⊂ Λ ). Then f ∗ h ∈

Tm(κ,α).

Proof. By the Marx-Strohhacker theorem [8], if h is convex univalent in U, then

<{
h(z)

z
} > 1/2.

Utilizing convolution properties, we obtain

<(Dmκ (f ∗h)(z))
′ = <

(h(z)
z
∗Dmκ f(z)

′
)
.

But <(Dmκ f(z)′) > α; thus, in view of Lemma 2, we have the desire conclusion. �

Theorem 7. Let f,h ∈ Tm(κ,α). Then f ∗h ∈ Tm(κ,β), where

β :=
κ(2α + 1) + 4α− 1

2(κ + 1)
, 0 ≤ κ ≤ 1.



Int. J. Anal. Appl. 17 (4) (2019) 657

Proof. Define a function h ∈ Λ as follows:

h(z) = z +

∞∑
n=2

ϑnz
n, z ∈ U.

Since h ∈ Tm(κ,α) then

<{1 +
∞∑
n=2

n[n +
κ

2
(1 + (−1)n+1)]m ϑnzn−1} > α.

Let ϕ0 = 1, and in general, we have

ϕn =
κ + 1

[(n + 1)(n +
κ

2
(1 + (−1)n+2) + 1)]m

, n ≥ 1, 0 ≤ κ ≤ 1, m = 1, 2, ....

Obviously, the sequence {ϕn}∞n=0 is a convex null sequence. Therefore, by Lemma 1, we conclude that

<{1 +
∞∑
n=2

κ + 1

[(n + 1)(n +
κ

2
(1 + (−1)n+2) + 1)]m

zn−1} >
1

2
.

Now the convolution

(
1 +

∞∑
n=2

n[n +
κ

2
(1 + (−1)n+1)]m ϑnzn−1

)
∗
(

1 +

∞∑
n=2

κ + 1

[(n)(n +
κ

2
(1 + (−1)n+1))]m

zn−1
)

= 1 +

∞∑
n=2

(κ + 1) ϑnz
n−1

satisfies the real

<{1 +
∞∑
n=2

(κ + 1) ϑnz
n−1zn−1} > α.

In other words, we have

<{
h(z)

z
} = <{1 +

∞∑
n=2

ϑnz
n−1} >

κ + α

α + 1
.

Thus,

<{
h(z)

z
} = <{1 +

∞∑
n=2

ϑnz
n−1 −

2α + κ− 1
2(κ + 1)

} >
1

2
.

But f,h ∈ Tm(κ,α), this implies that

<{
(h(z)

z
−

2α + κ− 1
2(κ + 1)

)
∗Dmκ (f)(z))

′} > α.

Consequently, we conclude that



Int. J. Anal. Appl. 17 (4) (2019) 658

<{
(h(z)

z

)
∗Dmκ (f)(z))

′} >
κ(2α + 1) + 4α− 1

2(κ + 1)
:= β.

Thus, by Lemma 2 and the fact

<(Dmκ (f ∗h)(z))
′ = <

(h(z)
z
∗Dmκ f(z)

′
)
,

we realize the requested result. �

Note that some applications of the Dunkl operator in a complex domain can be found in [9].

Acknowledgments

The work here is partially supported by the Universiti Kebangsaan Malaysia grant: GUP ( Geran Uni-

versiti Penyelidikan)-2017-064.

References

[1] G.S. Salagean, Subclasses of univalent functions, Complex Analysis-Fifth Romanian-Finnish Seminar, Part 1 (Bucharest,

1981), Lecture Notes in Math., vol. 1013, Springer, Berlin, (1983), 362–372.

[2] C.F. Dunkl, Differential-difference operators associated with reflections groups, Trans. Amer. Math. Soc. 311 (1989), 167–

183.

[3] R. W. Ibrahim, New classes of analytic functions determined by a modified differential-difference operator in a complex

domain, Karbala Int. J. Modern Sci. 3(1) (2017), 53–58.

[4] J.M. Jahangiri, K. Farahmand, Partial sums of functions of bounded turning, J. Inequal. Pure Appl. Math. 4(4) (2003),

Article 79.

[5] M. Darus M, R.W Ibrahim, Partial sums of analytic functions of bounded turning with applications, Comput. Appl. Math.

29 (1) 2010), 81–88.

[6] R.W Ibrahim, M. Darus, Extremal bounds for functions of bounded turning, Int. Math. Forum. 6 (33) (2011), 1623–1630.

[7] R.W Ibrahim, Upper bound for functions of bounded turning. Math. Commun. 17(2) (2012), 461–468.

[8] S.S. Miller, and P. T. Mocanu. Differential subordinations: theory and applications. CRC Press, 2000.

[9] R.W Ibrahim, Optimality and duality defined by the concept of tempered fractional univex functions in multi-objective

optimization. Int. J. Anal. Appl. 15 (1) (2017), 75–85.


	1. Introduction
	2. Results
	Acknowledgments
	References